The structure of mode-locking regions of piecewise-linear continuous maps
aa r X i v : . [ m a t h . D S ] O c t The structure of mode-locking regions of piecewise-linearcontinuous maps
D.J.W. SimpsonInstitute of Fundamental SciencesMassey UniversityPalmerston NorthNew ZealandApril 10, 2018
Abstract
The mode-locking regions of a dynamical system are the subsets of the parameter spaceof the system within which there exists an attracting periodic solution. For piecewise-linear continuous maps, these regions have a curious chain structure with points of zerowidth called shrinking points. In this paper we perform a local analysis about an arbitraryshrinking point. This is achieved by studying the symbolic itineraries of periodic solutionsin nearby mode-locking regions and performing an asymptotic analysis on one-dimensionalslow manifolds in order to build a comprehensive theoretical framework for the local dynam-ics. We obtain leading-order quantitative descriptions for the shape of nearby mode-lockingregions, the location of nearby shrinking points, and the key properties of these shrinkingpoints. We apply the results to the three-dimensional border-collision normal form, non-smooth Neimark-Sacker-like bifurcations, and grazing-sliding bifurcations in a model of adry friction oscillator.
This paper concerns piecewise-linear continuous maps of the form x i +1 = f ( x i ; µ, ξ ) := ( A L ( ξ ) x i + B ( ξ ) µ , s i ≤ A R ( ξ ) x i + B ( ξ ) µ , s i ≥ , (1.1)where x i ∈ R N ( N ≥
2) and s i denotes the first component of x i , i.e., s i := e T x i . (1.2)In (1.1), A L and A R are real-valued N × N matrices and B ∈ R N . A L , A R and B are C K functions of a parameter ξ ∈ R M , and µ ∈ R is another parameter.1he assumption that (1.1) is continuous on the switching manifold s = 0 implies that A L and A R differ in only their first columns, i.e., A R = A L + Ce T , (1.3)for some C ∈ R N . Furthermore, (1.1) satisfies the linear scaling property f ( γx ; γµ, ξ ) ≡ γf ( x ; µ, ξ ) , (1.4)for any γ >
0. For this reason, the structure of the dynamics of (1.1) is independent of themagnitude of µ , and the size of any bounded invariant set of (1.1) is proportional to | µ | .Maps of the form (1.1) arise in diverse contexts. The tent map and the Lozi map, oneand two-dimensional examples of (1.1), are instructive prototypical maps exhibiting chaos [1,2]. Maps that can be put in the form (1.1) through a change of variables have been used tomodel phenomena involving a switch or abrupt event, particularly in social sciences [3]. Mostimportantly, maps of the form (1.1) describe the dynamics near border-collision bifurcations.A border-collision bifurcation occurs when a fixed point of a piecewise-smooth map collideswith a switching manifold under certain regularity conditions [4, 5, 6]. Except in special cases,(1.1) has a border-collision bifurcation at µ = 0. The dynamics of (1.1) for µ < µ > µ is the primarybifurcation parameter. By varying ξ in a continuous fashion, we can investigate how the dynamicscreated in the border-collision bifurcation changes with respect to other parameters.This paper concerns mode-locking regions of (1.1). A mode-locking region of a map is a regionof parameter space within which the map has an attracting periodic solution of a given period.We consider two-dimensional cross-sections of parameter space as these are simple to visualiseand informative. For (1.1) we always consider cross-sections with fixed µ = 0, so that we avoidthe border-collision bifurcation at µ = 0 and degeneracies due to the scaling property (1.4).Fig. 1 shows an example using A L = τ L − σ L δ L , A R = τ R − σ R δ R , B = , (1.5)where τ L , σ L , δ L , τ R , σ R , δ R ∈ R . The map (1.1) with (1.5) is the border-collision normal formin three dimensions [5, 6]. Here parameter space (not including µ ) is six-dimensional (we couldwrite ξ = ( τ L , σ L , δ L , τ R , σ R , δ R )). The two-dimensional cross-section of parameter space used inFig. 1 is defined by the restriction τ L = 0 , σ L = − , σ R = 0 , δ R = 2 . (1.6)The mode-locking regions are coloured by the period, n , of the corresponding stable periodicsolution. Only mode-locking regions up to n = 50 are shown. The periodic solutions are “rota-tional”, in a symbolic sense defined in § mn . As wemove from left to right across Fig. 1, the rotation number decreases roughly monotonically. Forthe parameters of Fig. 1 there also exist mode-locking regions corresponding to non-rotationalperiodic solutions, but these regions are small, relative to the rotational ones, and not shown inFig. 1 or studied in this paper. 2 τ R δ L
37 512 25 718 513 821 38
Figure 1: Mode-locking regions of (1.1) with (1.5)-(1.6) and µ > § n ≤
50. Selected mode-locking regions are labelledby their rotation number, mn . This figure was computed by numerically checking the admissibilityand stability of rotational periodic solutions on a 1024 ×
256 grid of τ R and δ L values. At gridpoints where multiple stable periodic solutions exist, the periodic solution with the highest period(less than 50) is indicated.The majority of the boundaries of the mode-locking regions shown in Fig. 1 are where onepoint of the corresponding periodic solution lies on the switching manifold ( s = 0). Theseboundaries may be viewed as curves of border-collision bifurcations for the n th -iterate of (1.1)[6]. The boundaries intersect at so-called shrinking points where (1.1) has a periodic solutionwith two points on the switching manifold.Many of the mode-locking regions of Fig. 1 have several shrinking points and an overall struc-ture that loosely resembles a string of sausages. This structure was first identified in piecewise-linear circle maps [7, 8], and subsequently described in piecewise-smooth models of a DC-DCpower converter [9] and a trade cycle [10]. The structure has also been identified in an integrate-and-fire neuron model [11], a model of an oscillator subject to dry friction [12], and a model ofthe synchronisation of breathing to heart rate [13].A rigorous study of shrinking points of (1.1) was performed in [14]. Here it was shownthat the four curves of border-collision bifurcations that bound the mode-locking region near ashrinking point correspond to where four different points of a periodic solution lie on the switchingmanifold. These four points admit a simple characterisation in terms of the symbolic itineraryof the periodic solution. Furthermore, a local analysis reveals that as the boundaries emanatefrom the shrinking point, they curve in a manner that favours the boundaries reintersecting atother shrinking points.However, the results of [14] do not provide us with any information about the dynamics of(1.1) outside the associated mode-locking region. In any neighbourhood of a typical shrinkingpoint there are infinitely many mode-locking regions corresponding to higher periods. These arenot readily apparent in Fig. 1, as only periods up to 50 are shown, but about shrinking pointscorresponding to relatively low periods, we can see the beginnings of sequences of mode-lockingregions converging to the shrinking points.The purpose of this paper is to describe nearby mode-locking regions and their associated3hrinking points. This is achieved by combining Farey addition, used to identify the sym-bolic itineraries of periodic solutions in nearby mode-locking regions, with calculations on one-dimensional slow manifolds in order to develop a comprehensive theoretical framework that ex-plains the dynamics. To benefit the reader we begin in § § § § § § N − N dimensions to one dimension.Finally, in § In this section we present the main results. We begin in § § § Let f L ( x ; µ, ξ ) := A L ( ξ ) x + B ( ξ ) µ , f R ( x ; µ, ξ ) := A R ( ξ ) x + B ( ξ ) µ , (2.1)denote the two affine half-maps of (1.1). As in [6, 15, 16], we work with symbol sequences, S : Z → { L, R } , and match periodic solutions of (1.1) to periodic symbol sequences. Thisis made precise by the following definition. (In this definition, and throughout this paper, S i denotes the i th element of S .) Definition 2.1.
Let S be a periodic symbol sequence of period n . We refer to an n -tuple, { x S i } n − i =0 , satisfying x S ( i +1) mod n = f S i (cid:0) x S i (cid:1) , for all i = 0 , . . . , n −
1, as an S -cycle .As explained in § x S i lies on the “correct” side of the switching manifold (or on theswitching manifold), then the S -cycle is a periodic solution of (1.1) and said to be admissible.4iven a periodic symbol sequence S of period n , let f S := f S n − ◦ · · · ◦ f S . (2.2)A straight-forward expansion leads to f S ( x ) = M S x + P S Bµ , (2.3)where M S := A S n − · · · A S , (2.4) P S := I + A S n − + A S n − A S n − + · · · + A S n − · · · A S . (2.5)Each x S i is a fixed point of f S ( i ) , where we use S ( i ) to denote the i th left shift permutation of S .If det ( I − M S ) = 0, then the S -cycle is unique and s S i = det ( P S ( i ) ) ̺ T Bµ det ( I − M S ) , (2.6)where ̺ T := e T adj ( I − A L ). In view of (2.6), it is useful to treat a mode-locking region boundaryon which s S i = 0 as a curve on which det ( P S ( i ) ) = 0 [6, 15]. Definition 2.2.
Given ℓ, m, n ∈ Z + , with ℓ < n , m < n and gcd( m, n ) = 1, we define a symbolsequence F [ ℓ, m, n ] : Z → { L, R } by F [ ℓ, m, n ] i := ( L , im mod n < ℓR , im mod n ≥ ℓ . (2.7)We say that F [ ℓ, m, n ], and any shift permutation of F [ ℓ, m, n ], is rotational .We refer to mn as the rotation number of F [ ℓ, m, n ]. The requirement gcd( m, n ) = 1 ensuresthat mn is an irreducible fraction and that each F [ ℓ, m, n ] is of period n [15]. Throughout this paperwe let d ∈ { , . . . , n − } denote the multiplicative inverse of m modulo n (i.e. md mod n = 1).For brevity we omit “mod n ” in subscripts where it is clear that modulo arithmetic is being used.Next we provide a definition of a shrinking point of (1.1). Each shrinking point correspondsto a particular rotational symbol sequence, S = F [ ℓ, m, n ], and is referred to as an S -shrinkingpoint. As explained in § S -shrinking point there are infinitely many S -cycles. For thisreason our definition of an S -shrinking point refers to an S -cycle (which is unique), where weuse S i to denote the symbol sequence that differs from S in only the indices i + jn , for all j ∈ Z .It is important to note that S is a shift permutation of F [ ℓ − , m, n ], and S ℓd = F [ ℓ + 1 , m, n ],see § Definition 2.3.
Consider (1.1) for some ξ ∈ R M and µ = 0, and suppose ̺ T B = 0. Let S = F [ ℓ, m, n ] be a rotational symbol sequence with 2 ≤ ℓ ≤ n −
2. Suppose det ( I − M S ) = 0and det (cid:0) I − M S ℓd (cid:1) = 0. If the S -cycle is admissible, and s S i = 0 only for i = 0 and i = ℓd ,then we say that ξ is an S -shrinking point . 5n Definition 2.3, s S = 0 and s S ℓd = 0 are the two codimension-1 conditions that specify an S -shrinking point. The remaining conditions of the definition ensure genericity. At an S -shrinkingpoint, we let y i := x S i , t i := s S i , (2.8) a := det ( I − M S ) , b := det (cid:0) I − M S ℓd (cid:1) . (2.9)As shown in § ab < S -shrinking point, M S has a unit eigenvalue of algebraic multiplicity 1 (see § M S ( i ) . For each of the four indices j = 0, ( ℓ − d , ℓd and − d (taken modulo n ), we let u T j and v j denote the left and right eigenvectors of M S ( j ) correspondingto the unit eigenvalue and normalised by u T j v j = 1 and e T v j = 1. The restriction to the fourgiven indices ensures this normalisation can always be achieved (see § § u T v − d a + u T ℓd v ( ℓ − d b = u T ( ℓ − d v ℓd a + u T − d v b = 1 c , (2.10)where c denotes the product of the nonzero eigenvalues of I − M S , that is c := N Y i =2 λ i , (2.11)where λ i are the eigenvalues of I − M S , counting multiplicity, and λ = 0.We now consider the properties of (1.1) near an S -shrinking point. For simplicity we assume ξ ∈ R and write ξ = ( ξ , ξ ). Suppose that (1.1) has an S -shrinking point at some ξ = ξ ∗ . Itfollows that there exists a neighbourhood of ξ ∗ within which the S -cycle exists and is unique.By Definition 2.3, s S = s S ℓd = 0 at ξ = ξ ∗ . We let η := s S ( ξ , ξ ) , ν := s S ℓd ( ξ , ξ ) , (2.12)and assume that the coordinate change ( ξ , ξ ) → ( η, ν ) is locally invertible, i.e. det( J ) = 0,where J := " ∂η∂ξ ∂η∂ξ ∂ν∂ξ ∂ν∂ξ ξ = ξ ∗ . (2.13)In ( η, ν )-coordinates the S -shrinking point ξ = ξ ∗ is located at ( η, ν ) = (0 , η, ν ) = (0 , S -cycle with s S j = 0, for j = 0, ( ℓ − d , ℓd and − d .The curves are orientated as in Fig. 2 and locally define two regions, Ψ and Ψ . In Ψ thereexist unique F [ ℓ, m, n ] and F [ ℓ − , m, n ]-cycles, and in Ψ there exist unique F [ ℓ, m, n ] and F [ ℓ + 1 , m, n ]-cycles. If, in both Ψ and Ψ , one of the two periodic solutions is stable, then (1.1)has a mode-locking region with zero width at ξ = ξ ∗ . In this case, the F [ ℓ, m, n ]-cycle is stableon exactly one side of the shrinking point, as determined by the sign of a , Table 1.Finally let σ ≥ M S , excluding theunit eigenvalue, at the S -shrinking point. That is, σ := max i =2 ,...,N | ρ i | , (2.14)6 ν Ψ Ψ s S = 0 s S ( ℓ − d = 0 s S ℓd = 0 s S− d = 0 Figure 2: The basic unfolding of a generic S -shrinking point of (1.1), as explained in the sur-rounding text and in more detail in § η, ν )-coordinates (2.12) provide a two-dimensionalcross-section of parameter space for which two of the border-collision bifurcation curves coincidewith the coordinate axes. The insets are schematic phase portraits indicating the location of thepoints of the S -cycle relative to the switching manifold. a < a > F [ ℓ, m, n ]-cycle in Ψ stable unstable F [ ℓ − , m, n ]-cycle in Ψ unstable stable F [ ℓ, m, n ]-cycle in Ψ unstable stable F [ ℓ + 1 , m, n ]-cycle in Ψ stable unstableTable 1: Cases for the stability of periodic solutions near an S -shrinking point, where S = F [ ℓ, m, n ], in the scenario that stable periodic solutions exist on both sides of the shrinkingpoint.where ρ i are the eigenvalues of M S , counting multiplicity, and ρ = 1. In general the stabilitymultipliers of an S -cycle are the eigenvalues of M S , § σ < S -cycle for some parameter values near the S -shrinking point. Each mode-locking region of Fig. 1 corresponds to stable F [ ℓ, m, n ]-cycles with fixed values of m and n , and values of ℓ that change by one each time we cross a shrinking point. Nearbymode-locking regions have rotation numbers close to mn . This motivates the following definition. Definition 2.4.
Given k ∈ Z + , χ ∈ Z with | χ | < k , and a rotational symbol sequence F [ ℓ, m, n ],7 ν θ +0 θ + − θ − θ − G + [ k, G + [ k, − G − [ k, G − [ k, G +5 G +4 G +3 G − G − G − F [ ℓ, m, n ] Figure 3: A sketch of typical G ± k -mode-locking regions (for k = 3 , ,
5) near an F [ ℓ, m, n ]-shrinking point in ( η, ν )-coordinates in the case a <
0. Each shrinking point is labelled by itsassociated symbol sequence. Formulas for the angles, denoted θ ± χ , about which sequences of G ± [ k, χ ]-shrinking points emanate from the F [ ℓ, m, n ]-shrinking point are given by (2.25)-(2.26).In the case a >
0, the relative location of the G + k and G − k -mode-locking regions is reversed.we let G ± [ k, χ ] := F (cid:2) ℓ ± k + χ, m ± k , n ± k (cid:3) , (2.15)where ℓ ± k := kℓ + ℓ ± , m ± k := km + m ± , n ± k := kn + n ± , (2.16)and m − n − and m + n + are the left and right Farey roots of mn , ℓ + := l ℓn + n m , and ℓ − := j ℓn − n k . We alsolet ˜ ℓ := ℓ ± k + χ , and let d ± k denote the multiplicative inverse of m ± k modulo n ± k .Each G ± [ k, χ ] is a rotational symbol sequence with rotation number m ± k n ± k = km + m ± kn + n ± . Theserotation numbers limit to mn , as k → ∞ , and are in the first level of complexity relative to mn [17]. Other rotational symbol sequences with rotation numbers near mn have rotation numbers ofhigher levels of complexity and are beyond the scope of this paper.Near a typical shrinking point, there exist mode-locking regions corresponding to G ± [ k, χ ]-cycles for several consecutive values of χ , Fig. 3. We refer to these as G ± k -mode-locking regions.For any χ max ∈ Z + , we define the collectionΞ χ max := (cid:8) G + [ k, χ ] (cid:12)(cid:12) k ∈ Z + , − χ max ≤ χ < χ max , | χ | < k (cid:9) ∪ (cid:8) G − [ k, χ ] (cid:12)(cid:12) k ∈ Z + , − χ max < χ ≤ χ max , | χ | < k (cid:9) . (2.17)We define polar coordinates ( r, θ ) by η = (cid:12)(cid:12)(cid:12)(cid:12) ct d a (cid:12)(cid:12)(cid:12)(cid:12) r cos( θ ) , ν = (cid:12)(cid:12)(cid:12)(cid:12) ct ( ℓ − d a (cid:12)(cid:12)(cid:12)(cid:12) r sin( θ ) . (2.18)8e also define a continuous function Γ : (cid:0) , π (cid:1) → R byΓ( θ ) := ( ln(cos( θ )) − ln(sin( θ ))cos( θ ) − sin( θ ) , θ ∈ (cid:0) , π (cid:1) \ (cid:8) π (cid:9) √ , θ = π , (2.19)and extend this definition to all non-integer multiples of π in a periodic fashion:Γ( θ ) := Γ (cid:16) θ mod π (cid:17) , θ = jπ , j ∈ Z . (2.20)The following theorem provides us with the location of the G ± k -mode-locking regions to leadingorder. Theorem 2.1.
Suppose (1.1) with K ≥ has an S -shrinking point satisfying σ < and det( J ) =0 , and write S = F [ ℓ, m, n ] . Then for all χ max ∈ Z + , there exists k min ∈ Z + and a neighbourhood N of ( η, ν ) = (0 , , such that for all T = G ± [ k, χ ] ∈ Ξ χ max with k ≥ k min , within N there exists aunique C K curve on which det ( P T ) = 0 and a unique C K curve on which det (cid:16) P T ( ( ˜ ℓ − ) d ± k ) (cid:17) = 0 ,and both curves lie within O (cid:0) k (cid:1) of r = 1 k Γ( θ ) , (2.21) where if T = G + [ k, χ ] and a < , or T = G − [ k, χ ] and a > , then θ ∈ (cid:0) π , π (cid:1) , and if T = G + [ k, χ ] and a > , or T = G − [ k, χ ] and a < , then θ ∈ (cid:0) π , π (cid:1) . (2.22)Theorem 2.1 tells us that if there exists a G ± k -mode-locking region, it has a width of at most O (cid:0) k (cid:1) , lies approximately on the curve (2.21) (sketched in Fig. 4), and is located an O (cid:0) k (cid:1) distance from the S -shrinking point. By (2.22), if a < G + k -mode-locking regions liein the fourth quadrant of the ( η, ν )-plane, and the G − k -mode-locking regions lie in the secondquadrant of the ( η, ν )-plane (as in Fig. 3). If a >
0, then the opposite is true.Next we investigate shrinking points on the G ± k -mode-locking regions. We find that G ± [ k, χ ]-shrinking points exist for arbitrarily large values of k only for certain values of χ , and that these ην θ = 0 θ = π θ = π θ = π r = k Γ( θ ) r = k Γ( θ ) Figure 4: A sketch of (2.21).9 < a > θ + χ ∈ (cid:0) π , π (cid:1) θ + χ ∈ (cid:0) π , π (cid:1) θ − χ ∈ (cid:0) π , π (cid:1) θ − χ ∈ (cid:0) π , π (cid:1) Table 2: This table indicates the interval to which each θ + χ (2.25) and θ − χ (2.26) belong, asdetermined by the sign of a .values of χ depend on the given S -shrinking point. To determine the appropriate values of χ , wedefine scalar quantities κ ± χ by κ + χ := ( u T ℓd M − χ − S ℓd ) (cid:12)(cid:12) ( η,ν )=(0 , v ( ℓ − d , χ ≤ − u T M χ S ℓd (cid:12)(cid:12) ( η,ν )=(0 , v − d , χ ≥ , (2.23) κ − χ := ( u T − d M − χ S (cid:12)(cid:12) ( η,ν )=(0 , v , χ ≤ u T ( ℓ − d M χ − S ℓd ( ℓd ) (cid:12)(cid:12) ( η,ν )=(0 , v ℓd , χ ≥ . (2.24)To clarify these expressions, the matrix M − χ − S ℓd ) , for example, is the ( − χ − th power of M S ℓd ) ,where M S ℓd ) is given by (2.4) using S ℓd ) – the ( ℓd ) th left shift permutation of S . In (2.23) thismatrix is evaluated at the S -shrinking point.We also define θ + χ := tan − (cid:18) t ( ℓ +1) d t ( ℓ − d | κ + χ | (cid:19) , χ ≤ − − (cid:18) t d t − d | κ + χ | (cid:19) , χ ≥ , (2.25) θ − χ := tan − (cid:18) t d | κ − χ | t − d (cid:19) , χ ≤ − (cid:18) t ( ℓ +1) d | κ − χ | t ( ℓ − d (cid:19) , χ ≥ , (2.26)assuming κ ± χ = 0, where the ambiguity of each tan − ( · ) is resolved by Table 2. Theorem 2.2.
Suppose (1.1) with K ≥ has an S -shrinking point satisfying σ < and det( J ) =0 , and write S = F [ ℓ, m, n ] . Then for all χ max ∈ Z + , there exists k min ∈ Z + , and a neighbourhood N of ( η, ν ) = (0 , , such that for all T = G ± [ k, χ ] ∈ Ξ χ max with k ≥ k min , if κ ± χ = 0 , then within N :i) there exists a unique point ( η T , ν T ) at which det ( I − M T ) = det ( P T ) = 0 if and only if κ ± χ > , and ( η T , ν T ) lies within O (cid:0) k (cid:1) of (2.21) with θ = θ ± χ ;ii) there exists a unique C K curve on which (1.1) has a T -cycle with an associated stabilitymultiplier of − if and only if κ ± χ < , and this curve intersects det ( P T ) = 0 at a pointwithin O (cid:0) k (cid:1) of (2.21) with θ = θ ± χ .
10y Theorem 2.2, each (cid:0) η G ± [ k,χ ] , ν G ± [ k,χ ] (cid:1) is a potential G ± [ k, χ ]-shrinking point and exists if κ ± χ >
0. If κ ± χ <
0, then no such points exist (for sufficiently large values of k ). For a fixed valueof χ , (cid:0) η G + [ k,χ ] , ν G + [ k,χ ] (cid:1) and (cid:0) η G − [ k,χ ] , ν G − [ k,χ ] (cid:1) are sequences of points that limit to the S -shrinkingpoint as k → ∞ . Each (cid:0) η G ± [ k,χ ] , ν G ± [ k,χ ] (cid:1) is a G ± [ k, χ ]-shrinking point if all the statements inDefinition 2.3 (applied to G ± [ k, χ ]) are satisfied. Naturally we would like identify a practical setof testable conditions that ensure (cid:0) η G ± [ k,χ ] , ν G ± [ k,χ ] (cid:1) is a G ± [ k, χ ]-shrinking point. However, thisis difficult (and not achieved in this paper) because in order to show that the G ± [ k, χ ] -cycleis admissible we have to determine the sign of s G ± [ k,χ ] i for all kn + n ± values of i . Numericalinvestigations reveal that the G ± [ k, χ ] -cycle is often admissible when κ ± χ >
0, but this is notalways the case.Already the identity (2.10) provides some restrictions on the combinations of signs possiblefor the κ ± χ . The next result tells us that, for large k , if (cid:0) η G ± [ k,χ ] , ν G ± [ k,χ ] (cid:1) is a G ± [ k, χ ]-shrinkingpoint (this requires κ ± χ > κ ± χ − > κ ± χ +1 > a ).Essentially this says that the existence of a G ± [ k, χ ]-shrinking point implies the existence of aneighbouring shrinking point in the G ± k -mode-locking region, if admissibility is satisfied. Theorem 2.3.
Suppose (1.1) with K ≥ has an S -shrinking point satisfying σ < and det( J ) =0 , and write S = F [ ℓ, m, n ] . For any χ ∈ Z , if κ ± χ > and the point (cid:0) η G + [ k,χ ] , ν G + [ k,χ ] (cid:1) (as specifiedby Theorem 2.2) is a G + [ k, χ ] -shrinking point for arbitrarily large values of k , then, if a < , then κ ± χ − > , and if a > , then κ ± χ +1 > . (2.27)Finally we provide some properties of nearby G ± [ k, χ ]-shrinking points. Here we use tildesto denote quantities of a G ± [ k, χ ]-shrinking point. For any T ∈ Ξ χ max with κ ± χ >
0, we let˜ a = det ( I − M T ) and ˜ b = det (cid:16) I − M T ˜ ℓdk (cid:17) , evaluated at ( η T , ν T ). We denote the T -cycle by { ˜ y i } and let ˜ t i = e T ˜ y i . We also let ˜ η = s T , ˜ ν = s T ˜ ℓd ± k , and˜ J := " ∂ ˜ η∂η ∂ ˜ η∂ν∂ ˜ ν∂η ∂ ˜ ν∂ν ( η T ,ν T ) . (2.28) Theorem 2.4.
Suppose (1.1) with K ≥ has an S -shrinking point satisfying σ < and det( J ) =0 , and write S = F [ ℓ, m, n ] . Then for all χ max ∈ Z + , there exists k min ∈ Z + , such that for all T ∈ Ξ χ max with k ≥ k min , if κ ± χ > and ( η T , ν T ) is a T -shrinking point, theni) sgn(˜ a ) = sgn( a ) and det (cid:16) ˜ J (cid:17) > ,ii) at ( η T , ν T ) , all eigenvalues of M T , excluding the unit eigenvalue, have modulus O (cid:0) σ k (cid:1) . By part (i) of Theorem 2.4, G ± [ k, χ ]-shrinking points exhibit the unfolding depicted in Fig. 2,and have the same orientation as the S -shrinking point. By part (ii) of Theorem 2.4, it followsthat there exists a stable G ± [ k, χ ]-cycle on one side of the G ± [ k, χ ]-shrinking point.11 .3 Examples illustrating the main results The three-dimensional border-collision normal form
The map (1.1) with (1.5) has an F [2 , , τ L = 0 , σ L = − , δ L = 0 . ,τ R = − , σ R = 0 , δ R = 2 , (2.29)and µ >
0. This point is located in the middle of Fig. 1. The corresponding mode-locking regionhas a stable F [2 , , δ L < . F [3 , , δ L > . τ R and δ L unfold the shrinking point generically in thesense that det( J ) = 0, where J is given by (2.13) using ξ = τ R and ξ = δ L . This implies that −2 −1.9 −1.80.050.10.150.20.250.3 −2.2 −2.1 −2 −1.9 −1.80.050.10.150.20.250.3 A τ R δ L η νk = 40 k = 20 k = 10 k = 5 G + [ k, − G + [ k, − G − [ k, − G − [ k, G − [ k, B τ R δ L η νk = 40 k = 20 k = 10 k = 5 θ +0 θ + − θ + − θ −− θ − θ − θ − Figure 5: Panel A shows mode-locking regions of (1.1) with (1.5), µ >
0, and with the remainingparameter values given by (2.29) (except τ R and δ L are variable). This figure is centred about the F [2 , , G ± [ k, χ ]-cycles. Panel B illustrates the predictions of Theorems 2.1 and 2.2, as discussed in thetext. 12nder the smooth transformation to ( η, ν )-coordinates (2.12), the mode-locking region conformsto Fig. 2 locally.Fig. 5-A shows a magnified area of Fig. 2 and indicates the η and ν axes. Parts of G ± k -mode-locking regions are also shown. These were computed by numerically continuing the bifurcationboundaries. The regions are shown for all k ≤
10 in order to illustrate the proximity of theregions to one another (indeed they overlap slightly) and also for k = 20 and k = 40 in order toillustrate the location and shape of the regions for relatively large values of k without clutteringthe figure. Triangles indicate T -shrinking points for T ∈ Ξ . Circles indicate T -shrinking pointsfor T ∈ Ξ \ Ξ . For clarity the mode-locking regions are not shown beyond these shrinkingpoints. The additional (purple) curves are boundaries at which stable periodic solutions losestability via an associated stability multiplier attaining the value − § G ± k -mode-locking regions behave in the limit k → ∞ based on various key quantities of the F [2 , , k = 5 , , ,
40. For simplicity we used a linear approximation to thecoordinate change ( τ R , δ L ) → ( η, ν ) to produce Fig. 5-B.Theorem 2.2 provides approximations to the locations of nearby shrinking points and stabilityloss bifurcation boundaries based on the scalar quantities κ ± χ (2.23)-(2.24) and θ ± χ (2.25)-(2.26).For the shrinking point (2.29) we have κ + − = 23633 , κ −− = 49455 ,κ + − = 3855 , κ − = 4355 ,κ +0 = − , κ − = 1033 ,κ +1 = 2633 , κ − = − . (2.30)We have used these values to generate Fig. 5-B. Recall, Theorem 2.2 tells us that κ ± χ > G + [ k, χ ]-shrinking points, whereas κ ± χ < − O (cid:0) k (cid:1) ofthe curve (2.21) at θ = θ ± χ . Only θ +1 is not shown in Fig. 5-B because at each (cid:0) η G + [ k, , ν G + [ k, (cid:1) the G + [ k, -cycle is virtual.In summary, Fig. 5-B illustrates Theorems 2.1 and 2.2 with χ max = 2. We see that Fig. 5-Bprovides a good approximation to the mode-locking regions shown in Fig. 5-A, including theshrinking points and stability loss boundaries, and the accuracy of the approximation increaseswith k . If we double the value of k , for example, then to leading order the distance of themode-locking region to the F [2 , , Alternate cross-sections of parameter space
The signs of the κ ± χ determine which G + [ k, χ ] have shrinking points and which G + [ k, χ ] havestability loss boundaries. The values of θ ± χ determine the relative location and spacing of thesefeatures on the curves (2.21). Yet each κ ± χ and θ ± χ is a property of the S -shrinking point, andso is independent to the parameters used to unfold the S -shrinking point. Therefore with any13on-degenerate two-dimensional slice of parameter space through the F [2 , , G + [ k, − G + [ k, −
1, and k these features are relatively far apart. Changes in the properties of an S -shrinking point A shrinking point is a codimension-two phenomenon, therefore within three-dimensional regionsof parameter space there exist curves of shrinking points. As we follow a curve of S -shrinkingpoints in a continuous manner, the values of κ ± χ and θ ± χ change continuously. Therefore thestructure of nearby mode-locking regions varies along the curve, and there may be critical pointsat which the structure changes in a fundamental way (e.g. at a point where one of the κ ± χ changes −0.1 −0.05 0 0.11.61.822.22.4 −0.2 −0.1 0 0.2−1.2−1.1−1−0.9−0.8 A τ L δ R η νk = 40 k = 20 k = 10 k = 5 G + [ k, − G + [ k, − G − [ k, − G − [ k, G − [ k, B σ R σ L ηνk = 40 k = 20 k = 10 k = 5 G + [ k, − G + [ k, − G − [ k, − G − [ k, G − [ k, Figure 6: Panel A shows mode-locking regions of (1.1) with (1.5) with ( σ L , δ L , τ R , σ R ) =( − , . , − ,
0) and µ >
0. Panel B shows mode-locking regions of (1.1) with (1.5) with( τ L , δ L , τ R , δ R ) = (0 , . , − ,
2) and µ >
0. 14ign). Here we show an example.For all δ R > . F [2 , , τ L = 0 , σ L = − , δ L = δ R + 2 δ R ( δ R + 2 δ R + 2) ,τ R = − δ R + δ R + 2 δ R + 2 , σ R = 0 , (2.31)and µ >
0, see [6]. The F [2 , , δ R = 2.As we decrease the value of δ R from δ R = 2 (at which the κ ± χ are given by (2.30)) to δ R = 1,the sign of κ − changes from negative to positive (at δ R ≈ . κ ± χ remains unchanged. With δ R = 1 the nearby mode-locking regions, as shownin Fig. 7-A, have the same structure as those in Fig. 5-A except that G − [ k, k . −1.5 −1.3 −1.20.50.60.7 −1.4 −1.3 −1.2 −1.10.70.80.91 A τ R δ L η νk = 40 k = 20 k = 10 k = 5 G + [ k, − G + [ k, − G − [ k, − G − [ k, G − [ k, G − [ k, B τ R δ L η νk = 40 k = 20 k = 10 k = 5 G + [ k, − G + [ k, − G − [ k, − G − [ k, G − [ k, G − [ k, Figure 7: Mode-locking regions of (1.1) with (1.5) about the F [2 , , τ L = 0, σ L = − σ R = 0 and µ > δ R = 1; in panel B, δ R = 0 .
8. Theseregions are shown for δ R = 2 in Fig. 5-A. 15pon a further decrease in the value of δ R , we have θ +0 = θ + − at δ R ≈ . δ R = 0 .
8. At this value of δ R the mode-locking regions do notextend beyond the G + [ k, −
1] shrinking points, for sufficiently large values of k . This is because θ +0 > θ + − and so the bifurcation boundaries on which an G + [ k, − Nonsmooth Neimark-Sacker-like bifurcations
As another example, consider (1.1) with A L = (cid:20) r L cos (2 πω L ) 1 − r L (cid:21) , A R = " s R cos (2 πω R ) 1 − s R , B = (cid:20) (cid:21) , (2.32)where r L , s R , ω L , ω R ∈ R are parameters. The map (1.1) with (2.32) was studied in [18] in orderto investigate nonsmooth Neimark-Sacker-like bifurcations.Fig. 8 shows mode-locking regions of (1.1) with (2.32). This figure can be interpreted asshowing the mode-locking dynamics created in the border-collision bifurcation at µ = 0, wherethis bifurcation is akin to a Neimark-Sacker bifurcation in that an invariant circle is usuallycreated as the values of µ passes through 0. In Fig. 8 there is a dominant curve of shrinking pointsrunning diagonally from the bottom-left of the figure to the top-right. Numerical computationsof Lyapunov exponents reveal that this curve appears to be a boundary for chaotic dynamics[18]. The geometric mechanism responsible for this boundary of chaos is not fully understood.The shrinking point in the large mode-locking region in the top-right of Fig. 8 correspondsto S = F [2 , , k . ω R s R Figure 8: Mode-locking regions of (1.1) with (2.32), r L = 0 . ω L = 0 .
09 and µ >
0. (as inFig. 13 of [18]). 16 pplication to grazing-sliding bifurcations
Dynamics near grazing-sliding bifurcations of ( N + 1)-dimensional piecewise-smooth systems ofordinary differential equations are captured by return maps of the form (1.1) for which eitherdet ( A L ) = 0 or det ( A R ) = 0 [19, 20]. In [12], the authors investigate mode-locking regions neara grazing-sliding bifurcation in a model of a mechanical oscillator subject to dry friction. Thereturn map that they analyse can, through an affine change of variables, be put in the form (1.1)with A L = (cid:20) ξ cos ( ξ ) 1 − e ξ (cid:21) , A R = (cid:20) e ξ cos ( ξ ) 10 0 (cid:21) , B = (cid:20) (cid:21) , (2.33)where ξ , ξ ∈ R . Mode-locking regions of (1.1) with (2.33) are shown in Fig. 10.Fig. 11-A shows a magnification of Fig. 10 about an F [8 , , A ω R s R ν ηk = 40 k = 20 k = 10 k = 5 G + [ k, − G + [ k, − G − [ k, − G − [ k, G − [ k, B ω R s R ν ηk = 40 k = 20 k = 10 k = 5 θ −− θ − θ − θ − θ + − θ + − Figure 9: Panel A shows mode-locking regions of (1.1) with (2.32), r L = 0 . ω L = 0 .
09 and µ >
0, obtained by numerically continuing bifurcation boundaries. Panel B shows the leading-order approximation to the mode-locking regions, as well as shrinking points and stability lossboundaries, as given by Theorems 2.1 and 2.2 (using a linear approximation to the coordinatechange ( ω R , s R ) ↔ ( η, ν )). For a further explanation refer to the discussion surrounding Fig. 5.17hat since det ( A R ) = 0, curves of shrinking points admit simple analytic expressions [12]. Forexample the dashed curve in Fig. 11-A is given bye ξ sin (4 ξ ) − e ξ sin (5 ξ ) + sin ( ξ ) = 0 . (2.34)As with the previous two examples, Fig. 11-B illustrates the predictions of Theorems 2.1 and2.2. Here a >
0, so G + k -mode-locking regions lie on the same side as the ν -axis (the previousexamples have a < Here we develop symbolic notation on the alphabet { L, R } , following [6, 14, 15]. A word is afinite list of the symbols L and R , e.g. S = LRR . We index the elements of a word from i = 0 to i = n −
1, where n is the length of the word. Thus for the previous example n = 3 and S = L , S = R , S = R .Given two words S and T , the concatenation ST is a word that has a length equal to thesum of the lengths of S and T . The power S k , where k ∈ Z + , is the concatenation of k instancesof S . A word is said to be primitive if it cannot be written as a power with k > S , for any j ∈ Z we let S ( j ) denote the j th left cyclic permutation of S . Thatis, S ( j ) i = S ( i + j ) mod n for all i = 0 , . . . , n −
1. Also we let S j denote the word that differs from S in only the symbol S j mod n . For example, with S = LRR , S (0) = S = LRR , S = RRR , S (1) = RRL , S = LLR , S (2) = RLR , S = LRL . A symbol sequence S is a bi-infinite list of the symbols L and R . S is periodic with period n ∈ Z + , if S i + jn = S i , for all i = 0 , . . . , n − j ∈ Z . Given a periodic symbol sequence S ξ ξ
17 213 16 21115
Figure 10: Mode-locking regions of (1.1) with (2.33) and µ < .05 1.06 1.070.240.250.260.27 1.05 1.06 1.070.240.250.260.27 A ξ ξ νηk = 40 k = 20 k = 10 k = 5 G + [ k, G + [ k, G − [ k, G − [ k, G − [ k, B ξ ξ νηk = 40 k = 20 k = 10 k = 5 θ − θ − θ − θ −− θ +1 θ +0 θ + − Figure 11: Panel A shows mode-locking regions of (1.1) with (2.33) and µ < ξ , ξ ) ↔ ( η, ν )).For a further explanation refer to the discussion surrounding Fig. 5.of minimal period n , the word S · · · S n − is primitive and completely determines S . Conversely,given a primitive word S of length n , the infinite repetition of this word generates a symbolsequence S of minimal period n . For these reasons, in order to minimise the complexity of ournotation, we use periodic symbol sequences and primitive words interchangeably and denote themwith the same symbol, e.g. S . In this paper we use periodic symbol sequences to describe periodic solutions to (1.1). We restrictour attention to rotational symbol sequences F [ ℓ, m, n ], defined in Definition 2.2, as these relateto shrinking points of (1.1), [14, 15, 16]. Rotational symbol sequences were originally studiedindependently of piecewise-linear maps by Slater in [21, 22].19otational symbol sequences can be interpreted as rigid rotation on a circle [14, 15]. To seethis, we treat the values im mod n , that appear in the definition (2.7), as points on a circle. Thisis shown in Fig. 12 for F [3 , , n nodes on the circle, with ℓ of them to the left of the line.3) Find the first node that lies to the left of the lower intersection point of the circle and theline, and call it 0.4) For each i = 1 , . . . , n −
1, step m nodes clockwise from node i −
1, and call it node i .5) Then F [ ℓ, m, n ] i is equal to L if node i is left of the line and equal to R otherwise.As we label the n nodes in this fashion we revolve clockwise around the circle m times. For thisreason we refer to mn as the rotation number of F [ ℓ, m, n ].It is a simple combinatorial exercise to show that (2.7) has the equivalent form, F [ ℓ, m, n ] jd mod n = ( L , j = 0 , . . . , ℓ − R , j = ℓ, . . . , n − , (3.1)where d is the multiplicative inverse of m modulo n . The next result is a simple consequence of(3.1) and we omit a proof. Proposition 3.1.
For any F [ ℓ, m, n ] , if ℓ = 1 then F [ ℓ, m, n ] ( ℓ − d = F [ ℓ − , m, n ] , (3.2) F [ ℓ, m, n ] = F [ ℓ − , m, n ] ( − d ) , (3.3) and if ℓ = n − then F [ ℓ, m, n ] ℓd = F [ ℓ + 1 , m, n ] , (3.4) F [ ℓ, m, n ] − d = F [ ℓ + 1 , m, n ] ( d ) . (3.5) Figure 12: A pictorial interpretation of the rotational symbol sequence F [3 , ,
7] =
LRRLRRL .20y combining (3.3) and (3.4) we obtain the following identity which is central to our under-standing of shrinking points.
Corollary 3.2.
For any F [ ℓ, m, n ] , F [ ℓ, m, n ] ℓd ( d ) = F [ ℓ, m, n ] . (3.6) Example 3.1.
As an example, let us illustrate (3.5) with ( ℓ, m, n ) = (3 , , i im mod n F [3 , ,
7] =
LRRLRRL (see also Fig. 12). Moreover, F [4 , ,
7] =
LRLLRRL . Here d = 3, and so to evaluate the left hand-side of (3.5) we use − d = 4 (in modulo 7 arithmetic).Flipping the symbol F [3 , , produces F [3 , , − d = LRRLLRL . Conversely, the right hand-side of (3.5) is given by the third left shift permutation of F [4 , , F [4 , , (3) = LRRLLRL , which we see is indeed the same as F [3 , , − d . Definition 3.2.
For any S = F [ ℓ, m, n ], let X = S · · · S ( ℓd −
1) mod n , Y = S ℓd mod n · · · S n − , (3.7)ˆ X = S · · · S ( − d −
1) mod n , ˆ Y = S − d mod n · · · S n − , (3.8)ˇ X = S ℓd mod n · · · S (( ℓ − d −
1) mod n , ˇ Y = S ( ℓ − d mod n · · · S ( ℓd −
1) mod n . (3.9)These six words are determined by the values of ℓ , m and n . We do not explicitly write themas functions of ℓ , m and n as it should always be clear which values of ℓ , m and n are being used.The word X , for instance, consists of the first ℓd mod n symbols of S , and Y consists of theremaining symbols of the word S . We can therefore write F [ ℓ, m, n ] = X Y . Further partitionsof F [ ℓ, m, n ] are provided below in Proposition 3.3.First let us resolve a minor ambiguity in the definitions of ˇ X and ˇ Y . To be precise, ˇ X consistsof the symbols of S in cyclical order starting from S ℓd mod n and ending with S (( ℓ − d −
1) mod n (andsimilarly for ˇ Y ). For example, with F [3 , ,
7] =
LRRLRRL , we have d = 3, thus ℓd mod n = 2,and (( ℓ − d −
1) mod n = 5. Thus ˇ X = RLRR , and similarly ˇ Y = LLR .Fig. 13 illustrates the words (3.7)-(3.9) pictorially. For instance, the word X “follows” S fromnode 0 to node ℓd mod n , and the word ˇ X “follows” S from node ℓd mod n to node ( ℓ − d mod n .The words (3.7)-(3.9) can be used to partition F [ ℓ, m, n ] in different ways and are useful tous for constructing the symbol sequences of periodic solutions in nearby mode-locking regions of(1.1). We omit a proof of Proposition 3.3 as it follows simply from (3.7)-(3.9) and Proposition3.1. 21 roposition 3.3. For any F [ ℓ, m, n ] , F [ ℓ, m, n ] = X Y , F [ ℓ, m, n ] = ˆ X ˆ Y , (3.10) F [ ℓ, m, n ] ( ℓd ) = Y X , F [ ℓ, m, n ] ( ℓd ) = ˇ X ˇ Y , (3.11) F [ ℓ, m, n ] ( − d ) = X Y , F [ ℓ, m, n ] ( − d ) = ˆ Y ˆ X , (3.12) F [ ℓ, m, n ] (( ℓ − d ) = Y X , F [ ℓ, m, n ] (( ℓ − d ) = ˇ Y ˇ X . (3.13)The next result equates various concatenations of the words (3.7)-(3.9). These can be under-stood intuitively by following the arrows in Fig. 13. For instance, roughly speaking, both X ˇ X and ˆ X X take us from node 0 to node ( ℓ − d mod n via an intermediary node. Proposition 3.4.
For any F [ ℓ, m, n ] , X ˇ X = ˆ X X , (3.14) Y ˆ X = ˇ X Y , (3.15)ˆ Y X = X ˇ Y , (3.16)ˇ YY = Y ˆ Y . (3.17) Proof.
Here we derive (3.14). The remaining identities can be derived similarly.Let S = F [ ℓ, m, n ]. By (3.7) and (3.9), X ˇ X = S · · · S ( ℓd −
1) mod n S ℓd mod n · · · S (( ℓ − d −
1) mod n . (3.18)Also S ( − d ) = X Y , (3.12). Therefore X consists of the first ℓd symbols of S ( − d ) , i.e. X = S − d mod n · · · S (( ℓ − d −
1) mod n . Thereforeˆ X X = S · · · S ( − d −
1) mod n S − d mod n · · · S ( ℓ − d −
1) mod n , (3.19) X YX Y ˆ X ˆ Y ˇ X ˇ Y − d mod nd ( ℓ − d mod n ℓd mod n ( ℓ + 1) d mod n Figure 13: A pictorial interpretation of the words (3.7)-(3.9).22here we have substituted the definition of ˆ X (3.8). Therefore X ˇ X and ˆ X X both consist of thesymbols of S in cyclical order starting from S and ending with S ( ℓ − d −
1) mod n . The words X and X both have length ℓd mod n , and the words ˇ X and ˆ X both have length − d mod n . Thus X ˇ X and ˆ X X consist of the same number of symbols, which verifies (3.14). We begin by reviewing Farey addition and the Farey tree, and then apply the results to thesymbol sequences G ± [ k, χ ] (2.15).The Farey tree is a graph with the rational numbers in [0 ,
1] as its vertices [23, 24]. The Fareytree can be constructed by starting with the numbers and and supposing that there is anedge between them. Then all rational numbers between 0 and 1 are incorporated into the tree byrepeatedly applying the following rule. Given any two fractions m − n − and m + n + that are connectedby an edge, we create the new fraction mn = m − + m + n − + n + (this is Farey addition ), and say that thisfraction is connected by an edge to both m − n − and m + n + . Assuming m − n − < m + n + , we refer to m − n − and m + n + as the left and right roots of mn , respectively.For any mn in the Farey tree, mn is irreducible, i.e. gcd( m, n ) = 1, and its left and right rootssatisfy m + n − − m − n + = 1. As a consequence, mn − − m − n = 1, m + n − mn + = 1, d = n − ,and − d mod n = n + (where again d is the multiplicative inverse of m modulo n ). By applyingthese observations to the quantities in Definition 2.4 we immediately obtain the following result(illustrated in Fig. 14). Lemma 3.5.
Write m ± = m ± and n ± = n ± (to accommodate the case k = 1 ). For all k ∈ Z + ,the left and right roots of m + k n + k are mn and m + k − n + k − , respectively, and the left and right roots of m − k n − k are m − k − n − k − and mn , respectively. Moreover d + k = n and − d − k mod n − k = n . The next result concerns ˜ ℓ – the number of L ’s in G ± [ k, χ ]. This result is useful in later m − n − m − n − m − n − m − n − mn m +3 n +3 m +2 n +2 m +1 n +1 m + n + Figure 14: Part of the Farey tree centred about an arbitrary irreducible fraction mn .23ections because ˜ ℓd ± k mod n ± k is one of the four indices corresponding to a curve of border-collisionbifurcations emanating from a G ± [ k, χ ]-shrinking point. Lemma 3.6.
For any k ∈ Z + , | χ | < k , and F [ ℓ, m, n ] , ˜ ℓd ± k mod n ± k = ( ℓd mod n ± χn ) mod n ± k , (3.20) Proof.
By (2.16) and the definition of ℓ + , ℓ + k = kℓ + (cid:24) ℓn + n (cid:25) = kℓ + ℓn + n + (cid:18) − ℓn + n mod 1 (cid:19) = ℓn + k n + (cid:18) ℓdn mod 1 (cid:19) , (3.21)where in the last step we substituted n + = n − d . Since d + k = n (see Lemma 3.5), (3.21) implies(3.20) in the “+ case”. Similarly, ℓ − k = kℓ + (cid:22) ℓn − n (cid:23) = kℓ + ℓn − n − (cid:18) ℓn − n mod 1 (cid:19) = ℓn − k n − (cid:18) ℓdn mod 1 (cid:19) , (3.22)which with − d − k mod n − k = n , leads to (3.20) in the “ − case”.The next result provides us with an alternative interpretation of ℓ + and ℓ − . Here we statethe result, present an example, then give a proof. Lemma 3.7.
For any F [ ℓ, m, n ] , ℓ + is equal to the number of L ’s in ˆ X , and ℓ − is equal to thenumber of L ’s in ˆ Y . Moreover, ℓ + + ℓ − = ℓ . Example 3.3.
Consider F [3 , ,
7] =
LRRLRRL . Here mn = , thus m + n + = and m − n − = .Therefore ℓ + = l ℓn + n m = (cid:6) (cid:7) = 2, and ℓ − = j ℓn − n k = (cid:4) (cid:5) = 1.Also d = 3, so by (3.8), ˆ X = LRRL and ˆ Y = RRL . Thus the number of L ’s in ˆ X is 2 andthe number of L ’s in ˆ Y is 1, in agreement with Lemma 3.7. Proof of Lemma 3.7.
Let ˆ ℓ + denote the number of L ’s in ˆ X . By (2.7),ˆ ℓ + = (cid:12)(cid:12)(cid:12) (cid:8) i = 0 , . . . , n + − (cid:12)(cid:12) im mod n < ℓ (cid:9) (cid:12)(cid:12)(cid:12) . (3.23)For each i = 0 , . . . , n + − im mod n = (cid:18) im + in + (cid:19) mod n = im + nn + mod n , (3.24)where we have used m + n − mn + = 1 in the last equality. Using (3.24) we can rewrite (3.23) asˆ ℓ + = (cid:12)(cid:12)(cid:12)(cid:12)(cid:26) i = 0 , . . . , n + − (cid:12)(cid:12)(cid:12)(cid:12) im + mod n + < ℓn + n (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) . (3.25)Since gcd( m + , n + ) = 1, (3.25) is the same as ˆ ℓ + = (cid:12)(cid:12)(cid:12)n j = 0 , . . . , n + − (cid:12)(cid:12)(cid:12) j mod n + < ℓn + n o(cid:12)(cid:12)(cid:12) .That is, ˆ ℓ + = l ℓn + n m = ℓ + , as required.Also, ℓ − = j ℓn − n k = ℓ + j ℓn − n − ℓ k = ℓ − l ℓ − ℓn − n m = ℓ − l ℓn + n m = ℓ − ℓ + , where we have used n − + n + = n . Therefore ℓ − is equal to the number of L ’s in F [ ℓ, m, n ] minus the number of L ’sin ˆ X . Since F [ ℓ, m, n ] = ˆ X ˆ Y , see (3.10), ℓ − equals the number of L ’s in ˆ Y .24he final result of this section provides explicit expressions for G ± [ k, χ ] in terms of S = F [ ℓ, m, n ], ˆ X and ˆ Y . Again we state the result, present an example, then give a proof. Proposition 3.8.
For any k ∈ Z + , | χ | < k , and F [ ℓ, m, n ] , G + [ k, χ ] = S k + χ ˆ X (cid:16) S (cid:17) − χ , χ = − k + 1 , . . . , − (cid:16) S ℓd (cid:17) χ S k − χ ˆ X , χ = 0 , . . . , k − , (3.26) G − [ k, χ ] = S (cid:16) S (cid:17) − χ ˆ Y S k + χ − , χ = − k + 1 , . . . , S ℓd ˆ Y S k − χ (cid:16) S ℓd (cid:17) χ − , χ = 1 , . . . , k − . (3.27) Example 3.4.
Consider again F [3 , ,
7] =
LRRLRRL . Here ˆ X = LRRL and ˆ Y = RRL . With k = 3, for example, ℓ +3 = 3 × , ℓ − = 3 × ,m +3 = 3 × , m − = 3 × ,n +3 = 3 × , n − = 3 × . Therefore with χ = 0, for example, G + [3 ,
0] = F [11 , ,
25] =
LRRLRRLLRRLRRLLRRLRRLLRRL = S ˆ X , G − [3 ,
0] = F [10 , ,
24] =
LRRLRRLRRLLRRLRRLLRRLRRL = S ˆ Y S , matching Proposition 3.8.These sequences are illustrated in Fig. 15. Notice that for both G + [3 ,
0] and G − [3 , χ = 0, by (3.20), ˜ ℓd ± k = ℓd mod n = 3 × d + k = n = 7, thus node 7 of G + [3 ,
0] lies immediately to the left of node 0. Similarly − d − k mod n − k = n = 7, thus node 7 of G − [3 ,
0] lies immediately to the right of node 0.
Proof of Proposition 3.8.
Here we prove the result for G + [ k, χ ]. The result for G − [ k, χ ] can beobtained similarly.By the definition of a rotational symbol sequence (2.7), G + [ k, χ ] i = L if im + k n + k mod 1 < ˜ ℓn + k , and G + [ k, χ ] i = R otherwise. To evaluate im + k n + k mod 1, for each i = 0 , . . . , n ± k −
1, we write i = jn + r with j = 0 , . . . , k and r = 0 , . . . , n − j = k , r = 0 , . . . , n + − m + k n − mn + k = 1 weobtain im + k n + k mod 1 = ( jn + r )( mn + k + 1) nn + k mod 1 = (cid:18) rmn + jn + k + rnn + k (cid:19) mod 1 . (3.28)Now let s = rm mod n . Then im + k n + k mod 1 = sn + jn + k + rnn + k , (3.29)25here the “mod 1” is omitted on the right hand side of (3.29) because the right hand side has avalue between 0 and 1.Next, by combining (3.21), (3.29) and ˜ ℓ = ℓ + k + χ , we obtain im + k n + k mod 1 − ˜ ℓn + k = s − ℓn + j − χn + k + r − ( ℓd mod n ) nn + k . (3.30)Notice that the sign of (3.30) determines the symbol G + [ k, χ ] i . In contrast, S i is determined bythe sign of rmn mod 1 − ℓn = s − ℓn . (3.31)In the case χ = 0, it is straight-forward to see that for all j and r the right hand-sides of (3.30)and (3.31) have the same sign, and so G + [ k, i = S i for all i . We therefore have (3.26) in thecase χ = 0.With s = ℓ , we have r = ℓd mod n . Substituting this into (3.30) produces im + k n + k mod 1 − ℓ + k + χn + k = j − χn + k , which is negative if and only if j < χ , and this implies (3.26) in the case χ > s = ℓ −
1, we have r = ( ℓ − d mod n . Since we can rewrite i = jn +( ℓ − d mod n as i = (( j − k − n + ℓd mod n ) mod n + k , into (3.30) we can substitute r = ℓd mod n , s = ℓ and j j − k − im + k n + k mod 1 − ℓ + k + χn + k = j − k +1 − χn + k . This is negative ifand only if j < k + 1 + χ , which implies (3.26) in the case χ < In this section we provide some essential algebraic results for periodic solutions of (1.1). Muchof this theory is also developed in [6, 14, 15]. We begin in § F [11 , , F [10 , , Figure 15: Pictorial interpretations of G + [3 ,
0] and G − [3 , F [3 , ,
7] =
LRRLRRL , givenin Example 3.4. 26inear algebra concepts. Then in § S -cycles anddescribe their properties. Lastly in § S -cycles whenit is known that the matrix M S is non-singular (as is the case at an S -shrinking point). Given an N × N matrix A , let m ij denote the determinant of the ( N − × ( N −
1) matrix formedby removing the i th row and j th column from A (the m ij are the minors of A ). The adjugate of A is then defined by adj( A ) ij = ( − i + j m ji . For any A ,adj( A ) A = A adj( A ) = det( A ) I , (4.1)and if A is nonsingular, A − = adj( A )det( A ) , see [25, 26, 27] for further details.The following result is known as the matrix determinant lemma. For a proof using partitionedmatrices, see [28]. Lemma 4.1.
Let A be an N × N matrix, and u, v ∈ R N . Then det (cid:0) A + vu T (cid:1) = det( A ) + u T adj( A ) v . (4.2)The next result is useful to us in view of the relationship between A L and A R , (1.3). Indeedin later sections we only require (4.3) with u = e , and in this case (4.3) follows immediatelyfrom the above definition of an adjugate matrix [6, 29]. For completeness we provide a proof ofLemma 4.2 in Appendix A. Lemma 4.2.
Let A be an N × N matrix, and u, v ∈ R N . Then u T adj (cid:0) A + vu T (cid:1) = u T adj( A ) . (4.3)The next result provides an explicit formula for the adjugate of a singular matrix. A proof isgiven in Appendix A. Related properties of adjugate matrices are discussed in [30, 31]. Lemma 4.3.
Let A be an N × N matrix. If rank( A ) = N − , then adj( A ) = cvu T , (4.4) where u T A = 0 , Av = 0 , u T v = 1 , and c is the product of all nonzero eigenvalues of A , countingmultiplicity. If rank( A ) < N − , then adj( A ) is the zero matrix. An S -cycle is a periodic solution { x S i } of the half maps of (1.1) in the order determined by S ,refer to Definition 2.1 for a formal statement. If s S i ≤ S i = L , and s S i ≥ S i = R , then the S -cycle is a periodic solution of (1.1) and said to be admissible , otherwise itis said to be virtual . The following result relates to border-collision bifurcations of S -cycles (atwhich s S j = 0, for some j ) and is an immediate consequence of the continuity of (1.1).27 emma 4.4. Let { x S i } be an S -cycle. If s S j = 0 , for some j , then { x S i } is also an S j -cycle. Each x S i is a fixed point of f S ( i ) ( x ) = M S ( i ) x + P S ( i ) Bµ , (4.5)see (2.3), where S ( i ) denotes the i th left shift permutation of S . By (2.4), changing i only changesthe cyclic order in which A L and A R are multiplied to produce M S ( i ) . This is the basis for thefollowing result which is a minor generalisation of a result proved in [14, 15], and so we omit aproof. Lemma 4.5.
The determinant of M S ( i ) , and its eigenvalues and the multiplicities of the eigen-values, are independent of i . In view of (4.5), the stability of an S -cycle is governed by the eigenvalues of M S ( i ) , and byLemma 4.5 it suffices to consider i = 0. These observations provide us with the following result. Proposition 4.6.
An admissible S -cycle, with s S i = 0 for all i , is attracting if and only if alleigenvalues of M S have modulus less than . Equation (4.5) provides us with an explicit expression for each x S i , as stated in the next result. Proposition 4.7.
The S -cycle is unique if and only if I − M S is nonsingular, and if I − M S isnonsingular then x S i = ( I − M S ( i ) ) − P S ( i ) Bµ . (4.6)Lastly, to obtain a useful explicit expression for each s S i (the first component of x S i ), we usethe row vector ̺ T := e T adj ( I − A L ) = e T adj ( I − A R ) , (4.7)where the second equality is a consequence of (1.3) and (4.3). The following identity, e T adj ( I − M S ( i ) ) P S ( i ) = det ( P S ( i ) ) ̺ T , (4.8)is a consequence of (1.3), see [15, 16] for a derivation. By combining (4.6) and (4.8) we obtaindet( I − M S ) s S i = det ( P S ( i ) ) ̺ T Bµ , (4.9)from which the next result follows immediately.
Proposition 4.8. i) If I − M S is nonsingular, then s S i = det ( P S ( i ) ) ̺ T Bµ det ( I − M S ) . (4.10) ii) If I − M S is singular, f S has a fixed point, µ = 0 , and ̺ T B = 0 , then P S ( i ) is singular forall i . .3 Consequences of det ( I − M S ) = 0 Our definition of a shrinking point (Definition 2.3) includes the assumption det ( I − M S ) = 0.By Proposition 4.7, this ensures that the S -cycle is unique. In this section we provide twoimportant results requiring the assumption det ( I − M S ) = 0. Lemma 4.9.
Suppose I − M S is nonsingular and I − M S is singular. Then the eigenvalue of M S has algebraic multiplicity , and the corresponding right eigenspace of M S is not orthogonalto e .Proof. By (1.3) and (2.4), we can write M S = M S + C S e T , (4.11)for some C S ∈ R N . By Lemma 4.1,det ( I − M S ) = det (cid:0) I − M S − C S e T (cid:1) = e T adj ( I − M S ) C S , (4.12)because det ( I − M S ) = 0. Since det ( I − M S ) = 0, by assumption, adj ( I − M S ) cannot be thezero matrix. Thus by Lemma 4.3, the eigenvalue 1 of M S has algebraic multiplicity 1.Let v ∈ R N be an eigenvector of M S corresponding to the eigenvalue 1. That is v = 0 and0 = ( I − M S ) v = ( I − M S ) v + C S e T v . (4.13)But ( I − M S ) v = 0, because I − M S is nonsingular, therefore e T v = 0 as required. Lemma 4.10.
Suppose I − M S is nonsingular, I − M S and P S are singular, µ = 0 , and ̺ T B = 0 .Then P S ( i ) is singular for all i .Proof. Since I − M S is nonsingular, there exists a unique S -cycle, n x S i o . The matrix P S issingular because P S = P S , (2.5). Therefore s S = 0, (4.10). By Lemma 4.4, n x S i o is also an S -cycle, and so x S is a fixed point of f S . Therefore, by Proposition 4.8(ii), P S ( i ) is singular forall i . Our definition of a shrinking point, Definition 2.3, is based on the conditions s S = 0 and s S ℓd = 0,where the s S i are the first coordinates of the points of the S -cycle, and S = F [ ℓ, m, n ] is arotational symbol sequence. Here we begin by providing additional motivation for our restrictionto rotational symbol sequences, and the choice of the indices i = 0 and i = ℓd in Definition 2.3. Conceptually, a shrinking point is a point in parameter space where (1.1) has an S -cycle, { x S i } ,with two points on the switching manifold. Without loss of generality we can suppose that oneof these points is x S , and that S = L . Let x S α , where 1 ≤ α ≤ n − n is the period, be29he other point of the S -cycle on the switching manifold. By a double application of Lemma 4.4,this S -cycle is also an S α -cycle. If S α = R , then S α has the same number of L ’s as S . In thiscase it is possible for there to exist an integer d such that S α ( d ) = S . (5.1)That is, if we flip the 0 th and α th symbols of S , then apply the d th left shift permutation, werecover the original symbol sequence.The next result tells us that if (5.1) holds, then S must equal F [ ℓ, m, n ] for some integers ℓ and m , and α = ℓd mod n . In other words, our restriction to rotational symbol sequences andthe choice of the indices i = 0 and i = ℓd in Definition 2.3 can be viewed as a consequence ofsupposing that the symbol sequence associated with a shrinking point satisfies (5.1) for somevalues of α and d . Proposition 5.1.
Let S be a periodic symbol sequence of period n with S = L , and suppose S α ( d ) = S , for some ≤ α ≤ n − and ≤ d ≤ n − with gcd( d, n ) = 1 . Let m denotethe multiplicative inverse of d modulo n , and ℓ = mα mod n . Then S = F [ ℓ, m, n ] , and α = ℓd mod n .Proof. The formula α = ℓd mod n is a trivial consequence of our definitions of ℓ and m in thestatement of the theorem. It remains to show that S = F [ ℓ, m, n ], which we achieve by verifying(3.1).For any j = 1 , . . . , ℓ − S jd mod n = S αjd mod n = S α ( d )( j − d mod n , (5.2)because j = 0 , ℓ , and where the second equality follows from the definition of a shift permutation.Then S α ( d )( j − d mod n = S ( j − d mod n , (5.3)because S α ( d ) = S , by assumption. By then starting with S = L , and recursively applying(5.2)-(5.3), we obtain S = S d = · · · = S ( ℓ − d mod n , matching (3.1).Equation (5.3) is true for all j ∈ Z , whereas (5.2) is false for j = ℓ (because ℓd mod n = α ). This implies S ℓd mod n = S ( ℓ − d mod n , and thus S ℓd mod n = R . Finally, (5.2) is true for all j = ℓ + 1 , . . . , n − S ℓd mod n = S ( ℓd +1) mod n = · · · = S − d mod n = R , which verifies (3.1), and hence S = F [ ℓ, m, n ]. Recall, at an S -shrinking point, y i denotes the i th point of the S -cycle and t i denotes the firstcoordinate of y i , see § t = 0 , t ℓd = 0 , (5.4)and t i = 0, for all i = 0 , ℓd . The S -cycle is assumed to be admissible, and this fixes the signs ofthe t i . In particular, t d < , t ( ℓ − d < , t ( ℓ +1) d > , t − d > , (5.5)see Fig. 16.The next three results provide key properties of shrinking points.30 roposition 5.2. At an S -shrinking point, { y i } has period n . This is proved in [15] and is a consequence of the rotational nature of F [ ℓ, m, n ]. Proposition 5.3.
At an S -shrinking point,i) { y i } is both an S -cycle and an S ( − d ) -cycle,ii) I − M S is singular,iii) P S ( i ) is singular for all i .Proof. By definition, { y i } is an S -cycle. Since t = 0, by Lemma 4.4, { y i } is also an S -cycle.Since t ℓd = 0, { y i } is similarly also an S ℓd -cycle, and so by (3.6), { y i } is also an S ( − d ) -cycle,which proves part (i).By part (i), y and y d are both fixed points of f S . By (5.4) and (5.5) these points are distinct,hence by Proposition 4.7, I − M S is singular. Finally, P S ( i ) is singular for all i by Proposition4.8(ii). Proposition 5.4.
For any S -shrinking point in the phase space of (1.1), let P denote the non-planar polygon formed by joining each y i to y ( i + d ) by a line segment. Then each point on P belongs to an admissible S -cycle of (1.1), and the restriction of (1.1) to P is homeomorphic torigid rotation with rotation number mn . Refer to [14, 15] for a proof.
By Proposition 5.3(ii), M S has a unit eigenvalue. Since det ( I − M S ) = 0, by Lemma 4.9 theunit eigenvalue is of algebraic multiplicity one. Furthermore, by Lemma 4.5 each M S ( i ) has aunit eigenvalue of algebraic multiplicity one. y ( ℓ − d y ( ℓ − d y ℓd y ( ℓ +1) d y ( ℓ +2) d − t ( ℓ − d t ( ℓ +1) d y d y d y y − d y − d − t d t − d s = 0 Figure 16: A schematic diagram showing the S -cycle at an S -shrinking point.31ecall, in § u T j and v j denote the left and right eigenvectors of M S ( j ) correspondingto the unit eigenvalue normalised by u T j v j = 1 and e T v j = 1, for j = 0 , ( ℓ − d, ℓd, − d . Thefollowing result provides explicit expressions for u T j and v j . Lemma 5.5.
At an S -shrinking point, for each j ∈ { , ( ℓ − d, ℓd, − d } , u T j = e T adj ( I − M S ( j ) ) c , v j = y j + d − y j t j + d − t j , (5.6) where c is given by (2.11).Proof. By applying Lemma 4.3 to the matrix A = I − M S ( j ) , we obtain adj ( I − M S ( j ) ) = cv j u T j .Since also e T v j = 1, this implies e T adj ( I − M S ( j ) ) c = u T j .Next let ˆ v j = y j + d − y j t j + d − t j . It remains to show that ˆ v j = v j . Trivially e T ˆ v j = 1. By Proposition5.3, y j and y j + d are both fixed points of f S ( j ) , and thus y j + d − y j = f S ( j ) ( y j + d ) − f S ( j ) ( y j ) = M S ( j ) ( y j + d − y j ) . (5.7)Therefore M S ( j ) ˆ v j = ˆ v j . That is, each ˆ v j satisfies the same properties as v j . But v j is unique,hence ˆ v j = v j , as required.The eigenvectors u T j and v j are sketched in Fig. 17. By Lemma 5.5, the eigenvector v , forexample, has the same direction as the line segment connecting y to y d , and e T v j = 1. Let z be any point on this line segment other than y and y d . By Proposition 5.4, z is a fixed point of f S . Moreover, there exists a neighbourhood of z that follows the sequence S under the next n iterations of (1.1). Within this neighbourhood, the hyperplane that intersects z and is orthogonalto u T is invariant. If all the eigenvalues of M S , other than the unit eigenvalue, have modulusless than 1 (i.e. σ <
1, see (2.14)), then within this neighbourhood the hyperplane is the stablemanifold of z for the map f S . In summary, iterates of f S approach the line segment connecting y to y d (which has direction v ) on a hyperplane orthogonal to u T . The remaining eigenvectors u T j and v j can be interpreted similarly.The next result indicates how the eigenvectors are related to one another algebraically. Lemma 5.6.
We have v ℓd = t d t ( ℓ +1) d M X v , u T ℓd = t ( ℓ +1) d t d u T M Y , (5.8) v = t ( ℓ +1) d t d M Y v ℓd , u T = t d t ( ℓ +1) d u T ℓd M X , (5.9) v ( ℓ − d = t − d t ( ℓ − d M X v − d , u T ( ℓ − d = t ( ℓ − d t − d u T − d M Y , (5.10) v − d = t ( ℓ − d t − d M Y v ( ℓ − d , u T − d = t − d t ( ℓ − d u T ( ℓ − d M X . (5.11)32 urthermore, v − d = − t d t − d M ˆ X v , u T − d = − t − d t d u T M ˆ Y , (5.12) v = − t − d t d M ˆ Y v − d , u T = − t d t − d u T − d M ˆ X , (5.13) v ( ℓ − d = − t ( ℓ +1) d t ( ℓ − d M ˇ X v ℓd , u T ( ℓ − d = − t ( ℓ − d t ( ℓ +1) d u T ℓd M ˇ Y , (5.14) v ℓd = − t ( ℓ − d t ( ℓ +1) d M ˇ Y v ( ℓ − d , u T ℓd = − t ( ℓ +1) d t ( ℓ − d u T ( ℓ − d M ˇ X . (5.15) Proof.
By Proposition 5.3, f X ( y ) = y ℓd and f X ( y d ) = y ( ℓ +1) d . Therefore M X v = 1 t d M X ( y d − y ) = 1 t d (cid:0) y ( ℓ +1) d − y ℓd (cid:1) = t ( ℓ +1) d t d v ℓd , (5.16) s = 0 y y d y ( ℓ − d y ℓd y ( ℓ +1) d y − d v v ( ℓ − d v ℓd v − d u u ( ℓ − d u ℓd u − d Figure 17: A schematic diagram illustrating the periodic solution { y i } and the eigenvectors u T j and v j . 33hich verifies the first part of (5.8). The first parts of the remaining equations can be derived inthe same fashion.By (3.10) and (3.11), M Y M S ( ℓd ) = M Y M X M Y = M S M Y . (5.17)Therefore u T M Y M S ( ℓd ) = u T M S M Y = u T M Y , (5.18)i.e. u T M Y is a left eigenvector of M S ( ℓd ) corresponding to the eigenvalue 1, and therefore is amultiple of u ℓd . Also, by using the first part of (5.9) we obtain t ( ℓ +1) d t d u T M Y v ℓd = u T v = 1 . (5.19)Therefore t ( ℓ +1) d t d u T M Y has the same magnitude and direction as u ℓd . This verifies the second partof (5.8), and second parts of the remaining equations can be demonstrated similarly. The behaviour of F [ ℓ, m, n ]-cycles and F [ ℓ ± , m, n ]-cycles near an S -shrinking point, where S = F [ ℓ, m, n ], was summarised in § ξ = ( ξ , ξ ) ∈ R , for simplicity, let ξ ∗ be an S -shrinking point, and introducelocal ( η, ν )-coordinates (2.12). The condition det( J ) = 0, where J is given by (2.13), ensuresthat the coordinate change ( ξ , ξ ) ↔ ( η, ν ) is invertible.The following result specifies curves of border-collision bifurcations, η = ψ ( ν ) and ν = ψ ( η ),along which F [ ℓ, m, n ] and F [ ℓ + 1 , m, n ]-cycles coincide. The subsequent result provides a usefulexpression for det ( I − M S ). Both results are proved in [14, 15], except that expressions for thecoefficients in terms of the t i are derived in [16]. Lemma 5.7.
Suppose (1.1) with K ≥ has an S -shrinking point at ξ = ξ ∗ and det( J ) = 0 .Then, in a neighbourhood of ξ = ξ ∗ ,i) there exists a unique C K function ψ : R → R , with ψ ( ν ) = − t d t ( ℓ − d t ( ℓ +1) d ν + o (cid:0) ν (cid:1) , (5.20) such that s S ℓd ℓd = 0 on the locus η = ψ ( ν ) ,ii) there exists a unique C K function ψ : R → R , with ψ ( η ) = − t ( ℓ − d t d t − d η + o (cid:0) η (cid:1) , (5.21) such that s S ℓd = 0 on the locus ν = ψ ( η ) . emma 5.8. Suppose (1.1) with K ≥ has an S -shrinking point at ξ = ξ ∗ and det( J ) = 0 .Then det( I − M S ) = at d η + at ( ℓ − d ν + O (cid:0) ( η, ν ) (cid:1) , (5.22) where a = det ( I − M S ) (cid:12)(cid:12) ξ = ξ ∗ . The next result identifies regions, Ψ and Ψ , within which F [ ℓ − , m, n ], F [ ℓ, m, n ] and F [ ℓ + 1 , m, n ]-cycles reside, and represents the basic unfolding of a shrinking point. The readeris referred to [14, 15] for a proof. Fig. 18 summarises the unfolding. If σ <
1, then some of theperiodic solutions are stable, see Table 1, but note that Theorem 5.9 does not concern stabilityand holds for any value of σ . Theorem 5.9.
Suppose (1.1) with K ≥ has an S -shrinking point at ξ = ξ ∗ and det( J ) = 0 .Let Ψ = (cid:8) ( η, ν ) (cid:12)(cid:12) η, ν ≥ (cid:9) and Ψ = (cid:8) ( η, ν ) (cid:12)(cid:12) η ≤ ψ ( ν ) , ν ≤ ψ ( η ) (cid:9) , where ψ and ψ arespecified by Lemma 5.7. Then there exists a neighbourhood N of ( η, ν ) = (0 , , such that (1.1)has unique admissible F [ ℓ, m, n ] and F [ ℓ − , m, n ] cycles in Ψ ∩ N \ { (0 , } and (1.1) hasadmissible F [ ℓ, m, n ] and F [ ℓ + 1 , m, n ] cycles in Ψ ∩ N \ { (0 , } . We conclude this section by deriving additional algebraic expressions regarding shrinking pointsthat are used in later sections.As implied by (5.22), I − M S is singular along a curve passing through the S -shrinking point.At points where I − M S is non-singular, the S -cycle is unique, and the following result providesus with an asymptotic expression for the location of the points of the S -cycle. Lemma 5.10.
Suppose (1.1) with K ≥ has an S -shrinking point at ξ = ξ ∗ and det( J ) = 0 .Then for all ( η, ν ) for which det( I − M S ) = 0 , for all i , x S i = y i + d t d η + y i t ( ℓ − d ν + O (( η, ν ) ) t d η + t ( ℓ − d ν + O (( η, ν ) ) . (5.23) Proof.
By (4.6), x S i = adj ( I − M S ( i ) ) P S ( i ) Bµ det( I − M S ) . Therefore we can write x S i ( η, ν ) = C + C η + C ν + O (( η, ν ) ) at d η + at ( ℓ − d ν + O (( η, ν ) ) , (5.24)for some C , C , C ∈ R N . When η = 0, x S i = x S i , thus x S i (0 , ν ) = y i + O ( η, ν ), hence C = 0and C = at ( ℓ − d y i . Similarly when ν = 0, x S i = x S i + d , thus x S i ( η,
0) = y i + d + O ( η, ν ), hence C = at d y i + d . By substituting these expressions for C and C into (5.24) and cancelling instancesof a , we obtain (5.23) as required. 35t the shrinking point, M S has a unit eigenvalue and so near the shrinking point M S hasan eigenvalue near 1. Throughout this paper this eigenvalue is denoted by λ . Locally λ is C K function of η and ν because the algebraic multiplicity of the unit eigenvalue at the shrinkingpoint is one, Lemma 4.9. Lemma 5.11.
Suppose (1.1) with K ≥ has an S -shrinking point at ξ = ξ ∗ and det( J ) = 0 .Then λ = 1 − act d η − act ( ℓ − d ν + O (cid:0) ( η, ν ) (cid:1) , (5.25) where c is the product of the nonzero eigenvalues of I − M S at ξ = ξ ∗ , (2.11). ην Ψ Ψ det ( P S ) = 0 η = 0 det ( P S (( ℓ − d ) ) = 0 ν = 0det ( P S ( ℓd ) ) = 0 η = ψ ( ν )det ( P S ( − d ) ) = 0 ν = ψ ( η ) det ( I − M S ) = 0 x S = x S x S ℓd = x S ℓd x S− d = x S x S ( ℓ − d = x S ℓd x S = x S ℓd x S ℓd = x S ℓd ℓd x S− d = x S ℓd x S ( ℓ − d = x S ℓd ℓd Figure 18: The basic unfolding of a shrinking point as specified by Theorem 5.9. In ( η, ν )-coordinates, the shrinking point is located at the origin, the positive axes are border-collisionbifurcation curves that bound the region Ψ , and η = ψ ( ν ) and ν = ψ ( η ) are border-collisionbifurcation curves that bound Ψ . We have also included sketches of the S -cycle in relation tothe switching manifold at a typical point on each of the four boundaries.36 roof. Write λ = 1 + k η + k ν + O (2), for some k , k ∈ R . Thendet( I − M S ) = (1 − λ ) ( c + O ( η, ν ))= (cid:0) − k η − k ν + O (cid:0) ( η, ν ) (cid:1)(cid:1) ( c + O ( η, ν ))= − k cη − k cν + O (cid:0) ( η, ν ) (cid:1) . (5.26)By matching (5.22) and (5.26) we obtain (5.25) as required.The last two results provide identities that connect various quantities associated with a shrink-ing point. For a proof of Lemma 5.12, refer to [16]. The proof involves expanding s S i and s S ℓd i in terms of η and ν (for certain values of i ) and matching coefficients. This assumes det( J ) = 0,but we expect that Lemmas 5.12 and 5.13 hold regardless of how (1.1) varies with ξ as the resultsconcern properties of the shrinking point itself. Lemma 5.12.
Suppose (1.1) with K ≥ has an S -shrinking point at ξ = ξ ∗ and det( J ) = 0 .Then ab = − t d t ( ℓ − d t − d t ( ℓ +1) d . (5.27) Lemma 5.13.
Suppose (1.1) with K ≥ has an S -shrinking point at ξ = ξ ∗ and det( J ) = 0 .Then, repeating (2.10), u T v − d a + u T ℓd v ( ℓ − d b = 1 c , (5.28) u T ( ℓ − d v ℓd a + u T − d v b = 1 c . (5.29) Proof.
Here we derive only (5.28). Equation (5.29) may be derived similarly.By (5.9), (5.11) and (5.27), u T v − d = (cid:18) t d t ( ℓ +1) d u T ℓd M X (cid:19) (cid:18) t ( ℓ − d t − d M Y v ( ℓ − d (cid:19) = − au ℓd M X M Y v ( ℓ − d b . (5.30)Therefore u T v − d a + u T ℓd v ( ℓ − d b = u T ℓd (cid:0) I − M X M Y (cid:1) v ( ℓ − d b . (5.31)By (5.6), u T ℓd = e T adj ( I − M S ( ℓd ) ) c , and by (3.11), M S ( ℓd ) = M X M Y . By (2.4) and Lemma 4.2, e T adj ( I − M X M Y ) = e T adj (cid:0) I − M X M Y (cid:1) , thus u T ℓd = e T adj (cid:0) I − M X M Y (cid:1) c . (5.32)By substituting (5.32) into (5.31) and using (4.1) we obtain u T v − d a + u T ℓd v ( ℓ − d b = e T v ( ℓ − d det (cid:0) I − M X M Y (cid:1) bc . (5.33)Finally, det (cid:0) I − M X M Y (cid:1) = b , because M X M Y = M S ( ℓd )0 = M S ℓd ( ℓd ) = M F [ ℓ +1 ,m,n ] ( ℓd ) , by (3.4).Thus (5.33) reduces to (5.28), because also e T v ( ℓ − d = 1, by definition.37 Locating nearby shrinking points
At an S -shrinking point, for each j , the line segment connecting y j to y j + d consists of fixed pointsof f S ( j ) , see Fig. 17. For parameter values near the shrinking point, the line segments persist asone-dimensional slow manifolds. These are described in § § G ± [ k, χ ]-shrinking points to leading order. In a neighbourhood of an S -shrinking point, for each j = 0 , ( ℓ − d, ℓd, − d , we let ω T j and ζ j denote the left and right eigenvectors of M S ( j ) corresponding to λ . More specifically, M S ( j ) ζ j = λζ j , e T ζ j = 1 , (6.1) ω T j M S ( j ) = λω T j , ω T j ζ j = 1 . (6.2)Each ω T j and ζ j is a C K function, of η and ν . Recall, u T j and v j denote the eigenvectors at theshrinking point, see § ζ j (0 ,
0) = v j , ω T j (0 ,
0) = u T j . (6.3)The following result relates the eigenvectors to one another based on the partitions of S introducedin Definition 3.2. Lemma 6.1.
For any matrix Q , ω T QM ˆ X ζ = ω T − d M ˆ X Qζ − d , (6.4) ω T M ˆ Y Qζ = ω T − d QM ˆ Y ζ − d , (6.5) ω T ℓd QM ˇ X ζ ℓd = ω T ( ℓ − d M ˇ X Qζ ( ℓ − d , (6.6) ω T ℓd M ˇ Y Qζ ℓd = ω T ( ℓ − d QM ˇ Y ζ ( ℓ − d . (6.7) Proof.
Here we derive (6.4). The remaining identities can be derived in the same fashion.Since M S = M ˆ Y M ˆ X and M S ( − d ) = M ˆ X M ˆ Y , refer to (2.4), (3.10) and (3.12), by (6.1) we have ζ = k M ˆ Y ζ − d , (6.8)for some k ∈ R . Similarly ω T = k ω T − d M ˆ X , (6.9)for some k ∈ R . By using (6.1)-(6.2), we then deduce1 = ω T ζ = k k ω T − d M ˆ X M ˆ Y ζ − d = k k ω T − d λζ − d = λk k . (6.10)Finally, by combining (6.8)-(6.10) we obtain ω T QM ˆ X ζ = k k ω T − d M ˆ X QM ˆ X M ˆ Y ζ − d = k k ω T − d M ˆ X Qλζ − d = ω T − d M ˆ X Qζ − d , (6.11)as required. 38et us now consider the dynamics of f S ( j ) (for any j = 0 , ( ℓ − d, ℓd, − d ). If det ( I − M S ) = 0,then f S ( j ) has a unique fixed point, x S j . The line intersecting x S j and of direction ζ j is a slowmanifold on which the dynamics of f S ( j ) is dictated by the value of λ .Since x S j is sensitive to changes in η and ν , see (5.23), instead of x S j it more helpful to usethe intersection of the slow manifold with the switching manifold, call it ϕ j , as a reference pointabout which we can perform calculations. If det ( I − M S ) = 0, then this intersection point isgiven by ϕ j = (cid:0) I − ζ j e T (cid:1) x S j . (6.12)The utility of ϕ j lies in the fact that it is well-defined even when det ( I − M S ) = 0, see Lemma6.2.Any point on the slow manifold can be written as ϕ j + hζ j , where h ∈ R is the first componentof this point (since e T ϕ j = 0 and e T ζ j = 1). If det ( I − M S ) = 0, then, since x S j is a fixed pointof f S ( j ) and M S ( j ) ζ j = λζ j , we have f S ( j ) ( ϕ j + hζ j ) = ϕ j + ( hλ + γ j ) ζ j , (6.13)where, γ j = (1 − λ ) e T x S j . (6.14)Equation (6.13) describes the dynamics on the slow manifold, see Fig. 19. The next result justifiesour use of ϕ j and γ j when det ( I − M S ) = 0. Lemma 6.2.
Suppose (1.1) with K ≥ has an S -shrinking point at ξ = ξ ∗ and det( J ) = 0 .Then there exists a neighbourhood N of ( η, ν ) = (0 , , such that for all j ∈ { , ( ℓ − d, ℓd, − d } ,there exists unique C K functions ϕ j : N → R N and γ j : N → R with e T ϕ j = 0 , γ j (0 ,
0) = 0 , and ϕ (0 ,
0) = ϕ − d (0 ,
0) = y , ϕ ( ℓ − d (0 ,
0) = ϕ ℓd (0 ,
0) = y ℓd , (6.15) such that (6.13) is satisfied for all h ∈ R and all ( η, ν ) ∈ N . Moreover, (6.12) and (6.14) aresatisfied for all ( η, ν ) ∈ N for which det ( I − M S ) = 0 .Proof. Consider the matrix equation (cid:0) ( I − M S ( j ) ) (cid:0) I − e e T (cid:1) + ζ j e T (cid:1) φ j = P S ( j ) Bµ , (6.16)where we wish to solve for the unknown vector φ j . Here we show that (6.16) defines φ j uniquely,and that the desired quantities ϕ j and γ j are given by ϕ j = (cid:0) I − e e T (cid:1) φ j , γ j = e T φ j . (6.17)To show that (6.16) has a unique solution, we apply Lemma 4.1 to writedet (cid:0) ( I − M S ( j ) ) (cid:0) I − e e T (cid:1) + ζ j e T (cid:1) = det ( I − M S ( j ) ) det (cid:0) I − e e T (cid:1) + e T adj (cid:0) I − e e T (cid:1) adj ( I − M S ( j ) ) ζ j . (6.18)Since det (cid:0) I − e e T (cid:1) = 0 and adj (cid:0) I − e e T (cid:1) = e e T , (6.18) reduces todet (cid:0) ( I − M S ( j ) ) (cid:0) I − e e T (cid:1) + ζ j e T (cid:1) = e T adj ( I − M S ( j ) ) ζ j . (6.19)39t ( η, ν ) = (0 , e T adj ( I − M S ( j ) ) = cu T j and ζ j = v j , see (5.6) and (6.3). Using also u T j v j = 1we obtain det (cid:0) ( I − M S ( j ) ) (cid:0) I − e e T (cid:1) + ζ j e T (cid:1) = c + O ( η, ν ) , (6.20)which is nonzero in a neighbourhood of (0 , φ j in thisneighbourhood.Next we define ϕ j and γ j by (6.17). These are C K functions of η and ν because the componentsof (6.16) are C K . By (6.17), e T ϕ j = 0. Also φ j = ϕ j + γ j e , and by substituting this into (6.16)and simplifying we obtain ( I − M S ( j ) ) ϕ j + γ j ζ j = P S ( j ) Bµ . (6.21)Thus for any h ∈ R M S ( j ) ( ϕ j + hζ j ) + P S ( j ) Bµ = ϕ j + ( hλ + γ j ) ζ j , (6.22)where we have used M S ( j ) ζ j = λ j ζ j . and therefore (6.13). This shows that (6.13) is satisfied.When ( η, ν ) = (0 , γ j = 0 and either ϕ j = y or ϕ j = y ℓd , asgiven in (6.15). This verifies (6.15) and γ j (0 ,
0) = 0 because φ j is unique. Similarly, whendet ( I − M S ) = 0, (6.21) is satisfied by (6.12) and (6.14) because( I − M S ( j ) ) (cid:0) I − ζ j e T (cid:1) x S j + (1 − λ ) ζ j e T x S j = ( I − M S ( j ) ) x S j , (6.23) s = 0 ϕ j ζ j ϕ j + ( γ j + hλ ) ζ j ϕ j + hζ j xf S ( j ) ( x ) ω j qM S ( j ) q Figure 19: An illustration of dynamics near the slow manifold of f S ( j ) , for any j = 0 , ( ℓ − d, ℓd, − d , for parameter values near an S -shrinking point. The slow manifold is a line ofdirection ζ j (the slow eigenvector (6.1)) that intersects the switching manifold at ϕ j In orderto study images of x under f S ( j ) , it is helpful to decompose x using the eigenspaces of M S ( j ) ,specifically (6.24). 40nd therefore ( I − M S ( j ) ) x S j = P S Bµ . Again, because of the uniqueness of φ j , (6.12) and (6.14)hold in a neighbourhood of ( η, ν ) = (0 , j , and any x ∈ R N , it is helpful to write x = ϕ j + hζ j + q , (6.24)where h ∈ R and q ∈ R N with ω T j q = 0. The vector hζ j represents the component of x − ϕ j in the ζ j direction. The vector q represents the component of x − ϕ j in the remaining eigendirectionsof M S ( j ) . The decomposition (6.24) is unique and enables us to express iterates of x under f S ( j ) succinctly, Lemma 6.3. This is illustrated in Fig. 19 and formalised by the following result. Weomit a proof as it is a straight-forward application of eigenspace decomposition. Lemma 6.3.
For all j ∈ { , ( ℓ − d, ℓd, − d } , and for all x ∈ R N , there exists unique h ∈ R and q ∈ R N with ω T j q = 0 such that x = ϕ j + hζ j + q . Moreover h = ω T j ( x − ϕ j ) , q = (cid:0) I − ζ j ω T j (cid:1) ( x − ϕ j ) , (6.25) and for any k ∈ Z + , f ( S ( j ) ) k ( x ) = ϕ j + γ j k − X j =0 λ j + hλ k ! ζ j + M k S ( j ) q . (6.26)The remaining results of this paper assume σ <
1, where σ denotes the maximum modulusof the eigenvalues of M S , excluding the unit eigenvalue, at the S -shrinking point, (2.14). Thisassumption ensures iterates under f S ( j ) converge to the slow manifold, which is central to thevalidity of our main results. Lemma 6.4. If σ < , then c > .Proof. As in (2.14), let ρ , . . . , ρ N be the eigenvalues of M S , counting multiplicity, and ρ = 1.By the definition of c (2.11), c = Q Ni =2 (1 − ρ i ). The assumption σ < | ρ i | <
1, foreach i = 1. If ρ i ∈ R , then 1 − ρ i >
0. Any ρ i / ∈ R appear in complex conjugate pairs with(1 − ρ i )(1 − ρ i ) >
0. Thus Q Ni =2 (1 − ρ i ) can be expressed as a product of positive numbers, andso c > q is a linear combination of the eigendirections of M S other than ζ j . If σ <
1, then the corresponding eigenvalues all have modulus less than 1, in which case (cid:13)(cid:13) M k S ( j ) q (cid:13)(cid:13) →
0, as k → ∞ . Moreover, we can write M k S ( j ) q = O (cid:0) σ k (cid:1) . This is true for any q of the form (6.25),thus we have the following result. Lemma 6.5. If σ < , then for each j ∈ { , ( ℓ − d, ℓd, − d } , M k S ( j ) (cid:0) I − ζ j ω T j (cid:1) = O (cid:0) σ k (cid:1) . (6.27)41 .2 Calculations for nearby shrinking points In this section we derive formulas for the border-collision bifurcation curves of G ± [ k, χ ]-cycles.The first result provides us with expressions for det (cid:0) ρI − M G ± [ k,χ ] (cid:1) , useful for large values of k ∈ Z + . Lemma 6.6.
Suppose (1.1) with K ≥ has an S -shrinking point at ξ = ξ ∗ and det( J ) = 0 and σ < . Choose any χ max ∈ Z + , k ∈ Z + , and ρ ∈ C . Then, in a neighbourhood of ( η, ν ) = (0 , , det (cid:0) ρI − M G + [ k,χ ] (cid:1) = ρ N (cid:16) − λ k + χ +1 ρ ω T ℓd ( M X M Y ) − χ − M ˇ X ζ ℓd (cid:17) + O (cid:0) σ k (cid:1) , − χ max ≤ χ ≤ − ρ N (cid:16) − λ k − χ ρ ω T (cid:0) M Y M X (cid:1) χ M ˆ X ζ (cid:17) + O (cid:0) σ k (cid:1) , ≤ χ ≤ χ max , (6.28)det (cid:0) ρI − M G − [ k,χ ] (cid:1) = ρ N (cid:16) − λ k + χ ρ ω T − d ( M Y M X ) − χ M ˆ Y ζ − d (cid:17) + O (cid:0) σ k (cid:1) , − χ max ≤ χ ≤ ρ N (cid:16) − λ k − χ +1 ρ ω T ( ℓ − d (cid:0) M X M Y (cid:1) χ − M ˇ Y ζ ( ℓ − d (cid:17) + O (cid:0) σ k (cid:1) , ≤ χ ≤ χ max . (6.29) Proof.
Here we derive (6.28) for 0 ≤ χ ≤ χ max . The remaining formulas can be derived similarly.By Lemma 4.5, M G + [ k,χ ] and M G + [ k,χ ] ( i ) share the same eigenvalues and multiplicities, for any i . The same is true for ρI − M G + [ k,χ ] and ρI − M G + [ k,χ ] ( i ) , thus, for any i ,det (cid:0) ρI − M G + [ k,χ ] (cid:1) = det (cid:0) ρI − M G + [ k,χ ] ( i ) (cid:1) . (6.30)For any 0 ≤ χ ≤ χ max , G + [ k, χ ] = (cid:16) S ℓd (cid:17) χ S k − χ ˆ X , (3.26). The word ˆ X has n − d symbols(3.2), thus G + [ k, χ ] ( d − n ) = ˆ X (cid:16) S ℓd (cid:17) χ S k − χ . (6.31) G + [ k, χ ] ( d − n ) is a particularly useful permutation of G + [ k, χ ] for the purposes of this proof because,by (6.31), it ends in a power involving k . By (6.31), M G + [ k,χ ] ( d − n ) = M k − χ S M χ S ℓd M ˆ X . (6.32)Since M S ζ = λζ , by Lemma 6.5 with j = 0 we can write M k − χ S = M k − χ S ζ ω T + M k − χ S (cid:0) I − ζ ω T (cid:1) = λ k − χ ζ ω T + O (cid:0) σ k (cid:1) , (6.33)with which (6.32) becomes M G + [ k,χ ] ( d − n ) = λ k − χ ζ ω T (cid:0) M Y M X (cid:1) χ M ˆ X + O (cid:0) σ k (cid:1) , (6.34)where we have also substituted M S ℓd = M Y M X . Finally, by using Lemma 4.1, (6.30) with i = n − d , and (6.34), we arrive at (6.28) for 0 ≤ χ ≤ χ max as required.42o motivate the next result, recall that boundaries of G ± k -mode-locking regions are pointswhere s G ± [ k,χ ] i = 0, for certain values of i . Each s G ± [ k,χ ] i can be evaluated using (2.6). Thefollowing result provides asymptotic expressions for the numerator of (2.6), applied to s G ± [ k,χ ] i . Inview of Proposition 3.1 and Lemma 4.4, different values of χ and i can give the same boundary s G ± [ k,χ ] i = 0. This gives us some choice as to the values of χ and i that we can use to describea given boundary. In Lemma 6.7 we choose the index i that provides the simplest algebraicexpression, leading to four different cases, as indicated. Lemma 6.7.
Suppose (1.1) with K ≥ has an S -shrinking point at ξ = ξ ∗ and det( J ) = 0 and σ < . Choose any χ max ∈ Z + and k ∈ Z + . Then, in a neighbourhood of ( η, ν ) = (0 , , det (cid:18) P G + [ k,χ ] ( ˜ ℓd + k ) (cid:19) ̺ T Bµ = γ ( ℓ − d k + χ X j =0 λ j + ω T ( ℓ − d (cid:16) f ( YX ) − χ − ˇ X (cid:0) ϕ ( ℓ − d (cid:1) − ϕ ( ℓ − d (cid:17) λ k + χ +1 + O (cid:0) σ k (cid:1) , for all − χ max ≤ χ ≤ − , (6.35)det (cid:0) P G + [ k,χ ] (cid:1) ̺ T Bµ = γ − d k − χ − X j =0 λ j + ω T − d (cid:16) f ( X Y ) χ ˆ X ( ϕ − d ) − ϕ − d (cid:17) λ k − χ + O (cid:0) σ k (cid:1) , for all 0 ≤ χ ≤ χ max , (6.36)det (cid:18) P G − [ k,χ ] ( − d − k ) (cid:19) ̺ T Bµ = γ k − χ − X j =0 λ j + ω T (cid:16) f ( X Y ) − χ ˆ Y ( ϕ ) − ϕ (cid:17) λ k + χ + O (cid:0) σ k (cid:1) , for all − χ max ≤ χ ≤ , (6.37)det (cid:18) P G − [ k,χ ] ( ( ˜ ℓ − ) d − k ) (cid:19) ̺ T Bµ = γ ℓd k − χ X j =0 λ j + ω T ℓd (cid:16) f ( Y X ) χ − ˇ Y ( ϕ ℓd ) − ϕ ℓd (cid:17) λ k − χ +1 + O (cid:0) σ k (cid:1) , for all 1 ≤ χ ≤ χ max . (6.38) Proof.
Here we derive (6.36). The remaining expressions can be derived in the same fashion.By (3.10), (3.12), (3.14) and (3.15), S ˆ X = X Y ˆ X = X ˇ X Y = ˆ X X Y = ˆ X S ( − d ) . (6.39)Thus, for 0 ≤ χ ≤ χ max , (3.26) can be rewritten as G + [ k, χ ] = (cid:16) S ℓd (cid:17) χ ˆ X (cid:0) S ( − d ) (cid:1) k − χ . (6.40)43hen, by Lemma 6.3 with j = − d , for any x ∈ R N , f G + [ k,χ ] ( x ) = f ( S ( − d ) ) k − χ (cid:16) f ( S ℓd ) χ ˆ X ( x ) (cid:17) = f ( S ( − d ) ) k − χ (cid:16) M ( S ℓd ) χ ˆ X x + P ( S ℓd ) χ ˆ X Bµ (cid:17) = ϕ − d + γ − d k − χ − X j =0 λ j + ω T − d (cid:16) M ( S ℓd ) χ ˆ X x + P ( S ℓd ) χ ˆ X Bµ − ϕ − d (cid:17) λ k − χ ! ζ − d + M k − χ S ( − d ) (cid:0) I − ζ − d ω T − d (cid:1) (cid:16) M ( S ℓd ) χ ˆ X x + P ( S ℓd ) χ ˆ X Bµ − ϕ − d (cid:17) . (6.41)Now suppose det (cid:0) I − M G + [ k,χ ] (cid:1) = 0. Then x G + [ k,χ ]0 is well-defined and is the unique fixedpoint of f G + [ k,χ ] . By Lemma 6.3 we can uniquely write x G + [ k,χ ]0 = ϕ − d + hζ − d + q , (6.42)where h ∈ R and ω T − d q = 0. By substituting (6.42) for x and f G + [ k,χ ] ( x ) in (6.41), multiplyingboth sides of (6.41) by e T on the left, and applying Lemma 6.5, we obtain h = γ − d k − χ − X j =0 λ j + ω T − d (cid:16) M ( S ℓd ) χ ˆ X ( ϕ − d + hζ − d ) + P ( S ℓd ) χ ˆ X Bµ − ϕ − d (cid:17) λ k − χ + O (cid:0) σ k (cid:1) , (6.43)and q = O (cid:0) σ k (cid:1) . (6.44)By (6.42) and (6.44), s G + [ k,χ ]0 = h + O (cid:0) σ k (cid:1) , and by solving for h in (6.43), we arrive at s G + [ k,χ ]0 = γ − d P k − χ − j =0 λ j + ω T − d (cid:16) f ( S ℓd ) χ ˆ X ( ϕ − d ) − ϕ − d (cid:17) λ k − χ − λ k − χ ω T − d M ˆ X (cid:0) M S ℓd (cid:1) χ ζ − d + O (cid:0) σ k (cid:1) , (6.45)Finally, by (6.4) with Q = M S ℓd = M Y M X , and (6.28) with 0 ≤ χ ≤ χ max and ρ = 1, weobtain det (cid:0) I − M G + [ k,χ ] (cid:1) = 1 − λ k − χ ω T − d M ˆ X (cid:0) M S ℓd (cid:1) χ ζ − d + O (cid:0) σ k (cid:1) . (6.46)Therefore, by (2.6), (6.45) and (6.46), we produce (6.36), as required. Above we assumeddet (cid:0) I − M G + [ k,χ ] (cid:1) = 0. By continuity, (6.36) also holds at points near ( η, ν ) = (0 ,
0) for whichdet (cid:0) I − M G + [ k,χ ] (cid:1) = 0.We have now developed the tools necessary to prove Theorem 2.1. The proof is given AppendixA. In this proof we obtain additional asymptotic expressions for the boundary curves that areuseful below. Specifically, on the curves det (cid:0) P G + [ k,χ ] (cid:1) = 0 and det (cid:18) P G + [ k,χ ] ( ( ˜ ℓ − ) d + k ) (cid:19) = 0, wehave λ k t d η + 1 t ( ℓ − d ν + O (cid:0) ( η, ν ) (cid:1) = 0 , (6.47)44nd on the curves det (cid:0) P G − [ k,χ ] (cid:1) = 0 and det (cid:18) P G − [ k,χ ] ( ( ˜ ℓ − ) d + k ) (cid:19) = 0, we have1 t d η + λ k t ( ℓ − d ν + O (cid:0) ( η, ν ) (cid:1) = 0 . (6.48)The next result concerns the eigenvalues of M G ± [ k,χ ] and is used below to prove Theorem 2.2.Theorem 2.3 follows from this and is proved in Appendix A. Lemma 6.8.
Suppose (1.1) with K ≥ has an S -shrinking point at ξ = ξ ∗ and det( J ) = 0 and σ < . Choose any χ max ∈ Z + and k ∈ Z + . Then, in a neighbourhood of ( η, ν ) = (0 , , for any | χ | ≤ χ max , M G ± [ k,χ ] has an eigenvalue ρ G + [ k,χ ] satisfying ρ G + [ k,χ ] = ( λ k + χ +1 ω T ℓd ( M X M Y ) − χ − M ˇ X ζ ℓd + O (cid:0) σ k (cid:1) , − χ max ≤ χ ≤ − λ k − χ ω T (cid:0) M Y M X (cid:1) χ M ˆ X ζ + O (cid:0) σ k (cid:1) , ≤ χ ≤ χ max , (6.49) ρ G − [ k,χ ] = ( λ k + χ ω T − d ( M Y M X ) − χ M ˆ Y ζ − d + O (cid:0) σ k (cid:1) , − χ max ≤ χ ≤ λ k − χ +1 ω T ( ℓ − d (cid:0) M X M Y (cid:1) χ − M ˇ Y ζ ( ℓ − d + O (cid:0) σ k (cid:1) , ≤ χ ≤ χ max , (6.50) and all other eigenvalues of M G ± [ k,χ ] are O (cid:0) σ k (cid:1) .Proof. Here we prove the result for G + [ k, χ ] with 0 ≤ χ ≤ χ max . The remaining parts can beproved similarly.By Lemma 4.5, the eigenvalues of M G + [ k,χ ] are the same as those of M G + [ k,χ ] ( d − n ) , where thelatter matrix is given by (6.34). Therefore the eigenvalues of M G + [ k,χ ] differ from those of therank-one matrix λ k − χ ζ ω T (cid:0) M Y M X (cid:1) χ M ˆ X , by O (cid:0) σ k (cid:1) , and the only non-zero eigenvalue of thismatrix is λ k − χ ω T (cid:0) M Y M X (cid:1) χ M ˆ X ζ . Proof of Theorem 2.2.
Here we prove the result for G + [ k, χ ], with 0 ≤ χ ≤ χ max . The remainingparts of Theorem 2.2 can be proved in a similar fashion.We look for intersections of det (cid:0) P G + [ k,χ ] (cid:1) = 0, with det (cid:0) ρI − M G + [ k,χ ] (cid:1) = 0, for ρ = ±
1. Alongdet (cid:0) P G + [ k,χ ] (cid:1) = 0, the parameters η and ν are O (cid:0) k (cid:1) , and satisfy (6.47). In polar coordinates(2.18), tan( θ ) = − t d νt ( ℓ − d η . Thus by (6.47), if det (cid:0) P G + [ k,χ ] (cid:1) = 0, thentan( θ ) = − λ k + O (cid:18) k (cid:19) . (6.51)Notice det (cid:0) ρI − M G + [ k,χ ] (cid:1) = 0 when ρ = ρ G + [ k,χ ] , as given by (6.49). Thus by rearranging(6.49) with 0 ≤ χ ≤ χ max and putting ρ G + [ k,χ ] = ±
1, we see that det (cid:0) ρI − M G + [ k,χ ] (cid:1) = 0 when λ k = ± u T (cid:0) M Y M X (cid:1) χ M ˆ X v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (0 , + O (cid:18) k (cid:19) , (6.52)where we have used the fact that ω T and ζ are equal to u T and v , to leading order (6.3). Since M S ℓd = M Y M X and v − d = − t d t − d M ˆ X v (5.12), by (2.23), (6.52) reduces to λ k = ∓ t d t − d κ + χ + O (cid:18) k (cid:19) , (6.53)45here κ + χ is given by (2.23), and we assume κ + χ = 0.Therefore, by (6.51) and (6.53), any intersection of det (cid:0) P G + [ k,χ ] (cid:1) = 0 and det (cid:0) I − M G + [ k,χ ] (cid:1) =0 must satisfy tan( θ ) = t d t − d κ + χ + O (cid:0) k (cid:1) . Since tan( θ ) < t d < t − d >
0, the intersectionpoint exists if and only if κ + χ > (cid:0) P G + [ k,χ ] (cid:1) = 0 and det (cid:0) − I − M G + [ k,χ ] (cid:1) = 0 must satisfytan( θ ) = − t d t − d κ + χ + O (cid:0) k (cid:1) . In this case the intersection point exists if and only if κ + χ < In this section we work towards a proof of Theorem 2.4.Recall, η = s S and ν = s S ℓd (2.12) provide a convenient local coordinate system in which tostudy the dynamics of (1.1) near an S -shrinking point. As in § η = s T and ˜ ν = s T ˜ ℓd ± k denotethe analogous coordinates for a nearby T = G ± [ k, χ ]-shrinking point, where the T -shrinking pointis located at ( η, ν ) = ( η T , ν T ), Theorem 2.2.In order to relate ˜ η and ˜ ν to η and ν , we first represent points ( η, ν ) as perturbations from( η T , ν T ), by writing ( η, ν ) = ( η T + ∆ η, ν T + ∆ ν ) . (7.1)At the T -shrinking point, ∆ η = ∆ ν = 0 and ˜ η = ˜ ν = 0. From Lemmas 6.6 and 6.7 and (2.6),it can be seen that ∂s T ∂η and ∂s T ∂ν are O ( k ), as are the first derivatives of s T ˜ ℓd ± k . It follows that ˜ η and ˜ ν admit the following expansion:˜ η = (cid:18) p k + p + O (cid:18) k (cid:19)(cid:19) ∆ η + (cid:18) p k + p + O (cid:18) k (cid:19)(cid:19) ∆ ν + O (cid:0) (∆ η, ∆ ν ) (cid:1) , (7.2)˜ ν = (cid:18) q k + q + O (cid:18) k (cid:19)(cid:19) ∆ η + (cid:18) q k + q + O (cid:18) k (cid:19)(cid:19) ∆ ν + O (cid:0) (∆ η, ∆ ν ) (cid:1) , (7.3)for some constants p i , q i ∈ R .The condition det( J ) = 0 ensures that the change of coordinates from ( ξ , ξ ) ↔ ( η, ν ) islocally invertible. Similarly, the condition det( ˜ J ) = 0, where ˜ J is given by (2.28), ensures thatthe change of coordinates ( η, ν ) ↔ (˜ η, ˜ ν ) is locally invertible. In view of (7.2) and (7.3), we canwrite ˜ J := " ∂ ˜ η∂η ∂ ˜ η∂ν∂ ˜ ν∂η ∂ ˜ ν∂ν ( η T ,ν T ) = (cid:20) p k + p p k + p q k + q q k + q (cid:21) + O (cid:18) k (cid:19) . (7.4)Below in Lemma 7.2 we derive identities involving the constants p i , q i ∈ R . First we showthat in the analogous expansion for det ( I − M T ) the leading order coefficients take a simple formand are independent to the choice of T . Lemma 7.1.
Suppose (1.1) with K ≥ has an S -shrinking point at ξ = ξ ∗ and det( J ) = 0 and σ < . Then for any T = G ± [ k, χ ] , det ( I − M T ) = (cid:18) act d k + O (1) (cid:19) ∆ η + (cid:18) act ( ℓ − d k + O (1) (cid:19) ∆ ν + O (cid:0) (∆ η, ∆ ν ) (cid:1) . (7.5)46 roof. By (5.25), λ k = (cid:18) − act d η − act ( ℓ − d ν + O (cid:0) ( η, ν ) (cid:1)(cid:19) k . (7.6)By (5.25), ∂λ∂η ( η T , ν T ) = − act d + O (cid:0) k (cid:1) , and ∂λ∂ν ( η T , ν T ) = − act ( ℓ − d + O (cid:0) k (cid:1) . Therefore by expanding(7.6) about ( η, ν ) = ( η T , ν T ), we obtain λ k ( η, ν ) = λ k ( η T , ν T ) (cid:18) − (cid:18) act d k + O (1) (cid:19) ∆ η − (cid:18) act ( ℓ − d k + O (1) (cid:19) ∆ ν + O (cid:0) (∆ η, ∆ ν ) (cid:1)(cid:19) , (7.7)where we have also substituted λ k − ( η T , ν T ) = λ k ( η T , ν T ) + O (cid:0) k (cid:1) .By Lemma 6.6 we can writedet ( I − M T ) = 1 − c λ k + O (cid:18) k (cid:19) , (7.8)where c ∈ R depends on T but is independent of k . We have det ( I − M T ) = 0 at ( η, ν ) =( η T , ν T ), thus λ k ( η T , ν T ) = 1 c ( η T , ν T ) + O (cid:18) k (cid:19) . (7.9)With (7.7) and (7.9), (7.8) reduces to (7.5) as required. Lemma 7.2.
For any T = G ± [ k, χ ] , the coefficients of (7.2) and (7.3) satisfy t ( ℓ − d ∓ sgn( a )Γ (cid:0) θ + χ (cid:1) sin (cid:0) θ + χ (cid:1) ! p − t d ± sgn( a )Γ (cid:0) θ + χ (cid:1) cos (cid:0) θ + χ (cid:1) ! p = 0 , (7.10)1 t ( ℓ − d ∓ sgn( a )Γ (cid:0) θ + χ (cid:1) sin (cid:0) θ + χ (cid:1) ! q − t d ± sgn( a )Γ (cid:0) θ + χ (cid:1) cos (cid:0) θ + χ (cid:1) ! q = 0 , (7.11) where Γ is given by (2.19)-(2.20) and the θ ± χ are given by (2.25)-(2.26).Proof. Here we prove the result for T = G + [ k, χ ]. The result for T = G − [ k, χ ] can be proved inthe same fashion.The curve ˜ η = 0 is a boundary of the G + k -mode-locking region emanating from the T = G + [ k, χ ]-shrinking point. Note that det (cid:0) P G + [ k,χ ] (cid:1) = 0, which is approximated by (6.47), definesthe same curve. By evaluating (6.47) at the T -shrinking point, we obtain λ k ( η T , ν T ) = − t d ν T t ( ℓ − d η T + O (cid:18) k (cid:19) . (7.12)By further expanding (6.47) about the T -shrinking point with (7.1), we determine that ˜ η = 0 isdescribed by1 t d (cid:18) − akη T ct d + O (cid:18) k (cid:19)(cid:19) ∆ η + 1 t ( ℓ − d (cid:18) λ k ( η T , ν T ) − akη T ct d + O (cid:18) k (cid:19)(cid:19) ∆ ν + O (cid:0) (∆ η, ∆ ν ) (cid:1) = 0 . (7.13)47y multiplying both sides of (7.13) by − ct d aη T k and substituting (7.12), we obtain (cid:18) t d − caη T k + O (cid:18) k (cid:19)(cid:19) ∆ η + (cid:18) t ( ℓ − d + caν T k + O (cid:18) k (cid:19)(cid:19) ∆ ν + O (cid:0) (∆ η, ∆ ν ) (cid:1) . (7.14)By then evaluating η T and ν T with (2.18) and (2.21), and taking care to accommodate differentcases depending on the sign of a , we determine that ˜ η = 0 is described by1 t d a )Γ (cid:0) θ + χ (cid:1) cos (cid:0) θ + χ (cid:1) + O (cid:18) k (cid:19)! ∆ η + 1 t ( ℓ − d − sgn( a )Γ (cid:0) θ + χ (cid:1) sin (cid:0) θ + χ (cid:1) + O (cid:18) k (cid:19)! ∆ ν + O (cid:0) (∆ η, ∆ ν ) (cid:1) . (7.15)By matching (7.2) and (7.15) we obtain (7.10) for T = G + [ k, χ ]. The curve ˜ ν = 0 is also givenby (6.47), hence the same result holds for q and q , i.e. (7.11).We complete this section by deriving a novel identity for the leading order term of det (cid:16) ˜ J (cid:17) .This is used to prove Theorem 2.4 in Appendix A. First note that by (5.22) we can writedet ( I − M T ) = ˜ a ˜ t d ± k ˜ η + ˜ a ˜ t ( ˜ ℓ − ) d ± k ˜ ν + O (cid:0) (˜ η, ˜ ν ) (cid:1) . (7.16)Moreover, from Lemmas 6.6 and 6.7 it can be seen that ˜ t d ± k and ˜ t ( ˜ ℓ − ) d ± k are O (cid:0) k (cid:1) , and ˜ a is O (1),and therefore we can writedet ( I − M T ) = (cid:18) r k + r + O (cid:18) k (cid:19)(cid:19) ˜ η + (cid:18) r k + r + O (cid:18) k (cid:19)(cid:19) ˜ ν + O (cid:0) (˜ η, ˜ ν ) (cid:1) , (7.17)for some constants r i ∈ R . Lemma 7.3.
For any T = G ± [ k, χ ] , the coefficients of (7.17) satisfy p r + q r = 0 , (7.18) and ˜ J (7.4) satisfies det (cid:16) ˜ J (cid:17) = acr (cid:18) q t d − q t ( ℓ − d (cid:19) k + O (1) . (7.19) Proof.
By (7.4), det (cid:16) ˜ J (cid:17) = ( p q + p q − p q − p q ) k + O (1) , (7.20)where the k -term has vanished because (7.10) and (7.11) imply p q − p q = 0 . (7.21)48y substituting (7.2) and (7.3) into (7.17) we obtaindet ( I − M S ) = (cid:0) ( p r + q r ) k + ( p r + p r + q r + q r ) k + O (1) (cid:1) ∆ η + (cid:0) ( p r + q r ) k + ( p r + p r + q r + q r ) k + O (1) (cid:1) ∆ ν + O (cid:0) (∆ η, ∆ ν ) (cid:1) . (7.22)By matching the k -terms of (7.5) and (7.22), we deduce that p r + q r = 0 (verifying (7.18))and p r + q r = 0. Note that these equations are equivalent in view of (7.21). By then matchingthe k -terms of (7.5) and (7.22), we obtain p r + p r + q r + q r = act d , p r + p r + q r + q r = act ( ℓ − d . (7.23)By combining (7.18), (7.21) and (7.23), we obtain ac (cid:18) q t d − q t ( ℓ − d (cid:19) = r ( p q + p q − p q − p q ) , (7.24)which by (7.20) yields (7.19), as required. Shrinking points are codimension-two points in the parameter space of piecewise-linear continuousmaps at which mode-locking regions have zero width. In this paper we have studied the N -dimensional map (1.1), which has a single switching manifold, s = 0. We have consideredmode-locking regions that, in a symbolic sense, can be assigned a rotation number, mn . At anyshrinking point in such a mode-locking region there exists an invariant polygon in the phasespace of (1.1). All orbits on the polygon have period n , rotation number mn , and, say, ℓ points tothe left of the switching manifold per period (except a special periodic orbit, labelled { y i } , thathas two points on the switching manifold).This paper provides the first rigorous study into the dynamics near an arbitrary shrinkingpoint, other than the period- n dynamics within the mode-locking region itself which was examinedin [14]. We refer to the shrinking point as an S -shrinking point, where S = F [ ℓ, m, n ] is thesymbol sequence associated with orbits on the invariant polygon. On each side of the mode-locking region connected to an S -shrinking point, there is a sequence of mode-locking regions.On one side the mode-locking regions have associated rotation numbers km + m − kn + n − , and on the otherside the mode-locking regions have associated rotation numbers km + m + kn + n + , where k ∈ Z + and m − n − and m + n + are the left and right Farey roots of mn . The local curvature and relative spacing ofthese mode-locking regions was described using polar coordinates and the nonlinear function Γ(2.19)-(2.20), as indicated in Theorem 2.1.The two sequences of mode-locking regions themselves have shrinking points. Thus sequencesof shrinking points converge to the S -shrinking point. We have characterised these shrinkingpoints with symbol sequences, G ± [ k, χ ]. But the G ± [ k, χ ]-shrinking points only exist for particularvalues of χ ∈ Z . We proved, subject to certain non-degeneracy conditions, see Theorem 2.2, thatthere exists a sequence of potential G ± [ k, χ ]-shrinking points, that converge to the S -shrinking49oint as k → ∞ , if and only if κ ± χ >
0, where κ ± χ are scalar constants associated with the S -shrinking point. The angular coordinates of the potential G ± [ k, χ ]-shrinking points are given,to leading order, by θ ± χ . Numerical investigations reveal that these points are commonly validshrinking points, but may not be due a lack of admissibility of the orbits on the associatedinvariant polygon. Theorem 2.3 and equation (2.10) show that there are some restrictions on thecombinations of signs possible for the κ ± χ . Theorem 2.4 tells us that nearby G ± [ k, χ ]-shrinkingpoints are non-degenerate and have the same orientation as the S -shrinking point.It remains to describe other dynamics near shrinking points, such as periodic, quasiperiodicand chaotic dynamics at points in parameter space between the nearby mode-locking regions thatwe have identified, and consider more general classes of piecewise-smooth maps. Such maps arisein diverse applications, and if there is only weak nonlinearity in the smooth pieces of the map(or if the relevant orbits are only traversing parts of phase space that involve weak nonlinearity),then the mode-locking regions can exhibit a sausage-string structure involving points of near-zerowidth [11, 12, 32]. Border-collision bifurcations are described by piecewise-smooth continuousmaps, and the influence of the nonlinearity in the pieces of the map increases with the distancein parameter space from the border-collision bifurcation. This influence on mode-locking regionboundaries emanating from shrinking points was explained in [16], but it remains to understandthe effect of such nonlinearities on other local dynamics. A Additional proofs
Proof of Lemma 4.2.
First, suppose that A and A + vu T are nonsingular. The identity (cid:0) A + vu T (cid:1) − = A − − A − vu T A − u T A − v , (A.1)is known as the Sherman-Morrison formula and be can verified directly. We use (4.1) to rewrite(A.1) as adj (cid:0) A + vu T (cid:1) det ( A + vu T ) = adj( A )det( A ) − adj( A ) vu T adj( A )det( A ) det ( A + vu T ) , (A.2)and therefore adj (cid:0) A + vu T (cid:1) = adj( A ) + adj( A ) u T adj( A ) v − adj( A ) vu T adj( A ) . (A.3)Upon multiplying (A.3) by u T on the left, the last two terms cancel leaving us with (4.3).The subset of triples ( A, u, v ) for which both A and A + vu T are nonsingular is dense in theset of all triples ( A, u, v ). Therefore since both sides of (4.3) are continuous functions of A , u and v , (4.3) holds in general. Proof of Lemma 4.3.
First, suppose rank( A ) = N −
1. Then 0 is an eigenvalue of A , and so thereexist u, v ∈ R N such that u T A = 0, Av = 0, and u T v = 1. By (4.1), adj( A ) must be of the formadj( A ) = ˆ cvu T , (A.4)for some ˆ c ∈ R . To demonstrate (4.4) it remains to show that ˆ c = c .50et ε ∈ R . Then by (4.1) and Av = 0, we havedet( A + εI ) v = adj( A + εI )( A + εI ) v = adj( A + εI ) vε = adj( A ) vε + O (cid:0) ε (cid:1) . (A.5)By substituting (A.4) into (A.5) and using u T v = 1, we obtaindet( A + εI ) v = ˆ cvε + O (cid:0) ε (cid:1) . (A.6)Notice, ε is an eigenvalue of A + εI . Let λ j ( ε ), for j = 2 , . . . , N , denote the remainingeigenvalues of A + εI , counting multiplicity. By definition, c = Q Nj =2 λ j (0), and det( A + εI ) isthe product of all eigenvalues of A + εI , thusdet( A + εI ) = ε N Y j =2 λ j ( ε ) = cε + O (cid:0) ε (cid:1) . (A.7)By matching (A.6) and (A.7), we deduce that ˆ c = c , and therefore (4.4) as required.Second, if rank( A ) < N − i and j , the ( N − × ( N −
1) matrix formed byremoving the i th row and j th column from A also has rank less than N −
1. Thus m ij = 0 for all i and j and so adj( A ) is the zero matrix. Proof of Theorem 2.1.
Here we construct C K curves along which det (cid:0) P G + [ k,χ ] (cid:1) = 0, for 0 ≤ χ ≤ χ max , and det (cid:18) P G + [ k,χ ] ( ( ˜ ℓ − ) d + k ) (cid:19) = 0, for 1 ≤ χ ≤ χ max , and verify (2.21)-(2.22) for these curves.The result for the remaining curves can be obtained in the same fashion.To solve det (cid:0) P G + [ k,χ ] (cid:1) = 0 we use (6.36). In (6.36) we substitute ϕ − d = y + O ( η, ν ), f S ℓd ( y ) = y + O ( η, ν ), and f ˆ X ( y ) = y − d + O ( η, ν ), to obtain ω T − d (cid:16) f ( S ℓd ) χ ˆ X ( ϕ − d ) − ϕ − d (cid:17) = u T − d ( y − d − y ) + O ( η, ν )= t − d + O ( η, ν ) . (A.8)Thus by (6.36), if det ( I − M S ) = 0 (in which case λ = 1), we can writedet (cid:0) P G + [ k,χ ] (cid:1) ̺ T Bµ = γ − d (cid:0) − λ k − χ (cid:1) − λ + λ k − χ ( t − d + O ( η, ν )) + O (cid:0) σ k (cid:1) . (A.9)By (5.25), 1 − λ = act d η + act ( ℓ − d ν + O (cid:0) ( η, ν ) (cid:1) . (A.10)To evaluate γ − d , we substitute (5.23) and (A.10) into (6.14) to obtain, after simplification, γ − d = at − d ct ( ℓ − d ν + O (cid:0) ( η, ν ) (cid:1) . (A.11)Also, since λ = 1 + O ( η, ν ) and χ is a constant (independent of k ), we can write λ k − χ = λ k (1 + O ( η, ν )). By substituting these expressions into (A.9) we arrive atdet (cid:0) P G + [ k,χ ] (cid:1) ̺ T Bµ = at − d c (1 − λ ) (cid:18) λ k t d η + 1 t ( ℓ − d ν + O (cid:0) ( η, ν ) (cid:1)(cid:19) . (A.12)51ote, the apparent singularity λ = 1 in (A.12) is spurious because det (cid:0) P G + [ k,χ ] (cid:1) ̺ T Bµ is C K ina neighbourhood of ( η, ν ) = (0 , k in the leading order term of (A.12) occurs in the quantity λ k , whereby (A.10) λ k = (cid:18) − act d η − act ( ℓ − d ν + O (cid:0) ( η, ν ) (cid:1)(cid:19) k . (A.13)Therefore, in the limit k → ∞ , λ k only can take O (1) values other than 0 and 1 if η, ν = O (cid:0) k (cid:1) ,as the limit is taken. For this reason it is appropriate to write η = ˆ ηk , ν = ˆ νk . (A.14)and treat ˆ η and ˆ ν as O (1) constants. By substituting (A.14) into (A.13) we obtain λ k = e − ac (cid:18) ˆ ηtd + ˆ νt ( ℓ − d (cid:19) + O (cid:18) k (cid:19) . (A.15)Then by substituting (A.10) and (A.15) into (A.12) we obtaindet (cid:0) P G + [ k,χ ] (cid:1) ̺ T Bµ = t − d e a ˆ νct ( ℓ − d ˆ ηt d e − a ˆ ηctd + ˆ νt ( ℓ − d e a ˆ νct ( ℓ − d ˆ ηt d + ˆ νt ( ℓ − d + O (cid:18) k (cid:19) . (A.16)Next we work in polar coordinates (2.18). For clarity, we consider only the case a <
0. Theresult for a > t d < t ( ℓ − d <
0, see (5.5), and c > a < η = ct d a ˆ r cos( θ ) , ˆ ν = ct ( ℓ − d a ˆ r sin( θ ) , (A.17)where we let ˆ r = rk . (A.18)Then by (A.17) and (A.16) we can writedet (cid:0) P G + [ k,χ ] (cid:1) ̺ T Bµ = t − d e ˆ r sin( θ ) H (ˆ r, θ ) , (A.19)where H is a C K function and H (ˆ r, θ ) = H (ˆ r, θ ) + O (cid:18) k (cid:19) , (A.20)where H (ˆ r, θ ) = cos( θ ) e − ˆ r cos( θ ) + sin( θ ) e ˆ r sin( θ ) cos( θ ) − sin( θ ) . (A.21)It is a straight-forward exercise to show that H (Γ( θ ) , θ ) = 0, where Γ is given by (2.19) and θ ∈ (cid:0) π , π (cid:1) . 52ext we employ the implicit function theorem to find where the right hand-side of (A.19) iszero. We define, H (ˆ r, θ, ε ) = kεH (ˆ r, θ ) + (1 − kε ) H (ˆ r, θ ) , (A.22)and we are interested in small values of ε ∈ R . Notice H is C K , H (Γ( θ ) , θ,
0) = 0, and H (ˆ r, θ, ε ) = H (ˆ r, θ ) + O ( ε ). Therefore, for any θ ∈ (cid:0) π , π (cid:1) , there exists a neighbourhoodof (ˆ r, ε ) = (Γ( θ ) ,
0) in which we can apply the implicit function theorem. That is, there existsa unique C K function ˜Γ( θ, ε ), such that H (cid:16) ˜Γ( θ, ε ) , θ, ε (cid:17) = 0, inside the neighbourhood, and˜Γ( θ,
0) = Γ( θ ). Then, assuming k is sufficiently large, H (cid:16) ˜Γ (cid:0) θ, k (cid:1) , θ (cid:17) ≡
0. By (A.19), thisshows that det (cid:0) P G + [ k,χ ] (cid:1) = 0 along a unique C K curve satisfying (2.21) and (2.22).To obtain the same result for det (cid:18) P G + [ k,χ ] ( ( ˜ ℓ − ) d + k ) (cid:19) , we begin by using (3.20) to write (cid:16) ˜ ℓ − (cid:17) d + k mod n + k = ℓd mod n + ( χ − n , (A.23)where “mod n + k ” is not needed on the right hand-side by assuming 1 ≤ χ ≤ χ max . Thus by(6.40), the first (cid:16) ˜ ℓ − (cid:17) d + k mod n + k symbols of G + [ k, χ ] are given by (cid:16) S ℓd (cid:17) χ − X , and so we canwrite x G + [ k,χ ] ( ˜ ℓ − ) d + k = f ( S ℓd ) χ − X (cid:16) x G + [ k,χ ]0 (cid:17) . (A.24)Substituting x G + [ k,χ ]0 = ϕ − d + hζ − d + q (6.42) into (A.24) gives x G + [ k,χ ] ( ˜ ℓ − ) d + k = f ( S ℓd ) χ − X ( ϕ − d ) + M X M χ − S ℓd ( hζ − d + q ) . (A.25)By then substituting h = s G + [ k,χ ]0 + O (cid:0) σ k (cid:1) , and q = O (cid:0) σ k (cid:1) (refer to the proof of Lemma 6.7)into (A.25), and multiplying both sides of (A.25) by e T det ( I − G + [ k, χ ]) on the left and using(2.6), we producedet (cid:18) P G + [ k,χ ] ( ( ˜ ℓ − ) d + k ) (cid:19) ̺ T Bµ = e T f ( S ℓd ) χ − X ( ϕ − d ) det (cid:0) I − M G + [ k,χ ] (cid:1) + e T M X M χ − S ℓd ζ − d det (cid:0) P G + [ k,χ ] (cid:1) ̺ T Bµ + O (cid:0) σ k (cid:1) . (A.26)The solution to det (cid:18) P G + [ k,χ ] ( ( ˜ ℓ − ) d + k ) (cid:19) = 0 is the same, to leading order, as the solution todet (cid:0) P G + [ k,χ ] (cid:1) = 0, because f ( S ℓd ) χ − X ( ϕ − d ) = y ℓd + O (cid:0) k (cid:1) and so the first term in (A.26) ishigher order than the term involving det (cid:0) P G + [ k,χ ] (cid:1) . Therefore near ( η, ν ) = (0 ,
0) there exists aunique C K curve satisfying (2.21) and (2.22) along which det (cid:18) P G + [ k,χ ] ( ( ˜ ℓ − ) d + k ) (cid:19) = 0. Proof of Theorem 2.3.
For brevity we restrict our attention to G + [ k, χ ] with 0 ≤ χ ≤ χ max .By Proposition 5.3(i), { ˜ y i } is a G + [ k, χ ]-cycle. Thus ˜ y maps to ˜ y ˜ ℓd + k under f L and f R inthe order specified by the first ˜ ℓd + k mod n + k symbols of G + [ k, χ ]. Since 0 ≤ χ ≤ χ max , by (3.20),53 ℓd + k mod n + k = ℓd mod n + χn . Thus, by (3.26), the first ˜ ℓd + k mod n + k symbols of G + [ k, χ ] are (cid:16) S ℓd (cid:17) χ X , and therefore ˜ y ˜ ℓd + k = f ( S ℓd ) χ X (˜ y ) . (A.27)Also, by Proposition 5.3(i), { ˜ y i } is a G + [ k, χ ]( − d + k )-cycle. Thus ˜ y d + k maps to ˜ y ( ˜ ℓ +1 ) d + k followingthe first ˜ ℓd + k mod n + k symbols of G + [ k, χ ]. That is,˜ y ( ˜ ℓ +1 ) d + k = f ( S ℓd ) χ X (cid:16) ˜ y d + k (cid:17) . (A.28)In addition ˜ y d + k = f S ( − d ) (˜ y ) , (A.29)because d + k = n (see Lemma 3.5) and the first n symbols of G + [ k, χ ] are S ℓd = X Y = S ( − d ) ,by (3.26) and (3.12).In the form ˜ y = ϕ − d + hζ − d + q (6.24), we have q = O (cid:0) σ k (cid:1) (for the same reasons as for x G + [ k,χ ]0 in the proof of Lemma 6.7), thus˜ y = ϕ − d + O (cid:0) σ k (cid:1) , (A.30)because also e T ˜ y = 0, e T ϕ − d = 0, and e T ζ − d = 1. By (6.26) and (A.29),˜ y d + k = ϕ − d + γ − d ζ − d + O (cid:0) σ k (cid:1) , (A.31)and by (A.28), (A.30) and (A.31),˜ y ( ˜ ℓ +1 ) d + k = f ( S ℓd ) χ X (cid:0) ˜ y + γ − d ζ − d + O (cid:0) σ k (cid:1)(cid:1) = f ( S ℓd ) χ X (˜ y ) + γ − d M X M χ S ℓd ζ − d + O (cid:0) σ k (cid:1) . (A.32)By then multiplying (A.32) by e T on the left and using (A.27) and e T ˜ y ˜ ℓd + k = 0, we obtain˜ t ( ˜ ℓ +1 ) d + k = γ − d e T M X (0 , M χ S ℓd (0 , v − d + O (cid:18) k (cid:19) , (A.33)where we have also used ζ − d (0 ,
0) = v − d .Our next step is to derive the following identity u T (cid:0) I − M S ℓd (0 , (cid:1) = bt d e T M X (0 , ct ( ℓ +1) d . (A.34)By substituting (5.6) for u T ℓd into (5.9) we obtain u T = t d e T adj ( I − M S ( ℓd ) (0 , M X (0 , ct ( ℓ +1) d . (A.35)54e have M S ( ℓd ) = M X M Y (3.10), and in view of (2.4) and (4.3) we can substitute M Y for M Y to obtain u T = t d e T adj (cid:0) I − M X (0 , M Y (0 , (cid:1) M X (0 , ct ( ℓ +1) d . (A.36)Note, det (cid:0) I − M X (0 , M Y (0 , (cid:1) = b , because M X M Y = M S ( ℓd )0 = M S ℓd ( ℓd ) , refer to (2.9),(3.11) and Lemma (4.5). Thus by (4.1) and (A.36), u T = bt d e T (cid:0) I − M X (0 , M Y (0 , (cid:1) − M X (0 , ct ( ℓ +1) d . (A.37)By substituting (cid:0) I − M X M Y (cid:1) − M X = M X (cid:0) I − M Y M X (cid:1) − and multiply both sides of (A.37)by I − M Y M X on the right we arrive at (A.34).By substituting (A.34) into (A.33) we obtain˜ t ( ˜ ℓ +1 ) d + k = γ − d ct ( ℓ +1) d u T (cid:0) I − M S ℓd (0 , (cid:1) M χ S ℓd (0 , v − d bt d + O (cid:18) k (cid:19) . (A.38)By using (5.27) and γ − d = at − d ν G +[ k,χ ] ct ( ℓ − d + O (cid:0) k (cid:1) (A.11), (A.38) simplifies to˜ t ( ˜ ℓ +1 ) d + k = ν G + [ k,χ ] (cid:0) κ + χ +1 − κ + χ (cid:1) + O (cid:18) k (cid:19) , (A.39)where we have substituted (2.23).If a >
0, then θ + χ ∈ (cid:0) π , π (cid:1) (see Table 2), thus sin (cid:0) θ + χ (cid:1) >
0, and so by (2.18) ν G + [ k,χ ] > k . Since κ + χ > t ( ˜ ℓ +1 ) d + k > (cid:0) η G + [ k,χ ] , ν G + [ k,χ ] (cid:1) is a G + [ k, χ ]-shrinking point), by (A.39) we must have κ + χ +1 >
0, as claimed.If χ ≥
1, we can similarly show that˜ t ( ˜ ℓ − ) d + k = ν G + [ k,χ ] (cid:0) κ + χ − − κ + χ (cid:1) + O (cid:18) k (cid:19) , (A.40)based on the knowledge that ˜ y and ˜ y d + k map to ˜ y ( ˜ ℓ − ) d + k and ˜ y ˜ ℓd + k , respectively, under f ( S ℓd ) χ − X .Then if a < ν G + [ k,χ ] <
0, for arbitrarily large values of k , and thus since κ + χ > t ( ˜ ℓ − ) d + k < κ + χ − >
0, as claimed.The remaining cases can be proved in the same fashion.
Proof of Theorem 2.4.
Part (ii) of the theorem is an immediate consequence of Lemma 6.8. Weprove part (i) for G + [ k, χ ] with 0 ≤ χ ≤ χ max . Other cases may be proved in a similar fashion.We first show that sgn(˜ a ) = sgn( a ). At ( η, ν ) = ( η T , ν T ), det (cid:0) I − M G + [ k,χ ] (cid:1) = 0 anddet (cid:0) I − M G + [ k,χ +1] (cid:1) = ˜ b . Thus, by (6.28) with ρ = 1,1 − (cid:16) λ k − χ ω T M χ S ℓd M ˆ X ζ (cid:17)(cid:12)(cid:12)(cid:12) ( η T ,ν T ) + O (cid:0) σ k (cid:1) = 0 , (A.41)1 − (cid:16) λ k − χ − ω T M χ +1 S ℓd M ˆ X ζ (cid:17)(cid:12)(cid:12)(cid:12) ( η T ,ν T ) + O (cid:0) σ k (cid:1) = ˜ b . (A.42)55y combining (A.41) and (A.42) and using (5.12) we obtain˜ b = 1 − u T M χ +1 S ℓd (0 , v − d u T M χ S ℓd (0 , v − d + O (cid:18) k (cid:19) . (A.43)Then by (2.23), ˜ b = 1 − κ + χ +1 κ + χ + O (cid:18) k (cid:19) , (A.44)and by (A.39), ˜ b = − ˜ t ( ˜ ℓ +1 ) d + k κ + χ ν G + [ k,χ ] + O (cid:18) k (cid:19) . (A.45)Since κ + χ > t ( ˜ ℓ +1 ) d + k >
0, (A.45) tells us that sgn(˜ b ) = − sgn (cid:0) ν G + [ k,χ ] (cid:1) . Hence by (2.18)and Table 2, sgn(˜ b ) = − sgn( a ). Since sgn( a ) = − sgn( b ), for any shrinking point (5.27), we havesgn(˜ a ) = − sgn(˜ b ), and therefore sgn(˜ a ) = sgn( a ) , (A.46)as required.Next we derive an explicit expression for p (a coefficient of a leading order term in (7.2))from which we can ascertain the sign of p . The desired result (det (cid:16) ˜ J (cid:17) >
0) then follows fromLemma 7.3 and some additional identities.By (3.3), G + [ k, χ ] = G + [ k, χ − ( − d + k ) . Therefore by (7.1) and (7.2) we can write p = lim k →∞ k ∂s G + [ k,χ − − d + k ∂η (cid:12)(cid:12)(cid:12)(cid:12) ( η G +[ k,χ ] ,ν G +[ k,χ ] ) . (A.47)Here we use Lemma 6.7 to evaluate s G + [ k,χ − − d + k . This requires separate calculations for the cases χ = 0 and χ ≥
1. If χ ≥
1, we express s G + [ k,χ − − d + k in terms of s G + [ k,χ − so that we can apply (6.36).If χ = 0, we express s G + [ k, − − d + k in terms of s G + [ k, − ( ℓ + k − ) d + k so that we can apply (6.35). For brevity herewe provide details only for the case χ ≥ x G + [ k,χ − lies within O (cid:0) σ k (cid:1) of the slow man-ifold with j = − d , see (6.42) and (6.44). The same is true for x G + [ k,χ − − d + k , because x G + [ k,χ − = f S ( − d ) (cid:16) x G + [ k,χ − − d + k (cid:17) . That is, x G + [ k,χ − i = ϕ − d + s G + [ k,χ − i ζ − d + O (cid:0) σ k (cid:1) , (A.48)for i = 0 and i = − d + k . By (6.26), we obtain s G + [ k,χ − − d + k = s G + [ k,χ − − γ − d λ + O (cid:0) σ k (cid:1) . (A.49)56ince λ = 1 + O (cid:0) k (cid:1) , and γ − d is independent of k , by (A.47) p = lim k →∞ k ∂s G + [ k,χ − ∂η (cid:12)(cid:12)(cid:12)(cid:12) ( η G +[ k,χ ] ,ν G +[ k,χ ] ) . (A.50)To evaluate (A.50) we use (4.10) to write s G + [ k,χ − = det (cid:0) P G + [ k,χ − (cid:1) ̺ T Bµ det (cid:0) I − M G + [ k,χ − (cid:1) . (A.51)Here it is sufficient to write the denominator of (A.51) asdet (cid:0) I − M G + [ k,χ − (cid:1) = ˜ a + O (∆ η, ∆ ν ) . (A.52)Since we assuming χ ≥
1, by (6.36) the numerator of (A.51) isdet (cid:0) P G + [ k,χ − (cid:1) ̺ T Bµ = γ − d k − χ X j =0 λ j + ω T − d (cid:18) f ( S ℓd ) χ − ˆ X ( ϕ − d ) − ϕ − d (cid:19) λ k − χ +1 + O (cid:0) σ k (cid:1) . (A.53)We now evaluate the components of (A.53). We have ϕ − d = y + O (cid:0) k (cid:1) , and so f ( S ℓd ) χ − ˆ X ( ϕ − d ) = f ( S ℓd ) χ − ˆ X ( y ) + O (cid:0) σ k (cid:1) = f ˆ X ( y ) + O (cid:0) σ k (cid:1) = y − d + O (cid:0) k (cid:1) . Thus ω T − d (cid:18) f ( S ℓd ) χ − ˆ X ( ϕ − d ) − ϕ − d (cid:19) = t − d + O (cid:18) k (cid:19) . (A.54)By (6.14), and the formula for the sum of a truncated geometric series, when λ = 1, γ − d k − χ X j =0 λ j = e T x S− d (cid:0) − λ k − χ +1 (cid:1) . (A.55)Since λ = 1 + O (cid:0) k (cid:1) , and χ is independent of k , we can write λ k − χ +1 = λ k + O (cid:0) k (cid:1) . By (7.7) and(7.12), λ k = − t d ν T t ( ℓ − d η T (cid:18) − (cid:18) act d k + O (1) (cid:19) ∆ η − (cid:18) act ( ℓ − d k + O (1) (cid:19) ∆ ν + O (cid:0) (∆ η, ∆ ν ) (cid:1)(cid:19) , (A.56)Also by (5.23), e T x S− d = t − d t ( ℓ − d ν + O (cid:0) ( η, ν ) (cid:1) t d η + t ( ℓ − d ν + O (cid:0) ( η, ν ) (cid:1) = t − d ν T t ( ℓ − d η T (cid:16) t d + ν T t ( ℓ − d η T (cid:17) + O (cid:18) k (cid:19) + − t − d ν T t d t ( ℓ − d η T (cid:16) t d + ν T t ( ℓ − d η T (cid:17) + O (1) ∆ η + t − d t d t ( ℓ − d η T (cid:16) t d + ν T t ( ℓ − d η T (cid:17) + O (1) ∆ ν + O (cid:0) (∆ η, ∆ ν ) (cid:1) . (A.57)57inally by (2.18) and (2.25), since κ + χ > ν T η T = t ( ℓ − d tan (cid:0) θ + χ (cid:1) t d + O (cid:18) k (cid:19) . (A.58)By substituting (A.54)-(A.58) into (A.53) we obtaindet (cid:0) P G + [ k,χ − (cid:1) ̺ T Bµ = at − d kct d − t − d η T ( θ + χ ) + O (1) ∆ η + at − d kct ( ℓ − d + t d t − d t ( ℓ − d η T tan ( θ + χ )1 + ( θ + χ ) + O (1) ∆ ν + O (cid:0) (∆ η, ∆ ν ) (cid:1) . (A.59)By substituting (A.52) and (A.59) into (A.51), and then evaluating (A.50), we arrive at p = at − d (cid:18) sgn( a )Γ ( θ + χ ) cos ( θ + χ ) (cid:19) ˜ act d (cid:18) ( θ + χ ) (cid:19) , (A.60)where we have used lim k →∞ kη T = (cid:12)(cid:12) ct d a (cid:12)(cid:12) Γ (cid:0) θ + χ (cid:1) cos (cid:0) θ + χ (cid:1) , which is due to (2.18) and (2.21).We now use (A.60) to show that p >
0. From the definition of Γ (2.19)-(2.20), and a bit ofcare with the different cases of the sign of a , see Table 2, we obtain1 + sgn( a )Γ ( θ + χ ) cos ( θ + χ )1 + ( θ + χ ) = tan (cid:0) θ + χ (cid:1) (cid:0) θ + χ (cid:1) + tan (cid:0) θ + χ (cid:1) ln (cid:0) − tan (cid:0) θ + χ (cid:1)(cid:1) , (A.61)which has a negative value. Since c > t d < t − d > a ) = sgn( a )(A.46), from (A.60) and (A.61) we conclude that p > t d k < t ( ˜ ℓ − ) d k <
0, by (7.16) and (7.17),sgn( r ) = sgn( r ) = − sgn(˜ a ) = − sgn( a ) . (A.62)Since p >
0, by (7.18) and (A.62), q <
0. From (7.11) we obtain q = t d t ( ℓ − d − tan ( θ + χ ) +1tan ( θ + χ ) ln ( − tan ( θ + χ ))1 + tan ( θ + χ ) +1ln ( − tan ( θ + χ )) q . (A.63)Since t d < t ( ℓ − d < q <
0, by (A.63) we must have q >
0. Therefore q t d − q t ( ℓ − d < (cid:16) ˜ J (cid:17) >
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