A stiction oscillator under slowly varying forcing: Uncovering small scale phenomena using blowup
AA stiction oscillator under slowly varying forcing: Uncovering small scalephenomena using blowup
Kristian Uldall Kristiansen
Abstract.
In this paper, we consider a mass-spring-friction oscillator with the friction modelled by a regularizedstiction model in the limit where the ratio of the natural spring frequency and the forcing frequencyis on the same order of magnitude as the scale associated with the regularized stiction model. Themotivation for studying this situation comes from [3] which demonstrated new friction phenomenain this regime. The results of [3] led to some open problems, see also [4], that we resolve in thispaper. In particular, using GSPT and blowup [24, 34] we provide a simple geometric descriptionof the bifurcation of stick-slip limit cycles through a combination of a canard and a global returnmechanism. We also show that this combination leads to a canard-based horseshoe and are thereforeable to prove existence of chaos in this fundamental oscillator system.
Key words. stiction, friction oscillator, GSPT, blowup, stick-slip, canards
AMS subject classifications.
DOI.
1. Introduction.
The importance of slow-fast systems:(1) ˙ x = f ( x, y, ε ) , ˙ y = εg ( x, y, ε ) , is well recognized in many areas of applied mathematics, perhaps most notably in mathemati-cal neuroscience [20, 45]. During the last decades these systems have been studied intensivelyusing the framework of Geometric Singular Perturbation Theory (GSPT) [12, 13, 24, 36]. Thepoint of departure from this theory, based upon Fenichel’s theory of singular perturbations,is the critical set S = { ( x, y ) : f ( x, y,
0) = 0 } of (1) ε =0 . If S is a compact submani-fold and the normal hyperbolicity conditions hold, i.e. D x f | S only have eigenvalues withnonzero real part, then Fenichel’s theory says that S , as well as its stable and unstable man-ifolds W s/u ( S ), perturb to S ε and W s/u ( S ε ), respectively, for all 0 < ε (cid:28)
1. To deal withpoints on S , where the normal hyperbolicity condition is lost, the blowup method [11] has– following the work of [34] – been used to extend GSPT. This extended theory has beenapplied in many different scientific contexts to describe complicated dynamics, including re-laxation oscillations [25, 26, 31, 37], as well as canard phenomena due to repelling criticalmanifolds [5, 35, 43, 44].More recently, systems that limit onto nonsmooth ones as ε →
0, e.g.(2) ˙ x = f ( x, y, φ ( yε − )) , ˙ y = g ( x, y, φ ( yε − )) , for x ∈ R n , y ∈ R , with φ a sigmoidal function lim s →±∞ φ ( s ) → ±
1, have been studied byadapting these methods, see e.g. [29, 30, 32, 40]. Systems of the form (2) also occur in many a r X i v : . [ m a t h . D S ] F e b ifferent scientific contexts, for example – as in electrical engineering and in biological systems– due to φ modelling a switch [9, 30]. More indirectly, systems (2) also occur through regular-izations of piecewise smooth systems [28], which are common in mechanics. For example, thesimplest friction laws are piecewise smooth and systems of the form (2) are therefore commonin this area too [3].As opposed to (1), the set Σ = { ( x, y ) : y = 0 } is a discontinuity set (also called a switchingmanifold) of (2) ε =0 . By working in the extended space ( x, y, ε ), the reference [29, 32], amongothers, gain smoothness through a cylindrical blowup of points ( x, , < ε (cid:28) In this paper, we consider the following model for the spring-mass-frictionoscillator illustrated in Figure 1:(3) ˙ x = y, ˙ y = − x − sin θ − µ d φ ( ε − y ) , ˙ θ = ω := εξ Notice, that the spring (or natural) frequency of the system (3) has been normalized to 1whereas the forcing frequency is ω = εξ ; we assume that ξ ∈ R (fixed) and 0 < ε (cid:28) f ( θ ) = sin θ is therefore “slowly varying”. At the same time, following the resultof [3], the friction force is modelled as a regularization of the nonsmooth stiction law, throughthe term µ d φ ( ε − y ), involving the same scale ε , by the following assumptions on the smoothfunction φ :(A1) φ ( s ) → ± s → ±∞ .(A2) φ is an odd function: φ ( − s ) = − φ ( s ).(A3) There is one δ > µ s > µ d , such that φ (cid:48) ( s ) > ⇔ s ∈ ( − δ, δ ), φ (cid:48) ( δ ) = 0 and φ ( δ ) = µ s /µ d > φ (cid:48)(cid:48) ( δ ) < µ d and µ s are proportional by a nondimensional factor n to the dynamicand static friction coefficients f d and f s , respectively, and µ s = nf s > µ d = nf d thereforereflects the well-known fact that f s > f d , which can be interpreted as follows: the forcerequired to keep the mass in motion is smaller than the force required to initiate the motion.The proportionality factor n is a ratio of the normal force to the amplitude of the externalforcing, see [3] for further details on the derivation. According to the numbers in [1], µ s /µ d = f s /f d ∼ − φ in Figure 2. We will also need the following technical assumption on φ :(A4) There exists a k ∈ N and an s > φ ( − s − ) = − − s k φ − ( s ) for all s ∈ [0 , s ]where φ − : [0 , s ] → R + is smooth with φ − (0) > S : ( x, y, θ ) (cid:55)→ ( − x, − y, θ + π ) . In fact, most of our results generalize to any forcing f ( θ ) with f ( θ + π ) = − f ( θ ) so that thesystem remains symmetric with respect to S . We primarily focus on f ( θ ) = sin θ for simplicity. A spring-mass-friction oscillator. We consider a nondimensionalized version where both the mass m and spring constant k are scaled to . We also use a time t nondimensionalized by the natural frequencyof the spring. The parameter ω is then the ratio of the frequency of the forcing f and the natural frequency.In this paper we shall consider ω = O ( ε ) with ε being a scale associated with the regularized stiction law forfriction µ d φ ( ε − y ) . Figure 2.
Regularization function satisfying (A1), (A2) and (A3) with φ (cid:48) ( ± δ ) = 0 and where φ ( y ) → ± as y → ±∞ . System (3) is a combination of (1) and (2), in the sense that θ is slowly varying by theassumption ω = O ( ε ). At the same time, (3) limits as ε → x (cid:48) = y,y (cid:48) = − x − sin θ ∓ µ d ,θ (cid:48) = 0 . (4)for y ≷
0, respectively, having Σ : y = 0 as a switching manifold. Within Σ, we find that x = − sin θ − µ d is a curve of tangencies (fold line) for the y > x = − sin θ + µ d is curve of tangencies (fold line) for the y < invisible [21]. Trajectories outside Σ are therefore simple arcs of a half circle.These sets are also red and blue, respectively, in Figure 3. The PWS system. The switching manifold Σ has three shades of grey. The darkest band isbounded by the fold lines in blue and red. On either side of this band, there is a different smaller band whichis related to the stiction law. In particular, according to this law trajectories “stick” to Σ until they leave thesebands. In the context of (3), special interest lies in the existence of different limit cycles: An ε -family of limit cycles Γ ε with lim ε → γ ε well-defined is said to be... • pure-slip if lim ε → Γ ε intersects Σ in a discrete set of points. • pure-stick if lim ε → Γ ε ⊂ Σ. • stick-slip otherwise.Since Γ ε has to intersect y = 0 for it to be a periodic orbit, a stick-slip limit cycle enters y ≷ ± c for c > < ε ≤ ε ( c ) small enough but lim ε → Γ ε ∩ Σ is not discrete.There will be no pure-slip (3) for all 0 < ε (cid:28)
The system (3) was studied in [3, 4] for ω = O (1) (using a slightlydifferent scaling) with the main focus on the connection between the smooth system and thePWS system limit with the following rule of stiction on Σ: If y (0) = 0, x (0) + sin θ (0) ∈ ( − µ s , µ s ) then y ( t ) = 0 until x ( t ) + sin θ ( t ) / ∈ ( − µ s , µ s ). The authors defined a physicalmeaningful notion of solutions at the PWS level ( stiction solutions ) and showed that thesesolutions could be forward nonunique ( singular stiction solutions ). They also demonstratedthat there are special (canard) solutions of the regularized system (3) for ω = O (1) thatproduce solutions that are not stiction solutions [3, Definition 4.3] , but yet they appearrobustly for any regularization function satisfying (A1)-(A4). These canard solutions providea resolution of the nonuniqueness at the PWS level. In fact, the dynamics is for ω = O (1)qualitatively independent of φ . On the other hand, [3] also performed numerical computationsthat showed that there are families of stick-slip periodic orbits which are connected for thesmooth system but disconnected for the PWS one. The connectedness of these cycles occurfor ω small enough due to a fold bifurcation, see Figure 12(b) (and the caption for furtherdetails) which reproduces the results in [3] in this parameter regime. [3] shows that there canbe no stick-slip orbits for ξ < δ for all 0 < ε (cid:28) ξ = δ is only a lower bound for the foldshown in Figure 12(b) and it does not explain the main mechanism for the bifurcation. Inthis paper, we will therefore focus on ξ > δ, (5) nd describe the limit ω = εξ as ε → y = 0 are independent of ω . Our analysis reveal asimple structure that allows for an almost complete description of the long term dynamics inthis limit including a simple geometric explanation for the fold bifurcation of limit cycles. Wewill see that in this limit, which can only be resolved at the regularized level, the dynamicsdepend upon φ at a qualitative level. Interestingly, the bifurcation of limit cycles that wedescribe is also shown to be associated with the existence of a horseshoe, in a new general(canard-based) mechanism which we lay out below, see Theorem 4.1. We will use ξ as ourprimary bifurcation diagram and restrict attention to parameters µ s , µ d and δ (and furtherdetails on φ ) that are consistent with the behaviour observed in [3, 4] (we formalize this inthe assumption (A5) below). Our results resolve all problems on the stiction model that wereleft open in [3, 4].The stiction oscillator (3) have also been studied at the PWS level in many other refer-ences, see e.g. [7, 19, 39]. For example, the references [19, 39] perform accurate numericalcomputations and demonstrate routes to chaotic dynamics through period doubling cascadeswithin ω ∈ (0 , µ s = µ d is the most classical one. Here a lot more is knownanalytically about existence of periodic orbits and bifurcations, see e.g. [15, 27, 38]. This casealso lends itself to the theory of Filippov system [9, 14]. In [10], for example, the authorsconnect the onset of chaotic dynamics to a PWS grazing sliding bifurcation. Our results com-plement these existing results, insofar that we describe bifurcation of periodic orbits and newroutes to chaotic dynamics for a regularized friction model in a limit that is inaccessible at thenonsmooth level. The results also provide a description of the onset of oscillatory dynamicsas we go from f = const (which corresponds to ω = 0) to f periodic. The article is organized as follows: In section 2 we describe our method,based upon blowup and GSPT, to study (3) for all 0 < ε (cid:28)
1. Then in section 3 we put this touse and describe simple geometric conditions that ensure existence of periodic orbits of bothstick-slip type and pure-stick. In section 4 these geometric conditions lead us to formulate anew canard-based mechanism for horseshoe chaos. This new mechanism is reminiscent of thechaos in the forced van der Pol [16] insofar that it relates to a folded saddle singularity, butthe global geometry and the details are different.
2. The blowup approach.
To describe the full dynamics of (3) in the limit ε →
0, wefollow [29, 32]: We augment ε (cid:48) = 0 and consider the fast time scale:(6) ˙ x = εy, ˙ y = ε (cid:0) − x − sin θ − µ d φ ( ε − y ) (cid:1) , ˙ θ = εω = ε ξ, ˙ ε = 0 . obtained by multiplying the right hand side of (3) by ε . For this system, every point ( x, y, θ, y = 0 is extra singular due to lack of smoothness. We therefore perform The result of blowing up the degenerate set Σ × { } of (6) . The figure is illustrated in a section θ = const. On the cylinder we find a critical manifold C = C + r ∪ F + ∪ C a ∪ F − ∪ C − r in green with two foldlines F ± separating the attracting sheet C a from two repelling ones C ± r . From the repelling sheets, the blowupapproach reveals a return mechanism from C ± r to C a . a blowup transformation: ( r, (¯ y, ¯ ε )) (cid:55)→ (cid:40) y = r ¯ y,ε = r ¯ ε, (7)for (¯ y, ¯ ε ) ∈ S and r ≥
0. In this way, we gain smoothness of the resulting vector-field X on ( x, θ, r, (¯ y, ¯ ε )) ∈ R × T × [0 , ∞ ) × S , obtained by pull-back of (6) through (7). Moreover,¯ ε ≥ X and consequently (cid:98) X := ¯ ε − X will have improved hyperbolicityproperties. It is (cid:98) X that we study in the following. We illustrate the result in Figure 4.To study (cid:98) X and perform calculations, we use directional charts. In particular, the scalingchart, obtained by setting ¯ (cid:15) = 1 y = r y ,ε = r , with chart-specific coordinates ( r , y ), produces the following slow-fast equations:(8) ˙ x = ε y , ˙ y = − x − sin θ − µ d φ ( y ) , ˙ θ = ε ξ, upon eliminating r = ε . Notice that (8) is a slow-fast system (1) with ε ≥ t denotesthe (fast) time in (8) we introduce the slow time τ = ε t so that(9) x (cid:48) = y ,ε y (cid:48) = − x − sin θ − µ d φ ( y ) ,θ (cid:48) = ξ, ith respect to this new time. The system (9) is called the slow system. These systems areobviously topologically equivalent for all ε >
0, but the ε = 0 limits are not. In particular,setting ε = 0 in (8) gives the layer problem(10) ˙ x = 0 , ˙ y = − x − sin θ − µ d φ ( y ) , ˙ θ = 0 , while ε = 0 in (9) gives the reduced problem(11) x (cid:48) = y , − x − sin θ − µ d φ ( y ) ,θ (cid:48) = ξ. In the following, we focus on (10) and delay the analysis of (11) to subsection 2.1.For (10) the set C defined by x = − sin θ − µ d φ ( y ) , θ ∈ T , y ∈ R , (12)is a critical manifold. By linearizing (10) about any point ( x, y , θ ) ∈ C we obtain one singlenontrivial eigenvalue: λ ( y ) := − µ d φ (cid:48) ( y ) . (13)Consequently, the sets F ± := C ∩ { y = ± δ } are nonhyperbolic fold lines where λ ( y ) = 0.These sets divide C into an attracting sheet: C a := C ∩ { y ∈ ( − δ, δ ) } , where λ ( y ) <
0, and two repelling sheets: C ± r := C ∩ { y ≷ ± δ } , where λ ( y ) > y = ±
1. ¯ y = 1 being identical (in fact we can just apply the symmetry S ) we focus on ¯ y = −
1, where we introduce the chart-specific coordinates ( r , ε ) defined by y = − r ,ε = r ε . Notice that we can change coordinates by y = − ε − , r = r ε . Dynamics in the chart ¯ y = − . Inserting this into (3) gives˙ x = − r , ˙ r = r (cid:16) x + sin θ + µ d ( − − ε k φ − ( ε )) (cid:17) , ˙ θ = r ε ξ, ˙ ε = − ε (cid:16) x + sin θ + µ d ( − − ε k φ − ( ε )) (cid:17) , where by assumption (A4), we have used that φ ( ε − ) = − − ε k φ − ( ε ). We summarise theproperties of this system in Figure 5. Notice, that for this system, r = ε = 0 is a set ofequilibria. In particular, the linearization about any point ( x , , θ ,
0) has only two non-zeroeigenvalues whenever x + sin θ − µ d (cid:54) = 0. This gives a saddle structure, as shown in thefigure, with heteroclinic connections between points ( x , , θ ,
0) with x + sin θ − µ d > x , , θ ,
0) with x = 2 µ d − θ − x . This follows from setting ε = 0 and dividing out the common factor r : x (cid:48) = − r ,r (cid:48) = x + sin θ − µ d , which is just (4) within θ = θ and upon setting r = − y . In particular, the forward orbitof x (0) = x , r (0) = 0 returns to r ( T ) = 0 after a time T = π with x := x ( T ) = 2 µ d − θ − x . emark x = − sin θ + µ d is a fixed point of theassociated mapping x (cid:55)→ x . This point within r = ε = 0 can be blown up to a sphere(using (A4)) in such a way that hyperbolicity is gained. We skip the details of such a blowupanalysis, since it is not important for our purposes, and just illustrate the result in Figure 6.Following this analysis, we use the hyperbolic structure to track orbits for 0 < ε (cid:28) C eventuallyreaches a neighborhood of C a for all 0 < ε (cid:28)
1. (2) Moreover, by patching together theanalysis in ¯ y = ± ε = 1 we obtain two maps from each repelling sheets C ± r to theattracting one C a . We will refer to these maps as return maps since orbits will follow theseupon leaving C a . In particular, for C − r we obtain one such a return mechanism by followingthe singular flow y -negative side of C − r ; we describe this by a map G − from C − r to C a givenby G − : ( x , y , θ ) (cid:55)→ ( x , y , θ ) , where x = 2 µ d − θ − x as above and where y is the least positive solution of x = − sin θ − µ d φ ( y ) . We can easily rewrite this equation as φ ( y ) = − − φ ( y ) , (14)from which it clearly follows that G − is well-defined. We have. Lemma 2.2. G − is well-defined for all ( x , y , θ ) ∈ C − r corresponding to points on therepelling sheet, i.e. x ∈ ( − sin θ + µ d , − sin θ + µ s ) , y < − δ . Furthermore, G − : C − r → C a is a diffeomorphism onto its image and the image value of y satisfies ( − δ, δ ) .Proof. Simple calculation, see also Figure 5. In particular, by (14) and since φ ( y ) ∈ ( − , − µ s /µ d ), we have that φ ( y ) ∈ ( − , − µ s /µ d ) ∈ ( − , µ s /µ d ) , using that µ s /µ d > y -negative side of the unstable manifold of C − r is foliated by stable fiberswith base points on C a , according to the assignment G − and vice versa. The map G + from C + r to C a is given G + := S ◦ G − ◦ S using the symmetry S . G − extends to the fold line F − . Here the image is contained within { y = y − } with y − defined by φ ( y − ) = − µ s /µ d , (15)using (14) and (A3).Clearly, as illustrated in Figure 5 there is also a simpler return mechanism from C − r , whichcan be described in the chart ¯ ε = 1 only, where the flow follows the y -positive side of theunstable manifold. This mapping L − from C − r to C a is given by( x , y , θ ) (cid:55)→ ( x , y , θ ) , The result of blowing up a degenerate point x = − sin θ + µ d . with ( x , y , θ ) ∈ C − r and where y is the unique solution of φ ( y ) = φ ( y ) , y > − δ. Again, the mapping L + from C + r to C a is given by the symmetry as L + = S ◦ L − ◦ S . Next, we describe the reduced problem on C working in thescaling chart ¯ ε = 1, recall (11). Seeing that C is naturally parameterized by y and θ , wedifferentiate (12) with respect to the fast time and rewrite (11) as(16) µ d φ (cid:48) ( y ) y (cid:48) = − y − ξ cos θ,θ (cid:48) = ξ, on (12). The fold lines F ± on C where φ (cid:48) ( y ) = 0 are singular for (16). We study theseequations in the classical way [43] using desingularization through multiplication of the righthand side by ξ − µ d φ (cid:48) ( y ). This gives(17) y (cid:48) = − ξ − y − cos θ,θ (cid:48) = µ d φ (cid:48) ( y ) . These systems are topologically equivalent on C a and topologically equivalent upon time-reversal on C ± r . The equilibria of (17) are folded singularities [43] and given by ( y , θ ) =( − δ, θ + ( ξ − )), ( y , θ ) = ( δ, θ − ( ξ − )) and z − : ( y , θ ) = ( − δ, θ − ( ξ − )) , z + : ( y , θ ) = ( δ, θ − ( ξ − )) , where θ ± ( ξ − ) := cos − ( ∓ δξ − ) . At ξ = δ there is a fold bifurcation of folded singularities, but as mentioned in the introduction,see (5), we restrict attention to ξ > δ where there are four folded singularities. Two of thesefolded singularities z ± ∈ F ± , related by the symmetry S , are of saddle-type whereas the tworemaining ones are either folded foci or folded nodes, depending on ξ . In particular, a simplecalculation shows that there is a ξ dn > δ such that these latter folded singularities are foldedfoci for all ξ > ξ dn ; this is the case shown in Figure 7.The fold lines F ± consist of the folded singularities and regular fold points. Let J − be theset of regular jump points on the fold line F − where ξ − δ − cos θ <
0, so that y is decreasing a) (b) Figure 7.
In (a): the reduced problem. For ξ > δ there are four folded singularities, two of which z ± are folded saddles whereas the two remaining ones are either folded foci or folded nodes, both stable for thedesingularized system (17) . In (b): the reduced problem upon blowing down to the ( x, y, θ ) -space within ε = 0 together with the returns obtained by the PWS flows. The flow within y = 0 is slow in the sense that it isobtained on a separate (slow) time scale. Orbits within y ≷ are therefore fast in comparison and are thereforegiven tripple headed arrows whereas in contrast orbits on y = 0 have a single arrow. Consequently, the limitcycles we obtain for < ε (cid:28) are generalized relaxation oscillations. Notice also that in this blow down versionin (b) we cannot distinguish C a from C ± r on the out-most bands, recall the discussion of these in Figure . for the reduced flow. We define J + in a similar way. In fact, it is given by J + = S J − . Forthese jump points, G ± give a return mechanism to C a , which we illustrate in blue in Figure 7.Consider the folded saddle z − on F − . This point, being a hyperbolic saddle for (17),produces a (singular) vrai canard γ − as a stable manifold W s ( z − ) for (17). This canardconnects C a with C − r in finite forward time for (11). Let ˜ γ − denote the subset of γ − on C − r . For these sets of points, G − and L − produce two different return mechanisms to C a byfollowing the associated unstable fibers on either side of C − r . These set of points are illustratedin Figure 7 along with their symmetric images for the set of points on ˜ γ + with γ + = S γ − being the vrai canard of the folded saddle z + on F + [43]. µ d and µ s . The global dynamics of the system depend on therelative position of γ ± , G ± ( J ± ), G ± (˜ γ ± ) and L ± (˜ γ ± ) on C a . We can describe γ ± in the limit ξ → ∞ of (11) since the system is Hamiltonian there with H ( θ, y ) = µ d φ ( y ) + sin θ . Inparticular, a simple calculation shows that for µ s >
1, the stable and unstable manifolds γ ∓ of z ∓ with θ ∓ (0) = π/ , π/
2, respectively, coincide on C a in this limit given that θ ∈ T . (Thereis a heteroclinic bifurcation at µ s = 1 where γ ± connects z ± .) Moreoever, upon using thatthe minimum and maximum values of y along γ ± occur at θ = π/ θ = 3 π/ ξ → ∞ ,it is also straightforward to show that these manifolds remain within y ≷
0, respectively, henever µ s >
2. Next, using H we can also show that the stable manifold γ + of z − inthis limit intersects G − ( J − ) if and only if µ d <
1, doing so transversally in the affirmativecase. (To prove this we simply compare the minimum y -value of γ + at θ = π/
2, given by µ d φ ( y ) = µ s − H , with the value of y − , recall (15).)Consider the ( y , θ )-plane and let Υ denote the section at { y = y − } with θ ∈ ( − cos − ( − ξ − y − ) , cos − ( − ξ − y − )) , such that ˙ y < G − ( J − ) ⊂ Υ and these sets coincide when ξ → ∞ .Now, suppose µ s > µ d <
1. Then upon combining the previous results, we have that γ + ⊂ { y > } and that γ + transversally intersects Υ in the limit ξ → ∞ . We now continuethis intersection point ( y − , θ Υ ( ξ − )) of γ + and Υ for ξ − > θ Υ ( ξ − ) issmooth by the implicit function theorem. In fact, a simple Melnikov calculation – using thefact that the integrand ( ˙ y , ˙ θ ) ∧ ∂ ξ − ( ˙ y , ˙ θ ) | ξ − =0 = y ˙ θ > , of the Melnikov integral has one sign by our assumption on µ s > θ (cid:48) Υ (0) > θ (cid:48) Υ ( ξ − ) > ξ > γ + with Υ exists. Consequently, there is a unique ξ pd > θ Υ ( ξ − pd ) = θ − ( ξ − pd ),i.e. where γ + intersects Υ in the point G − ( z − ), see Figure 8(b) for an illustration of thissituation. (Notice in particular that the θ -coordinate of z − , θ − ( ξ − ), moves in the oppositedirection since θ (cid:48) Υ ( ξ − ) < y remains negative on ˜ γ − on the repelling side of C , theMelnikov-calculation also provides a monotonicity condition on the set G (˜ γ − ), which in turnfor µ d < µ s > ξ t > ξ pd , such that γ + for ξ = ξ t intersects Υ transversally while being tangent to the set G − (˜ γ − ) at a point G − (˜ γ − ) ∩ γ + .In the following, we consider all φ -functions, specifically all δ > µ d < µ s > θ Υ ( ξ − ), defined as the θ -coordinate of the intersection of γ + with Υ, which iswell-defined for all ξ − small enough, satisfies these properties:(A5) There exist (a) a ξ > θ Υ ( s ) is well-defined for all s ∈ [0 , ξ − ) and satisfies θ (cid:48) Υ ( s ) >
0, (b) a ξ pd < ξ such that θ Υ ( ξ − pd ) = θ − ( ξ − pd ) and finally (c) a ξ t < ξ pd suchthat γ + for ξ = ξ t is tangent to G − (˜ γ − ) a single point G − (˜ γ − ) ∩ γ + , see Figure 8(d)for an illustration of this situation.From the preceding discussion we have (we leave out further details for simplicity): Lemma 2.3. µ s > and µ d < are sufficient conditions for (A5) to hold for any δ > (and any further details of φ ). But the conditions in this lemma are (clearly) not necessary; it is even clear by a continuityargument that we for any µ d < µ s > − c for c > φ ( s ) = x √ x + 1 (cid:18) α βx (cid:19) , (18)for which k = 2, recall the assumption (A4), and where α and β are given by the complicatedexpressions α = δ (cid:0) δ + 1 (cid:1) − δ √ δ + 1 µ , β = 2 δµ (cid:0) δ + 1 (cid:1) / − δ (cid:0) δ + 1 (cid:1) + 2 δ √ δ + 1 µ , nsuring that φ (cid:48) ( ± δ ) = 0 and φ ( ± δ ) = ± µ with µ := µ s /µ d . This holds for any δ > µ >
1. More specifically, in Figure 8(a), we fix δ = 0 . µ s = 1 . µ d between 0 . .
925 we visualize the graph of the corresponding θ Υ (which wecompute numerically in Matlab using ODE45 and a simple shooting method). We see that θ Υ is monotone and that there are two ξ -values, ξ pd (circles) and ξ t (squares) with the propertiesin (A5) for each µ d ∈ [0 . , . δ = 0 . , µ s = 1 . , µ d = 0 . , (19)which are the values used in [3], and for ξ = ξ pd ≈ . ξ = 0 . ξ = ξ t ≈ . γ ± are in green, the faux canards arein black, the sets G ± ( J ± ) in blue and finally G ± (˜ γ ± ) and L ± (˜ γ ± ) are all in green and dashed.Notice that γ + in green passes through G − ( z − ) for ξ = ξ pd whereas it is tangent to G − (˜ γ − ) for ξ = ξ t , see Figure 8(b) and Figure 8(d). For ξ = 0 . γ + and G − (˜ γ − ),see Figure 8(c). Additional numerical experiments (not shown) lead us to speculate that (A5)holds for any µ d < µ s >
1, but we have not find a way of proving this. (In particular, theonly reason to restrict to µ d ∈ [0 . , . By Fenichel’s theory [12, 13, 24, 36], any compact submanifold S a of C a perturbs to an attracting slow manifold S a,ε (which we may take to be symmetric [17]such that S S a,ε = S a,ε ) having a stable manifold which is foliated by perturbed fibers, eachsmoothly O ( ε )-close to the unperturbed ones of C a . A similar result holds for C + r and C − r ,producing repelling slow manifolds S ± r,ε for all 0 < ε (cid:28)
1. Moreover, the reduced flows onthese manifolds are regular perturbations of the corresponding reduced flow on the criticalmanifolds. Finally, points on S a,ε that reach a regular jump point in forward time, follows,see [43] and the previous analysis, the singular flow in such way that these points return to S a,ε through base-points obtained as a small perturbation of the images under G − . On theother hand, see [43], the singular vrai canards γ − , γ + = S γ − of the folded saddles perturb toperturbed canards γ − ,ε and γ + ,ε = S γ − ε , respectively, connecting S a,ε with S ± r,ε . These orbitshave an unstable foliation of fibers on the side of S ± r,ε , which following the perturbations of G ± and L ± produce stable foliations of base points on S a,ε . Since each point reaches a neighborhood of C , we considera section Π at θ = θ ∗ < cos − ( δξ − ) in the scaling chart, using the ( x, y , θ )-coordinates,defined in a neighborhood of S a ∩ { θ = θ ∗ } where S a ⊂ C a is a compact submanifold. We thendefine the associated stroboscopic mapping P ε : { θ = θ ∗ } → { θ = 2 π + θ ∗ } for all 0 < ε (cid:28) x, y ). By the symmetry of the system, we have P ε = R ε for R ε = S ◦ Q ε where Q ε is the “half-map” obtained from { θ = θ ∗ } to { θ = π + θ ∗ } .We now describe the singular map R . Let ( x, y , θ ∗ ) ∈ Π be a point in the aforementionedneighborhood of S a . Then project ( x, y , θ ∗ ) (cid:55)→ ( x, y b , θ ∗ ) onto S a using the smooth fiberprojections. Next, flow ( x, y b , θ ∗ ) forward using the reduced flow until either θ = π + θ ∗ (inwhich case R ( x, y ) is obtained upon applying the symmetry S to the resulting end-point) a) (b)(c) (d) Figure 8.
In (a) we show θ Υ for δ = 0 . , µ s = 1 . and different values µ d ; the red arrow indicatesthe increasing direction of µ d . The values of µ d are equidistributed within µ d ∈ [0 . , . , with µ d = 0 . corresponding the values used in [3]. The circles indicate the points with ξ = ξ pd whereas the squares indicate ξ = ξ t . These points trace out a curve in purple and cyan, respectively. (b), (c) and (d) are for µ d = 0 . and show the corresponding (computed) phase portraits for ξ = ξ pd ≈ . , ξ = 0 . and ξ = ξ t ≈ . ,respectively. or until we reach J − . In the latter case, we apply G − to obtain a new point on S a , whichwe again flow forward until θ = π + θ ∗ or until we reach either J − or J + . In the latter case,we apply either G − or G + , respectively. This process concludes after finitely many steps (itis clear that there can be no accumulation points) and the image R ( x, y ) is then obtainedupon applying the symmetry S to the resulting end-point. It is also clear that R is piecewisesmooth. The set of discontinuities of R is closed and includes the set of points that reach γ − or γ + = S γ − under the process describing R . Lemma 2.4.
Consider any open set (cid:101) Π ⊂ Π not including the points of discontinuity of R .Then R ε | (cid:101) Π → R | (cid:101) Π in C l for any l ∈ N as ε → .Proof. Follows from the analysis above, Fenichel’s theory and [43, 44].
3. Existence of stick-slip and slip periodic orbits.
In this section, we use our geometricapproach, based upon GSPT and specifically blowup, to prove existence of various families
Examples of limit cycles in the blown down ( x, y, θ ) -space (see also Figure ) in the singular limit ε = 0 . The stick-slip orbit Γ ss is in green, the canard orbit Γ c is purple and finally the pure-stick orbit Γ ps isin black, see the rigorous statements on the existence of these limit cycles in Theorem . Notice that Γ ss and Γ c are (generalized) relaxation oscillations with slow pieces (the stick phase) on y = 0 interspersed with fastjumps (the slip phase) within y ≷ . For Γ ss the transition to slip from stick occurs at points corresponding toa regular jump points upon blowup (i.e. points on J ± ), whereas for Γ c the onset of slip is delayed [4]. Noticethat Γ c can also be pure-stick if at the singular level it follows L ± rather than G ± . Γ ps in contrast is purely“slow”; for < ε (cid:28) it is contained within an attracting slow manifold S a,ε . of symmetric limit cycles of (3) – including stick-slip Γ ss , canard Γ c and pure-stick Γ ps limitcycles, see illustration in the blown down ( x, y, θ )-space in Figure 9 – as fixed-points of R ε . Theorem 3.1.
Consider (3) and suppose (A1)-(A5). Fix any compact intervals I ss ⊂ ( ξ pd , ξ ] , I ssc ⊂ ( ξ t , ξ pd ) , I c ⊂ ( ξ t , ξ ] and I ps ⊂ (0 , ξ ] , ξ i , i = t, pd, being described in(A5). Then for all < ε (cid:28) there exist four continuous families of hyperbolic symmetric limitcycles: I ss (cid:51) ξ (cid:55)→ Γ ss ( ξ, ε ) ,I ssc (cid:51) ξ (cid:55)→ Γ ssc ( ξ, ε ) ,I c (cid:51) ξ (cid:55)→ Γ c ( ξ, ε ) ,I ps (cid:51) ξ (cid:55)→ Γ ps ( ξ, ε ) . (Here subscripts indicate the following: ss =stick-slip, ssc =stick-slip with canards, c =canardlimit cycles which can be either stick-slip or pure-stick depending on whether it follows G ± or L ± , respectively, ps =pure-stick). For each ξ ∈ I ss , I ssc , I c , lim ε → Γ ss ( ξ, ε ) , lim ε → Γ ssc ( ξ, ε ) , lim ε → Γ c ( ξ, ε ) , lim ε → Γ pc ( ξ, ε ) arePWS cycles (see Figure ); in particular, lim ε → Γ c ( ξ, ε ) as well as lim ε → Γ ssc ( ξ, ε ) containcanard segments for all ξ ∈ I c , I ssc , respectively, whereas lim ε → Γ ss ( ξ, ε ) is of stick-slip typeintersecting the set of jump points J ± each once within one period. Finally, lim ε → Γ ps ( ξ, ε ) is pure-stick for all ξ ∈ I ps .
3. Γ ps ( ξ, ε ) and Γ ss ( ξ, ε ) are hyperbolically attracting for any ξ ∈ I ps , I ss , respectively. . Finally, fix any ξ ∈ (0 , ξ t ) and any large compact set B in the phase space. Then for all < ε (cid:28) there are no stick-slip orbits and Γ ps ( ξ, ε ) attracts each point in B .Proof. The existence of the family Γ c ( ξ, ε ) was proven in [4, Proposition 6.6]. Indeed,suppose first that ξ > ξ pd such that γ + by assumption (A5) intersects (the closure of) L − (˜ γ − ) ∪ G − (˜ γ − ) once. Then there is a singular cycle of the blown up system with de-sirable hyperbolicity properties, consisting of a segment of γ − , a fast jump on the repellingside at a point determined by the aforementioned intersection point, and then a symmetricsegment of γ + and symmetric jump. Using the geometric construction based upon GSPTin [4, Proposition 6.6] we then obtain the canard limit cycle Γ c ( ξ, ε ) for any 0 < ε (cid:28)
1. Wecan continue such a limit cycle continuously in ξ for any ξ > ξ t since by assumption (A5)the transverse intersection of γ + and L − (˜ γ − ) ∪ G − (˜ γ − ) persists up until ξ = ξ t . Γ ssc ( ξ, ε ) ishandled similarly from the intersection of γ + with G − (˜ γ − ) appearing after ξ = ξ pd .We therefore proceed to prove the existence of the families Γ ss ( ξ, ε ) and Γ ps ( ξ, ε ) and theproperties described in the theorem. For this we use Lemma 2.4 and proceed to analyze themapping R . It is without loss of generality to restrict R to C a and in this way R justbecomes a mapping of y . We shall adopt this convention henceforth and therefore write R as a mapping on y : y (cid:55)→ y = R ( y ). It is obvious that for a y ∈ ˜Π (i.e. the set where R is smooth), then R is decreasing and it is contracting there, i.e. R (cid:48) ( y ) ∈ ( − , R is obtained from motion along C a , jumps at either J − or J + and areflection y (cid:55)→ − y due to S . The maps G ± act like “translations” and does not affect thestability. In particular, we may locally identify points on J ± with those on G ± ( J ± ) through themappings G ± . Then the reduced dynamics becomes continuous at J ± and piecewise smooth.The contracting properties of R then follows from that the divergence of (17) is − ss ( ξ, ε ) for ξ > ξ pd . For this we define thesection Π by setting θ ∗ = θ Υ ( ξ − ) . (20)Let y ,c denote the y -value of the first intersection of γ − ∩ Π obtained by flowing the localstable manifold backwards on C a . It is a point of discontinuity of R , where by assumptionlim y → y − ,c R ( y ) < y ,c , lim y → y +2 ,c R ( y ) > y ,c . By (20), we have lim y →− δ + R ( y ) = y ,c , see also Figure 10. Let K = ( − δ, y ,c ). Then since R contracts it follows that at the singular level, the forward flow of any ( y , θ ∗ ) with y ∈ K first follows the slow flow on C a , jumps at J − to a point on G − ( J − ), and then finally followsthe slow on C a again up until θ = θ ∗ + π . Therefore R ( K ) ⊂ ( − δ, y ,c ) and − δ < lim y → y − ,c R ( y ) < y ,c , There is therefore a unique fixed point of R inside K by the contraction mapping theorem. Wemay also see this more directly by the intermediate value theorem: The continuous function V on K defined by V ( y ) := R ( y ) − y satisfieslim y →− δ + V ( y ) > > lim y → y − ,c V ( y ) . Construction of limit cycles through the mapping R . Using Lemma 2.4 we perturb the fixed-point of R into an attracting fixed-point of R ε for all0 < ε (cid:28)
1. It corresponds to a stick-slip periodic orbit Γ ss ( ξ, ε ). Seeing that the limit cycleis obtained from an implicit function theorem argument, Γ ss depends smoothly on ξ ∈ I ss .The family Γ ps ( ξ, ε ) is obtained by the same approach, except that we now fix the sectionΠ using θ ∗ = θ − ( ξ − ). Let K = ( − δ, b ) with b defined as follows: If the backward flow ofthe local stable manifold γ + intersects Π before it intersects { y = δ } then b is defined as the y -coordinate of the intersection Π ∩ γ + . Otherwise, b = δ . In any case, by construction ofthe section Π and by the symmetry S no points in K reach J ± and we have that R ( K ) ⊂ ( y ,c , b ), that R is monotonically decreasing and that it is contracting on K . We thereforeobtain a unique and attracting fixed-point by the contraction mapping theorem, which we byLemma 2.4 perturb into an attracting limit cycle Γ ps ( ξ, ε ) for all 0 < ε (cid:28) ps ( ξ, ε ) is pure-stick since it belongs to a slow manifold S a,ε . Clearly, thisargument also holds for 0 < ξ ≤ δ in which case J ± disappear and C a is forward invariantand hence the ω -limit set is Γ ps for any fixed ξ ∈ (0 , ξ t ) and all 0 < ε (cid:28) ss ( ξ, ε ), Γ ssc ( ξ, ε ) and Γ c ( ξ, ε ) are connected in a one-parameter family of limit cycles, that contains each of the cycles Γ ss , Γ ssc and Γ c withintheir domain of existence provided by Theorem 3.1, having a fold bifurcation at ξ = ξ t + o (1)and a period-doubling bifurcation for ξ = ξ pd + o (1). See Figure 11 for an illustration. But wehave not pursued a rigorous statement of this kind. However, since there can be no stick-sliplimit cycles for ξ < ξ t it follows indirectly that there has to be some sort of fold bifurcationbut it is not obvious whether the branches really are connected and whether there could beadditional folds. Certainly, the fold bifurcation cannot be a saddle-node bifurcation in anymeaningful sense of the word due to the chaotic dynamics we prove in Theorem 4.1 in thefollowing section. Nevertheless, we can relatively easy explain why we expect to find a pe-riod doubling bifurcation: When γ + intersects G − ( J − ) for ξ > ξ pd , the stick-slip limit cycleΓ ss ( ξ, ε ) is attracting cf. Theorem 3.1, but just on the other side of ξ < ξ pd , the stick-slip Illustration of the different families of limit cycles obtained by Theorem using ξ as ourbifurcation parameter and representing each limit cycle by max y . We expect the limit cycles Γ ss , Γ ssc and Γ c belong to one family of limit cycles that undergo a period doubling bifurcation at ξ = ξ pd and a fold bifurcationat ξ = ξ t in the limit ε → . limit Γ ssc ( ξ, ε ) is of saddle-type. In fact, in the latter case, we almost immediately see fromthe singular structure that O ( e − c/ε )-displacements from the corresponding fixed point of R ε along the negative y -direction tangent to S a,ε , leave a vicinity of C − r before the correspondingjump point of the limit cycle. Upon following G − these displacements then evolve “below”the corresponding limit cycle and therefore produce O (1)-displacements at θ = π + θ ∗ in thenegative y -direction. Upon applying the symmetry S , we see that these displacements thenlead to very negative eigenvalues of the linearization of R ε . Although, we have not pursuedthis in details, the transition from attracting to very repelling is expected to be monotone. Incombination this then explains the observed existence of a (locally unique) period doublingbifurcation near the the ξ -value ξ pd where γ + intersects the G − image of the folded saddle z − .We demonstrate that these considerations are in agreement with our numerical computationsin the following section. In Figure 12, we illustrate the results of numerical com-putations (using the bifurcation software system AUTO [8] as well as Matlab) of (3) for theregularization function (18) with the parameters (19) also used in [3, 4]. More specifically,Figure 12(b) shows a bifurcation diagram of limit cycles for ε = 0 .
01 using ξ as a bifurcationparameter. Figure 12(a) shows the corresponding symmetric periodic orbits in the ( θ, y )-plane in red and magenta (the corresponding points are also indicated in Figure 12(b)) for ξ = 0 . γ ± in green, the faux canard n black, the sets G ± ( J ± ) as well as G ± (˜ γ ± ) and L ± (˜ γ ± ) (green and dashed). The existenceof the three limit cycles in magenta (stick-slip, i.e. belonging to the branch Γ ss ( ξ, ε )), black(pure-stick, i.e. Γ ps ( ξ, ε )) and red (canard type, i.e. Γ c ( ξ, ε )) follow from Theorem 3.1. Thecomputations done in [4], for a slightly different regularization function and a separate scaling,show that the lower branch (i.e. Γ c ( ξ, ε )) consisting of canard orbits like the one in red inFigure 12(a) is connected to pure-slip periodic orbits. These computations also demonstrateda fold bifurcation of limit cycles, which we for our regularization function indicate as the cyansquare in Figure 12(b). In Figure 12(c) we show the corresponding nonhyperbolic periodicorbit for the parameter value ξ = 0 . ξ at the fold is close to the value of ξ t ≈ . γ + and is almost tangent to G − (˜ γ − ).Next, we emphasize that, as we follow the magenta limit cycle for smaller values of ξ towards the fold bifurcation, a period-doubling bifurcation occurs around ξ = 0 . ε . See figure caption for further explanation. Notice in particular, that for this valueof ε = 0 .
005 we find that the period doubling bifurcation occurs at ξ = 0 . ξ pd = 0 . ξ = 0 . ξ t = 0 . ε > Remark µ s <
1, see the figure caption for details, in which case there is a single stick-slip periodic orbit visiting y < − c and y > c for c > ξ = 0 . γ + leaves C a at F − without intersecting G − (˜ γ − ) ∪ L − (˜ γ − )there cannot be any canard-type limit cycles of the form described in Theorem 3.1 and nopure-stick orbits either. Figure 14(b) shows the corresponding time history x ( θ ). However,what we see as we decrease ξ slightly (not shown) from the value of ξ in Figure 14(a) is thatfirst one of each of the transitions into y < − c and y > c approaches the two canards γ ± (in green in Figure 14(a)), respectively, and subsequently, through the canard and the returnmappings G ± and L ± , there is an apparent smooth connection to stick-slip limit cycles withjust two excursions into y < − c and y > c during each period. This occurs around ξ ≈ . J ± land quite close to γ ± ,respectively. The bifurcation we describe in words is reminiscent of spike-adding bifurcationsin bursting models of neuroscience, see e.g. [6] which has a rigorous GSPT-based treatment of a) (b)(c) (d) Figure 12.
In (a) we use a projection onto ( θ, y ) to illustrate five coexisting periodic orbits for ξ = 0 . .In magenta and black are shown stick-slip and pure-stick periodic orbits, respectively, the existence of whichcan be predicted by Theorem . The red orbit is a symmetric periodic orbit having a (long) canard segment,the existence of such orbits were predicted in [4, Proposition 6.6]. Finally, in blue and cyan we illustratenonsymmetric periodic orbits that appear as a result of a period doubling bifurcation, shown in the bifurcationin (b), see zoom. This bifurcation diagram represents each stick-slip periodic orbit as a point using the samecolour for the corresponding value of ξ = 0 . . It also shows the fold bifurcation (cyan square) at ξ = 0 . (which is close to the value of ξ t ≈ . , recall Figure ); (c) shows the corresponding bifurcating periodicorbit using the same colour. We see that the bifurcation occurs to good accuracy when the canard (in green) istangent to the set G − (˜ γ − ) (dashed green line). Finally, (d) shows x ( θ ) for the different periodic orbits in (a).The remaining parameters are defined by (19) and ε = 0 . . this situation. In our friction setting a “spike” corresponds to a slip phase. As for Figure 12this canard-phenomena is accompanied by a period-doubling bifurcation. We have not pursuedany of this rigorously in our setting. Finally, we note that as we decrease ξ further then thereis one more transition from two excursions into each region y < − c and y > c during eachperiod to just one single excursion. This occurs around ξ ≈ . ξ = 0 .
61 slightly below this valueand close to the critical δ = 0 .
6, where the folded singularities disappear and beyond whichonly pure-stick orbits persist. Another interesting aspect of Figure 14(c) is that two foldedsingularities are now folded nodes and we see in Figure 14(c) the classical small oscillations a) (b) Figure 13.
Same in Figure but for ε = 0 . . Specifically, (b) shows the bifurcation diagram, the redpoint corresponding to a period doubling bifurcation of R ε at ξ = 0 . . The associated degenerate periodicorbit is shown in (a) also in red along with the black stick orbit. As indicated the period doubling bifurcationoccurs near ξ = ξ pd ≈ . which is the value of ξ for when the G − -image of z − intersects γ + (where thegreen and blue dashed lines meet). The fold bifurcation indicated by the cyan square occurs at ξ = 0 . , whichis also closer to the expected value to ξ t = 0 . . that occur here [33, 46].
4. Canard-induced chaos.
In this final section, we prove existence of chaos through asimple geometric mechanism based upon the folded saddle that produces a horseshoe. Thebasic geometry is shown in Figure 15, which under assumption (A5) occurs for ξ ∈ ( ξ t , ξ pd ),see also Figure 8(c) for the regularization function (18) and parameter values (19). In thecase illustrated in Figure 15, γ + transversally intersects G − (˜ γ − ) twice and by Theorem 3.1there are two limit cycles Γ ssc ( ξ, ε ) and Γ c ( ξ, ε ) with canard segments. (These coexists withthe pure-stick orbit Γ ps ( ξ, ε ) but this will play little role in the following. ) In principle,one intersection could be due to L − (˜ γ − ), without essential changes to our approach, but forsimplicity we will focus on G − (˜ γ − ) here (also to avoid the generic case in-betwen L − (˜ γ − )and G − (˜ γ − ). In particular, there are two singular periodic orbits γ = lim ε → Γ ssc ( ξ, ε ) and γ = lim ε → Γ c ( ξ, ε ), one for each transverse intersection, for ε = 0. Notice that each γ i on C is given by two symmetric copies, one of which we parameterize by θ as follows γ i : y = m ( θ ) , θ ∈ ( θ i − π, θ i ) , where θ i is the value of θ at the “jump point” on C r . The function m is smooth if γ i does notreach the the fold on the interval θ ∈ ( θ i − π, θ i ). Otherwise, by the mapping G + , it becomespiecewise smooth. Notice that θ > θ consisting with the definition of Γ c and Γ ssc , see alsoFigure 9.Consider a canard segment γ ε := { ( x ( τ ) , y ( τ ) , θ ( τ )) : τ ∈ [0 , T ] } of (9), connecting( x (0) , y (0) , θ (0)) ∈ S a,ε with ( x ( T ) , y ( T ) , θ ( T )) ∈ S r,ε . Then the variational equations along γ ε have a solution ( x (cid:48) , y (cid:48) , θ (cid:48) ) with y (cid:48) ∼ exp (cid:18)(cid:90) T ε − λ ( y ( τ )) dτ (cid:19) (21) a) (b)(c) Figure 14.
Same as in Figure but for µ d = 0 . , µ s = 0 . . In (a) and (b) we have ξ = 0 . whereas ξ = 0 . (just slightly above the critical value δ = 0 . ) in (c). In (a), we see that γ + intersects F − andnot G − ( J − ) and as a result no pure-stick orbits exists. We see that the stick-slip periodic orbits are morecomplicated with several excursions into y ≷ ± − c for c > sufficiently small for all < ε (cid:28) . Specifically,in (a) the red stick-slip orbit has four such fast excursions. Figure (b) shows the corresponding x ( θ ) . Finally,in (c) we see that the limit cycle in red passes through the folded nodes and consequently it is of mixed-modetype with small oscillations in this region. where λ is defined in (13). In particular, if (cid:82) T λ ( y ) dτ = ξ − (cid:82) θ T θ λ ( y ) dθ < θ = θ (0) , θ T = θ ( T ), then the contraction gained along S a,ε dominates the the expansion along S r,ε and the expression (21) is exponentially small. Following on from this we then ...(A6) suppose that (cid:90) θ θ − π λ ( m ( s )) ds < . (22)The interpretation of this condition is then that the contraction gained along the fast fibersof γ from θ = θ − π on the attracting side dominates the expansion on the repelling side upuntil θ . Seeing that θ > θ this condition ensures that the same holds for γ from θ − π to θ . We could also replace < by > < An illustration of a situation where the assumptions of Theorem are satisfied.
Theorem 4.1.
Suppose (A1)-(A6) all hold and let ξ ∈ ( ξ t , ξ pd ) be so that γ + = S γ − transver-sally intersects G − (˜ γ − ) twice as illustrated in Figure . Then there exists an ε > smallenough such that for all ε ∈ (0 , ε ) , R ε has an invariant cantor-set Λ ε where the restrictedmapping R ε | Λ ε is topologically conjugated to a shift of two symbols.Proof. As before we fix S a,ε and S r,ε and let γ − ,ε be a perturbation of γ − connecting thesemanifolds [43]. We take S a,ε to be symmetric [17] so that γ + ,ε = S γ − ,ε also belongs to S a,ε . Wenow redefine θ ∗ (cid:38) θ − π in such way, that by the Exchange Lemma [42], upon flowing a smallneighborhood of γ − ,ε ∩ { θ = θ ∗ } on S a,ε forward, we obtain a(n) (specific) unstable manifold W u ( γ − ,ε ) of γ − ,ε on the repelling side. This set reaches, by following G − , a vicinity of γ + ,ε on S a,ε by assumption. By the stable foliation of fibers of S a,ε we can therefore extend W u ( γ − ,ε )in this way so that it is now foliated by stable fibers with base points on S a,ε . By transversalityof G − (˜ γ − ) with γ + on S a in two points, as assumed, it follows that the foliation of W u ( γ − ,ε )intersects γ + ,ε transversally in two points q ε and p ε on S a,ε . We consider two sufficiently small“intervals” I ε and J ε of base points of W u ( γ − ,ε ) that act as disjoint neighborhoods of q ε and p ε on S a,ε . Upon restricting I ε and J ε further if necessary the forward flow of these points on S a,ε coincide as S a,ε ∩ { θ = π + θ ∗ } on a sufficiently small neighborhood of γ + ,ε ∩ { θ = π + θ ∗ } .Therefore, on this neighborhood W u ( γ − ,ε ) becomes two one-dimensional disjoint “strips” H ,ε and H ,ε that are exponentially close to S a,ε ∩ { θ = π + θ ∗ } . By construction, the “preimages”on θ = θ ∗ of these strips are exponentially small intervals K ε and M ε on S a,ε ∩ { θ = θ ∗ } , bothexponentially close to γ a,ε .Now, consider the stable manifolds V ,ε := W s ( K ε ) and V ,ε := W s ( M ε ) of K ε and M ε , espectively, within { θ = θ ∗ } . These sets are then each O (1) in the “verticle” direction,transverse to S a , and exponentially thin in the “horizontal” direction tangent to S a . By (22),the forward flow of these sets produce exponential thickenings of the horizontal strips H ,ε and H ,ε , being the images of K ε and M ε , respectively. We continue to denote these “thickened”objects by the same symbols H ,ε and H ,ε . It then follows that S H ,ε and S H ,ε are twohorizontal strips at θ = θ ∗ that intersect V ,ε and V ,ε in four exponentially small rectangles.This gives the desired horseshoe mechanism. We then proceed to verify the cone properties ofthe Conley-Moser conditions using [47, Theorem 25.2.1]. For this, we again use the slow-faststructure and define an “expanding cone” K u of opening angle d u > x (cid:48) , y (cid:48) ) of C a at γ − ∩{ θ = θ ∗ } . Indeed, consider then a point q ∈ V i,ε and let p = R ε ( q ) ∈ H i,ε . Then, since the intersection of S a,ε and S r,ε is transverse, and since G − (˜ γ − )is transverse to the slow flow on C a , it follows that K u is invariant for DR ε and vectors isexpanding exponentially due the motion near C r . In summary, DR ε ( q )( K u ) ⊂ K u and | DR ε ( q )( v ) | ≥ e c u /ε | v | , v ∈ K u , for some c u > < ε (cid:28)
1. Similarly, we define a “contracting cone” K s of openingangle d s > , y (cid:48) ) of the leaves of the stablefoliation of C a ∩ { θ = θ ∗ } . Then by the Exchange Lemma and the transverse intersection of S a,ε and S r,ε , we have that DR − ε ( p )( K s ) ⊂ K s and | DR − ε ( p )( v ) | ≥ e c s /ε | v | , v ∈ K s , for some c s > < ε (cid:28)
1. Having established the cone properties, the result thereforefollows from classical results on Smale’s horseshoe, see e.g. [47, Theorem 25.1.5].
5. Discussion.
In this paper, we have solved some open problems on the bifurcation ofstick-slip limit cycles for a regularized model (3) of a spring-mass-friction oscillator in the limitwhere the ratio ω of the forcing frequency and the natural spring frequency is comparable withthe scale associated with the friction. In particular, using GSPT and blowup we showed thatthe existence (and nonexistence) of limit cycles is directly related to a folded saddle and thelocation of the canard relative to the image of the fold lines under a return mechanism to theattracting critical manifold, see also Theorem 3.1.There are parameter regimes with µ s < ω → φ ) of the friction force. For ω = O (1),we know from [3], and also from more abstract results on the connection between Filippov ystems [14] and the regularization of piecewise smooth systems [28, 40], that the dynamicsis qualitatively independent of φ . Based upon this observation, we are led to the conclusionthat the details of friction is to be determined on “diverging time scales”. Although it iswell established that our friction model (depending solely on the relative velocity) is oversimplified [48], this conclusion is consistent with experimental results [18] as well as withfriction modelling in regimes of low and high relative velocity, see e.g. [41, 48]. REFERENCES [1]
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