Global heteroclinic rebel dynamics among large 2-clusters in permutation equivariant systems
Bernold Fiedler, Sindre W. Haugland, Felix P. Kemeth, Katharina Krischer
GGlobal 2-cluster dynamics under large symmetric groups
Bernold Fiedler*, Sindre W. Haugland**,Felix Kemeth***, Katharina Krischer**version of August 24, 2020 *Institut f¨ur MathematikFreie Universit¨at BerlinArnimallee 314195 Berlin, Germany**Institut f¨ur PhysikTechnische Universit¨at M¨unchenJames-Franck-Straße 185748 Garching, Germany***Department of Chemical and Biomolecular EngineeringWhiting School of EngineeringJohns Hopkins University3400 North Charles StreetBaltimore, MD 21218, USA a r X i v : . [ n li n . C D ] A ug bstract We explore equivariant dynamics under the symmetric group S N of all permu-tations of N elements. Specifically we study cubic vector fields which commutewith the standard real ( N − S N .All stationary solutions are cluster solutions of up to three clusters. The resultingglobal dynamics is of gradient type: all bounded solutions are cluster equilibriaand heteroclinic orbits between them. In the limit of large N , we present adetailed analysis of the web of heteroclinic orbits among the plethora of 2-clusterequilibria. Our focus is on the global dynamics of 3-cluster solutions with onerebel cluster of small size. These solutions describe slow relative growth anddecay of 2-cluster states.Applications include oscillators with all-to-all coupling and electrochemistry. Forillustration we consider synchronization clusters among N Stuart-Landau oscil-lators with complex linear global coupling.
Contents S N
74 Two-cluster dynamics 8 c = − c < − c > − −∞ < c < − − < c < − / − / < c < − / − / < c < − / − / < c < − − < c < − / − / < c < + ∞ . . . . . . . . . . . . . . . . . . . . . . . . . 30 i Introduction
Networks of identical oscillators with global all-to-all coupling are a ubiquitous sourceof dynamical systems which are equivariant under the symmetric group S N of all per-mutations of N elements { , . . . , N } .In section 7 below, we address the specific example of coupled Stuart-Land oscillators.See also the companion paper [KHK19]. We will eliminate a global averaged phaseoscillation. Near the trivial periodic solution of total synchrony, we consider loss ofsynchrony, and of stability, by bifurcation at a zero transverse eigenvalue. We reducethe complex ODE dynamics from C N = R N to a local center manifold of real dimension N −
1. In particular we study the resulting reduced dynamics of 2-cluster solutionsand their heteroclinic transitions, up to and including third order. Our main tool willbe equivariance under the permutation group S N .For a general background on dynamics and equivariance see for example [GoSt86,GoSt02, GuHo83, Van82]. For a background on S N -equivariance see [Elm01, GoSt02,SEC03].Permutations π ∈ S N act linearly on vectors x ∈ X := R N by permutations of theircomponents x n . This linear representation of S N is given by(1.1) ( π x ) n := x π − ( n ) . Group invariants I : R N → R satisfy(1.2) I ( π x ) = I ( x )for all π ∈ S N and all x ∈ R N , by definition. The ring of polynomial S N invariants I is freely generated by the power sums(1.3) p m := N (cid:88) n =1 x mn , for m = 1 , . . . , N . We may subsume the case of constant I as m = 0. Equivariant vector fields f : R N → R N , here under the group S N , commute with thelinear group action:(1.4) f ( π x ) = π f ( x ) , for all π ∈ S N and all x ∈ R N . For Lipschitz continuous f , the solutions x = x ( t ) ofthe associated ordinary differential equation (ODE)(1.5) ˙ x = f ( x )are unique. Therefore the equivariance condition (1.4) means, equivalently, that π x ( t )is a solution of (1.5), whenever x ( t ) itself is a solution.1ne example of group equivariant vector fields f ( x ) are the negative gradients(1.6) f n ( x ) := − ∂ n I ( x )of group invariants I . Here ∂ n denotes the partial derivative with respect to x n , for n = 1 , . . . , N. The consequences of a gradient structure (1.6) are striking, even without any groupinvariance. Stationary solutions, alias equilibria f ( x ) = 0 of the ODE (1.5), become critical points ∇ I ( x ) = − f ( x ) = 0. The energy , or Lyapunov function , I ( x ( t )) de-creases strictly with time t , along any nonstationary solution x ( t ). In particular, anysolution x ( t ) which is bounded for all real times −∞ < t < + ∞ is heteroclinic betweenstationary solutions, i.e. x ( t ) becomes stationary for t → ±∞ . The energy I at thetarget equilibrium (or equilibria), for t → + ∞ , is always strictly lower than at thesource, i.e. for t → −∞ .Note that the linear zero sum space(1.7) X := { x ∈ X | p := x + . . . + x N = 0 } is an ( N − X = R N which is invariant under theaction (1.1) of S N . The standard representation of S N on X is given by the restrictionof the linear representation (1.1) to X . That representation is irreducible : there doesnot exist any nontrivial proper subspace of X which would also be invariant under all S N .The following cubic S N -equivariant vector field, with arbitrary real parameters λ and c , is the main object of our present study(1.8) ˙ x n = f n ( x ) := ( λ + c · (cid:104) x (cid:105) ) x n + (cid:102) x n + (cid:102) x n . Here we use the abbreviations(1.9) (cid:104) x m (cid:105) := N p m ( x ) , (cid:102) x mn := x mn − (cid:104) x m (cid:105) for the averages and the offsets of m -th powers. It is a simple, but useful, exerciseto check that the zero sum space X is indeed invariant, not only under the linearaction (1.1) of the group S N but also under the nonlinear dynamics of (1.8). Indeed (cid:104) x (cid:105) = (cid:104) (cid:102) x mn (cid:105) = 0.It turns out that, up to scaling and possible time-reversal, the ODE (1.8) on X rep-resents the most general cubic vector field which is equivariant under the standardrepresentation (1.1), (1.7); see [GoSt02], 2.4–2.7. A much more detailed source, whichis difficult to obtain, is the thesis [Elm01].To derive (1.8) from these results we first recall that all S N -equivariant polynomialvector fields f of the standard representation on X , up to and including order three,are in fact gradients (1.6) of polynomial invariants I up to order four. In the notation(1.9), this provides the general form(1.10) ˙ x n = f n ( x ) = λx n + A (cid:102) x n + B (cid:102) x n + C (cid:104) x (cid:105) x n
2f (1.8). For
A, B (cid:54) = 0, indeed, linear rescalings t → τ t , x n → σx n amount to thereplacements(1.11) λ → τ λ, A → τ σA, B → τ σ B, C → c := τ σ C .
Renaming τ λ as λ , the choices σ = A/B, τ = B/A then lead to (1.8) with(1.12) c := C/B .
Note that negative B , in particular, are associated with time reversal in (1.8).The importance of the dynamics (1.8) reaches far beyond any direct interpretation asa network of N identical scalar “cells” with all-to-all coupling via power sums. Indeed,the bifurcation analysis of any fully permutation-symmetric network, at eigenvalue 0,typically leads to irreducible eigenspaces. Beyond total synchrony x = . . . = x N ,the standard representation on X provides the simplest interesting case. Any centermanifold reduction, and subsequent truncation to cubic terms will then lead to ourreference bifurcation problem (1.8) with one or the other value of the one remainingcoefficient c . See section 7 for an explicit example.In fact, the cubic case (1.8) possesses a gradient structure (1.6). Indeed, f n ( x ) := − ∂ n I ( x ) holds on X , as required in (1.6), with the quartic polynomial(1.13) − I ( x ) := ( λ · p + N c · p ) + ( p − N p p ) + ( p − N p p ) . Here we have used p = 0 on X .Our main results on the bifurcation diagrams of (1.8), with respect to the bifurcationparameter λ , are presented and discussed in section 6. We distinguish seven zones ofqualitatively different global heteroclinic dynamics. The diagrams are distinguished byseven parameter ranges for the cubic coefficient c .The remaining sections are organized as follows.In section 2 we study , i.e. solutions x ( t ) of our reference ODE (1.8)which feature at most three different values of the components x n . More generally, an M -cluster features at most M < N values(1.14) { x , . . . , x N } = { ξ , . . . , ξ M } . Note how M -clusters degenerate to M (cid:48) -clusters, for some M (cid:48) < M , when some of the ξ -components still coincide.As Kuramoto noticed long ago [NK95], all stationary solutions of (1.8) are (at most)3-clusters. The reason is simple: any stationary component ξ = ξ k = x n must satisfythe same cubic equation(1.15) 0 = f n ( x ) = ( λ + c · N p ) ξ + ( ξ − N p ) + ( ξ − N p ) , with the same coefficients c, λ, p , p . This admits at most three distinct cluster values ξ = ξ , ξ , ξ . 3e aim at the dynamics of certain 3-cluster solutions which become heteroclinic be-tween 2-cluster equilibria. In section 2 we simplify this task as follows. For k = 1 , , N k count the number of components n which satisfy x n = ξ k ; note N + N + N = N .We then pass to the limit N → ∞ of large clusters N + N with a remaining rebelcluster N of uniformly bounded size; for example we may fix N = 1. Heteroclinicorbits between 2-cluster equilibria are then characterized by(1.16) ξ ( t ) − ξ ( t ) → ξ ( t ) − ξ ( t ) → , for t → ±∞ .In section 3 our heteroclinic objective gets simplified, in the limit N = ∞ , by thesomewhat surprising appearance of a skew product structure over the scalar quantity s ( t ) := ( ξ ( t ) − ξ ( t )) / ( α + 1). Here α := N /N denotes the relative populationfraction of components in the large clusters. The gradient structure (1.13) leads toasymptotically stationary s ∗ , (1.17) s ( t ) := ( ξ ( t ) − ξ ( t )) / ( α + 1) −→ s ∗ = const , for t → ±∞ . See section 4 for a detailed analysis of this dynamics, which drives theskew product.In section 5 we pass to the asymptotic states of stationary s = s ∗ = const. In suitablecoordinates y = ξ − ξ , this reduces our task to the discussion of a single scalar ODE(1.18) ˙ y = y ( y − ( α + 1) s ∗ )( y − ¯ y ( s ∗ ))on the real line; see (3.8), (5.2).Rebel heteroclinic solutions between 2-clusters will easily be identified. Indeed, the 2-cluster stationary solutions ξ = ξ and ξ = ξ correspond to the stationary solutions y = 0 and ( α + 1) s ∗ , respectively. At the crucial 3-cluster equilibrium ¯ y ( s ∗ ), the smallrebel cluster ξ ( t ) might get stuck in its transition between the two major clusters ξ , ξ . We call this phenomenon blocking of 2-cluster heteroclinicity.We thus arrive at the alternative of 2-cluster heteroclinicity, versus blocking of hetero-clinicity by a 3-cluster. The six critical parameters(1.19) c = − , − , − , − , − , − mark transitions between qualitatively different bifurcation diagrams of our cubic ref-erence equation (1.8). In section 6 we illustrate the resulting seven intermediate casesby diagrams of the stationary 2-clusters in a plane ( N /N, s ); see figures 6.1–6.8.Each diagram is foliated by the parameters λ , as level curves, where such stationary2-clusters appear. A transversality assumption suggests the heteroclinic dynamics, inthe non-blocking regions, to indicate a slow drift of the population fraction α , alongconstant parameter levels λ , by successive transitions of small rebel population fractions N between the major clusters. Contrary to standard intuition, these rebel transitionsdo not always favor the larger cluster. The seven cases which we discuss in fact indicate4ow cluster dynamics is an exceedingly subtle phenomenon, even in our simplistic cubicsetting.In section 7 we conclude with the promised application to clustering in Stuart-Landauoscillators with global complex linear coupling.So, where are the theorems? The present paper is a rather detailed case study of S N -equivariant 3-cluster dynamics in the standard representation on X , as is. Ourmain focus is the rebel dynamics among the plethora of coexisting 2-cluster solutionsof size ratios α = N /N , for large N . One novelty is our unusual presentation of theheteroclinic rebel dynamics as a formal flow on the level set diagrams λ = λ ( α, s ), insection 5, where s = ( ξ − ξ ) / ( α + 1) measures asynchrony. All of section 6 can thenbe read as a long theorem, which establishes the pairwise inequivalence of these formalflows in the seven intervals(1.20) c (cid:54)∈ {− , − , − , − , − , − } . We conjecture, conversely, equivalence of the formal flows in each of the seven comple-mentary intervals. Alas, we did not embark on the, more cumbersome than enlighten-ing, proof of this somewhat academic question.
Acknowledgment.
The first author gratefully acknowledges the deep inspiration by,and hospitality of, his coauthors at M¨unchen who initiated this work. Ian Stewartpersonally provided us with a copy of the extensive thesis [Elm01], which saved usquite some duplication of effort. Extensive corrections of ever so many revisions weremost diligently typeset by Patricia Habasescu. This work has also been supported bythe Deutsche Forschungsgemeinschaft, SFB910, project A4 “Spatio-Temporal Patterns:Control, Delays, and Design”, and by KR1189/18 “Chimera States and Beyond”.
Let ˙ x = f ( x ), on the zero sum space x ∈ X , denote any vector field which is equiv-ariant under the standard irreducible action of the symmetric group S N on X . See(1.1)–(1.5) and (1.7). The M -clusters are defined as those vectors x ∈ X which possessat most M distinct components x n ; see (1.14). After applying a suitable permutation π ∈ S N to x if necessary, we may assume without loss of generality that the indicesare sorted as(2.1) x = . . . = x N , . . . , x N + ... + N M − +1 = . . . = x N . We call N k the size of cluster k , for k = 1 , . . . , M . In other words, x is fixed underthe direct product S N := S N × . . . × S N M of permutation subgroups, where the firstfactor S N acts on the first N components of x , and so on. Any other M -cluster isfixed under a group suitably conjugate to S N .By (1.4), the linear space of S N -fixed vectors x is invariant under the ODE flow of f . Inparticular, nondegenerate M -clusters remain nondegenerate M -clusters, for all time.Only for t → ±∞ , an M -cluster x ( t ) may possibly limit onto an M (cid:48) -cluster with fewer5lusters, i.e. M (cid:48) < M . Since any stationary solutions are at most 3-clusters, by (1.15),this is precisely the situation which we plan to study, for M = 3 and M (cid:48) = 2.Specifically, consider the dynamics of any nondegenerate 3-cluster(2.2) { x , . . . , x n } = { ξ , ξ , ξ } in our reference ODE (1.8). Then the power sums p m of (1.3) become(2.3) p m = N ξ m + N ξ m + N ξ m . The cluster sizes N k ≥
1, respectively, count the number of times the distinct values ξ k occur among the x n .With these weighted power sums p m , the resulting dynamics of the ξ k is of course givenby the ODE(2.4) ˙ ξ k = ( λ + c · N p ) ξ k + ( ξ k − N p ) + ( ξ k − N p ) , for k = 1 , ,
3. Here we have simply replaced x n by ξ k , in (1.8).Taking differences ξ j − ξ k of any two equations in (2.4) we obtain(2.5) ddt ( ξ j − ξ k ) = ( ξ j − ξ k ) (cid:0) λ + c · N p + ( ξ j + ξ k ) + ( ξ j + ξ j ξ k + ξ k ) (cid:1) . We now introduce the redundant scaled difference variables(2.6) y := N N ( ξ − ξ ) , y := N N ( ξ − ξ ) , y := N N ( ξ − ξ ) = − ( y + y ) . The flow invariant zero sum space X of (1.7) becomes planar, in the variables ξ k :(2.7) 0 = p = N ξ + N ξ + N ξ . Therefore it is not surprising that we can invert the transformation from the redundantcoordinates ( ξ , ξ , ξ ) on X to ( y , y ) ∈ R by(2.8) ξ = − (1 + N N ) y − y ; ξ = N N y − y ; ξ = N N y +( N N + N N ) y . In principle, this allows us to rewrite the 3-system (2.4), i.e. a planar system on X , interms of the two new variables y , y . Since the general expressions are a little messywe simplify this calculation for the limit N → ∞ of large symmetric groups S N , in thenext section. 6 The limit of large symmetric groups S N As announced in the introduction, we now consider the S N -equivariant 3-cluster dy-namics (2.4) of (1.8), in the limit of large N . Specifically, we assume that the clustersize N remains small compared to N + N = N − N , i.e.(3.1) N /N → , for N → ∞ . We consider a fixed finite asymptotic size ratio(3.2) N /N → α ∈ (0 , ∞ )of the two large clusters sizes N and N , in the limit N → ∞ . Note how (3.1) isequivalent to N /N →
0, and likewise to N /N → , for N → ∞ . We therefore callthe comparatively tiny cluster ( N , ξ ) the rebel cluster .Inserting these limits into the transformation (2.8) above provides the simplified ex-pressions(3.3) ξ = − y − y ; ξ = αy − y ; ξ = αy + αy . In the above limit N → ∞ , this allows us to rewrite the 3-cluster ODE (2.4) in thestill slightly unwieldy planar form ˙ y = y (cid:16) λ + ( α − y − y + ( α − α + 1) y + 3(1 − α ) y y + 3 y + αc ( y + y ) (cid:17) (3.4) ˙ y = y (cid:16) λ + 2 αy + ( α − y + 3 α y + 3 α ( α − y y + ( α − α + 1) y + αc ( y + y ) (cid:17) (3.5)Just for academic completeness – or so it seems at first – let us also write the resultingODE for the sum(3.6) s := y + y = − y = ( ξ − ξ ) / ( α + 1)which redundantly appears in (2.6):(3.7) ˙ s = s (cid:0) λ + ( α − s + qs (cid:1) ; q := α + ( c − α + 1 . This is a scalar ODE for the sum s alone. In particular, bounded solutions s ( t ) convergeto some equilibria s ≡ s ∗ of (3.7) for t → ±∞ , respectively. Substitution of y = s − y in (3.4) provides the complementing ODE(3.8) ˙ y = ( α + 1) y ( y − s ) ( y − ¯ y ( s )) + y ˙ s/s . Here we have abbreviated(3.9) ¯ y ( s ) := ( α + 1) − ((2 − α ) s − . The polynomial ˙ s/s abbreviates the quadratic parenthesis of (3.7).7n conclusion, we observe a skew product structure , in the limit N → of two largeclusters N , N , and one comparatively small cluster N . Indeed the two ODEs (3.7)and (3.8) identify the asymptotic 3-cluster dynamics (2.4) in the zero sum subspace X of (2.7) as a system where the autonomous dynamics (3.7) of s drives the scalardynamics (3.8) of y .Perhaps this structure of our restricted 3-cluster problem should not surprise us, afterall. In fact, (3.7) describes the dynamics s of the two large clusters ξ , ξ which isnot affected by the comparatively small number N of rebels ξ . Ignoring N , indeed,the zero sum condition (2.7) implies conservation of N ξ + N ξ = 0, and hence aone-dimensional autonomous dynamics for the difference variable s = ( ξ − ξ ) / ( α + 1)of (3.6). Because s = 0 indicates synchrony of the two large clusters, i.e. effectivelya one-cluster dynamics, we also call s the asynchrony variable . The rebel dynamics(3.8) describes the remaining deviation y = ( ξ − ξ ) / ( α + 1) of the rebels ξ in thesmall cluster ( N , ξ ) from the state ξ of the large cluster ( N , ξ ), once the two largeclusters ( N , ξ ) , ( N , ξ ) have reached a status quo equilibrium s = s ∗ according totheir size ratio α = N /N . In this section we discuss the autonomous two-cluster dynamics. By (3.7), we onlyhave to study the asynchrony sum s = ( ξ − ξ ) / ( α + 1) = y + y defined in (3.6), i.e.(4.1) ˙ s = s (cid:0) λ + ( α − s + qs (cid:1) . Here the asymptotic ratio 0 < α = lim N /N < ∞ of the sizes N and N of the twolarge clusters, for N → ∞ , is a fixed parameter, in addition to the cubic coefficient c and the bifurcation parameter λ . Also from (3.7), we recall the abbreviation(4.2) q = q ( α ) := α + ( c − α + 1for the quadratic coefficient q .The scalar ODE (4.1) is cubic in s with trivial equilibrium s = 0. The remainingequilibria s = s ∗ are characterized by the vanishing quadratic parenthesis in (4.1) atbifurcation parameters λ , i.e. at parameters(4.3) λ = λ ( α, s ) := s (1 − α − qs ) . Explicit and elementary calculations reveal the standard bifurcation diagrams withrespect to λ , for fixed parameters α and c . For example we obtain˙ s = s ( λ − s (1 − s )) at α = 0 , q = 1;(4.4) ˙ s = s (cid:0) λ + ( c + 1) s (cid:1) at α = 1 , q = c + 1 . (4.5)We discuss three cases depending on the sign of c + 1, below. See section 6 for manyadditional examples. 8 igure 4.1: Global equilibrium bifurcation diagrams of the ODE flow (4.1) for the compactifiedasynchrony s defined in (3.6), with c = − . < −
1; see subsection 4.2. The compactified horizontal λ -axis (black) represents the one-cluster case of synchrony s = 0, for the large clusters. All 2-clusterbifurcation curves coexist, in the same phase space x ∈ X , for realizable ratios α = N /N ∈ [0 , α increasing from α = 0 (yellow) to α = 1 (blue), along eachbifurcation curve in the ( λ, s ) plane. The quadratic coefficient q in (4.3) changes sign at α = α c ∈ (0 , /α = N /N ∈ (0 ,
1) are omitted. Red: the two branches of extreme saddle-node values of ( λ, s ) = ( λ minmax ( α ) , s minmax ( α )) on each bifurcation curve; see (4.10),(4.11). Positive q , for 0 ≤ α < α c , imply positive s minmax . Negative q , for α c < α ≤
1, imply s minmax <
0. In-/stability of each stationary solution s ∗ can easily be derived from exchange of stability, at λ = 0 andthe saddle-nodes, or explicitly from (4.1). Figure 4.2:
The bifurcation diagram of figure 4.1, rotated such that the cluster ratios α ∈ [0 ,
1] canbe visualized as a second “parameter”. Color coding as before, but with yellow in front and blue inthe background. Note the red fold curves, for the projection into the horizontal plane (arctan λ, α ). .1 The degenerate transition case c = − q = ( α − is nonnegative and vanishes at α = 1, only. Note how the s -dynamics becomes linear, ˙ s = λs , at α = 1; see (4.5). For0 ≤ α < s to (cid:101) s := (1 − α ) s and obtain the α independent ODE(4.6) ˙ (cid:101) s = (cid:101) s (cid:0) λ − (cid:101) s + (cid:101) s (cid:1) . which coincides with the case α = 0 of (4.4).Stability of the stationary solutions s ∗ = (cid:101) s ∗ / (1 − α ) for any 0 ≤ α < λ < /
4, we have three stationary solutions (cid:101) s ∗ . Since(4.7) ˙ s = qs + . . . with q = ( α − >
0, the top and bottom equilibrium are unstable, while the interme-diate equilibrium is stable. At λ = 1 /
4, of course, we obtain a saddle-node equilibrium s ∗ = (1 − α ) − . For λ > / s ∗ = 0 remains, whichis unstable for all λ > c < − q = q ( α ) in (4.1), (4.2) changes sign strictly, at(4.8) α = α c := (1 − c − (cid:112) (1 − c ) − ∈ (0 , . Specifically q >
0, for 0 ≤ α < α c , and q <
0, for α c < α ≤
1. Interchanging N with N , we omit the redundant cases α = N /N >
1, at first.For an example we fix the cubic coefficient c = − . s = s ∗ at fixed λ = λ ∗ and size ratio α = α ∗ appear as the intersections s = s ∗ of the bifurcation curve forparameter α with the vertical line λ = λ ∗ , in this plot. The size ratio α = N /N may be considered as a fixed parameter, in any of the invariant cluster subspaces (2.1).We therefore plot the bifurcation diagrams as a family of curves, parametrized overdiscrete values α . Color coding is from yellow, at α = 0, to blue, at α = 1. Since allthese bifurcation diagrams coexist, in the large ( N − X ,we superimpose all bifurcation curves in figure 4.1.The less standard contour plot of figure 4.3 tracks the level sets of the parameter λ = λ ( α, s ), as a function of 0 ≤ α = N /N ≤ − π/ < arctan s < π/ s = s ∗ occur; see (4.3). We usearctan s again, rather than s itself, for compactification of the unbounded range of s ∈ R . The level sets are in fact level curves because the only critical point F of λ ,located at ( N /N = 1 , s = 0), is a nondegenerate saddle. This accounts for the twolevel curves of λ = 0, one solid black and one dotted yellow, which intersect at F . Thethird level curve of λ = 0 emanates from the left boundary as a dotted yellow curve.10 igure 4.3: Level curves of λ = λ ( α, s ) for nontrivial stationary solutions s = s ∗ > c = − . < −
1. Here α = N /N is horizontal, and arctan s is plotted vertically.Colors from yellow to blue indicate increasing values of −∞ < λ < + ∞ , this time. Note the black andthe two dotted yellow level curves of λ = 0 which intersect at the only critical point F of λ ( α, s ). Inparticular, any level curve begins and terminates at the boundary, as described in the text. Anotherexample is the dashed yellow level curve of the value λ = 1 /
4. Restricted to the left vertical s -axis,at α = 0, this is the maximal value of λ . As in figure 4.1, the two red curves indicate the values s = s minmax ( α ) where saddle-node bifurcations occur at the levels λ = λ minmax ( α ). Equivalently, theyindicate extremal values of α , on level curves of λ in that region. The region of stable equilibria s = s ∗ is located between the two red curves. Figure 4.4:
The substitution N ↔ N , y ↔ − y allows for a gluing identification s ↔ − s at theright boundary α = N /N = 1 of figure 4.3. The new horizontal axis N /N = α/ ( α + 1) ∈ [0 , α ∈ [0 , ∞ ] and allows us to omit s < N /N = 1 /
2, alias α = 1, of equal cluster size N = N is marked by a vertical dashedwhite line. Again, the region of stable equilibria s = s ∗ is located between the two red curves. α = N /N ≤ N , N of thetwo clusters. Without loss of generality, we may then label the large clusters ( N , ξ )and ( N , ξ ) such that the asynchrony(4.9) s = ( ξ − ξ ) / ( α + 1) > s <
0, apriori. Caution is required because our choice admits any cluster ratio α = N /N ∈ (0 , ∞ ). To represent α , we therefore use the percentage N /N = α/ ( α + 1) ∈ [0 ,
1] asa compactification of the horizontal axis, in figure 4.4 and all subsequent level plots ofthe same style. The important break-even point α = N /N = 1 of equal cluster parity N = N , alias N /N = 1 /
2, is marked by a thin white vertical line.Each level curve of λ ( α, s ) = λ terminates at two points on the boundary of figure 4.3.Any termination at the upper or lower boundary s = ±∞ must occur at α = α c , where q = 0. Indeed, λ = − qs + . . . in (4.3) implies limits λ = − (sign q ) · ∞ , for s = ±∞ and q (cid:54) = 0 . At the left and right boundaries α = 0 and α = 1 we encounter the values λ (0 , s ) = s (1 − s ) and λ (1 , s ) = − ( c + 1) s , respectively. See (4.4), (4.5).Along each level curve λ ( α, s ) = λ ∗ , we may also determine the local extrema of α ,i.e. the vertical tangents of the level curves. Equivalently, these are the local extremaof λ ( α, s ), for any fixed α = α ∗ . An elementary calculation shows that these curvesare given by the level sets of 0 = ∂ s λ ( α, s ) = 1 − α − qs , i.e. s = s minmax ( α ) := (1 − α ) /q > , (4.10) λ = λ minmax ( α ) := λ ( α, s minmax ( α )) = (1 − α ) /q . (4.11)These locations are marked in figures 4.1, 4.3, 4.4 as two red curves. Comparing (4.3)and (4.10), the red curves of saddle-nodes occur at half the s -value of the nontrivialdotted yellow level curve λ = 0, for each α .In-/stability of each stationary solution s = s ∗ can be derived easily from exchange ofstability, at λ = 0, or explicitly from (4.1). As in the previous subsection 4.1, positive q = q ( α ) > s ∗ , and stabilityof any intermediate s ∗ , on each level curve λ and for each fixed α . This identifies theregion of s between the two red saddle-node curves s = s minmax ( α ) as the only region ofstable stationary solutions s = s ∗ . We call such regions of α, s ∗ where the equilibrium s ≡ s ∗ is stable an s -stable region . The s -unstable region consists of the two partsbelow and above the two red saddle-node curves.Negative sign q ( α, c ) < α c ≤ α ≤ /α c , indicatesstability of the largest and smallest equilibria s = s ∗ , and instability of any intermediate s ∗ , there. In particular, this also identifies α c < α < /α c as the region where thedynamics of 4.1 is dissipative , i.e. where solutions s ( t ) are attracted to a boundedregion in forward time. 12 igure 4.5: Bifurcation diagrams of the ODE flow (4.1) for the asynchrony s defined in (3.6) at c = − . > −
1; see subsection 4.3. The quadratic coefficient q remains positive for all 0 ≤ α ≤ s ∗ . Note the single red branchof saddle-nodes at extremals λ = λ minmax ( α ), according to (4.10),(4.11). Indeed q > λ minmax > s minmax > c > − q = q ( α ) in (4.1), (4.2) is strictly positive, for all0 < α < ∞ . Therefore our discussion follows the part of the previous subsection 4.2for the case q >
0. For an explicit example we fix the cubic coefficient c = − .
75 . Seefigure 4.5 for the resulting bifurcation diagrams of (4.1).In the less standard contour plots of figures 4.6, 4.7, analogously to figures 4.3, 4.4,we present the level curves of the parameter λ = λ ( α, s ). The only critical point of λ is still the nondegenerate saddle F , with two associated level curves λ = 0 (solidblack and dotted yellow). This time, −∞ < λ ≤ / λ is attainedon the left boundary α = 0, at s = 1 / λ = −∞ at s = + ∞ , this time, all terminations occur at the right and leftboundaries. In the left region, delimited by the black and yellow level curves of λ = 0,both terminations are located on the left boundary. The maximal value of α , alongeach of the interior level curves of 0 < λ < /
4, occurs on the red curve of saddle-nodes,of course. Again that red curve is located at half the s -value of the dotted yellow level13 igure 4.6: Level curves of λ = λ ( α, s ) for nontrivial stationary solutions s = s ∗ > c = − . > −
1, in analogy to the case c = − . −∞ < λ ≤ / λ = 0, one black andone dotted yellow. Again they intersect at the only critical point F of λ ( α, s ). The maximal values of α , on level curves of λ , form a single red curve of saddle-node bifurcations, this time. See also figure4.5. All level curves of λ still begin and terminate at the boundary, as described in the text. Theregion of stable equilibria s = s ∗ is located between the black horizontal axis and the red saddle-nodecurve. Figure 4.7:
Glued version of figure 4.6. The horizontal axis is N /N = α/ ( α + 1), and s < s = s ∗ > α -axis s = 0,and the red saddle-node curve s = s minimax ( α ). λ = 0, for each α . Above the region delimited by the dotted yellow curve λ = 0,in figure 4.7, level curves connect the two vertical boundaries α = 0 and α = ∞ .In-/stability of each stationary solution s = s ∗ > q >
0, the region of s between the black horizontal α -axis and the red saddle-node curve s = s minmax ( α ) is the only s -stable region, for any 0 < α < ∞ . We address the remaining ODE for y := ξ − ξ = ( α + 1) y next; see (2.6), (3.8). Inthe previous section we have seen how the asynchrony variable s = ( ξ − ξ ) / ( α + 1)tends to total 1-cluster synchrony s ≡ s ≡ s ∗ (cid:54) = 0, for t → ∞ , where the two large clusters ( N , ξ ) and ( N , ξ )compete for the rebels ξ in size. In the present section, we study that remaining rebeldynamics of ( N , ξ ), when the two large clusters have already equilibrated.The synchrony case s ≡ y = y ( λ + y + y )for y = ( α + 1) y . To derive (5.1) we directly replace y = s − y = − y in (3.4), or weformally replace ˙ s/s by λ in (3.8) due to (3.7).For λ > /
4, we obtain global instability of the fully synchronous 1-cluster equilibrium0 ≡ s = ( ξ − ξ ) / ( α + 1) towards rebels y = ξ − ξ , which escape to ±∞ . For0 (cid:54) = λ < /
4, in contrast, we obtain a unique stable equilibrium y ≡ y ∗ . The domain ofattraction is delimited by the remaining two linearly unstable equilibria, beyond whichrebels y escape to ±∞ , respectively, as before. Only for λ < y ∗ = 0 against rebellion, in this sense. For 0 < λ < /
4, where 0 > y ∗ > − /
2, rebellioncan lead to the gradual formation of a tiny stable rebel cluster at y ∗ = − (1 −√ − λ ),at least as long as its tiny size N remains small compared to N ≈ N + N . Also notethe presence of a linearly unstable rebel cluster at y ≡ − (1+ √ − λ ), for all λ < / s ≡ s ∗ (cid:54) = 0, where the two large clusters compete for the rebels( N , ξ ) , is much more interesting. From (4.9) we recall s >
0, without loss of generality.Scaling (3.8), (3.9) to y := ( α + 1) y = ξ − ξ again, we obtain the cubic ODE˙ y = y ( y − ( α + 1) s ) ( y − ¯ y ( s )) , (5.2) ¯ y ( s ) := (2 − α ) s − . (5.3)We repeat that s ≡ s ∗ > s/s from (3.8)drops out in (5.2). The equilibrium y = 0 indicates ξ = ξ : the rebels ξ are atthe cluster ( N , ξ ). The equilibrium y = s , i.e. y = ( α + 1) s , in contrast, indicates ξ = ξ : the rebels are with the other cluster ( N , ξ ). Indeed, y = s is equivalentto y = 0, by (3.6), and hence to ξ = ξ , by (2.6). The third equilibrium y = ¯ y ( s )denotes a 3-cluster equilibrium where, in general, the tiny rebel cluster achieves its ownequilibrium balance, holding out against both large clusters.15uppose a nonstationary solution y = y ( t ) of the scalar ODE (5.2) remains bounded forall positive and negative times t ∈ R . Then y ( t ) is heteroclinic. Consider a heteroclinicorbit of (5.2) from y = 0 to y = ( α + 1) s , as t increases from t = −∞ to t = + ∞ . Thismeans that rebels leave the cluster ( N , ξ ) in favor of the cluster ( N , ξ ). The discrete-valued parameter α = N /N , along with N /N = α/ ( α + 1), becomes continuous andreal-valued in our asymptotics of large N → ∞ . Even though α = N /N is actuallyconstant, we indicate the above rebel migration by a magenta arrow towards smaller N /N , along the level curves of constant λ , in figures 5.2– 5.5 further below.In the opposite direction, a heteroclinic orbit of (5.2) from y = ( α + 1) s to y = 0indicates how rebels leave the cluster ( N , ξ ) in favor of the cluster ( N , ξ ). Weindicate this migration in favor of N by an arrow towards larger N /N , in figures 5.3and 5.5.The arrows on level curves of λ carry meaning beyond the merely formal level. Forlarge N < ∞ , each such heteroclinic orbit amounts to a discrete step in the rationalvalue of α = N /N .We illustrate this fundamental observation by numerical integration of Eq. (1.8) for c = − . λ = 0 .
18 and N = 32 units. As initial condition, a two cluster solution x = · · · = x N = ξ , x N +1 = · · · = x N = ξ was chosen, with ξ and ξ as insection 3. For N = 4, initially, this corresponds to an initial 2-cluster proportionof N /N = 0 . x n , n = N + 1 , in cluster ξ ,and integrate forward in time until the dynamics no longer changes. As a result, weobserve heteroclinic rebel dynamics, that is, the perturbed unit x n changes its clusteraffiliation from ξ to ξ . In other words, N = 5, after the rebel transient. In Fig. 5.1 werepeat this process, for ever increasing cluster sizes N . Note the successive heteroclinictransients of the rebels x n , from ξ down to ξ < ξ . After 12 transients, of course, equalcluster parity N = N = 16 is reached. After 15 transients, the dynamics enters ablocking region and finally settles on a three cluster solution; see the bottom right partof Fig. 5.1. At this stage, the third coexisting cluster at ξ < ξ near ξ < ξ consists ofjust one single rebel element. This trajectory is also visualized in the ( N /N, s ) planeof Fig. 5.2, with the color coding corresponding to the rebel cluster color in Fig. 5.1.For numerical integration, we employed the implicit Adams method provided by SciPy;see [VG&al]. After each perturbation, we subtracted the mean of the ensemble to en-sure the constraint p = 0 in the phase space X of (1.8). Note that by choosing initialconditions in the 2-cluster subspace with just a single unit perturbed, we suppress tran-sitions in which multiple units might change their cluster affiliation, and instabilitiesthat might break up the clusters altogether.The direction of the heteroclinic transients partially determines the ordering, by de-creasing energy or Lyapunov function I ( x ) of (1.13), of the two asymptotic large 2-clusters. This would require a nontrivial calculation, otherwise. The transitivity of thatorder, simply following the level curves of λ along our arrows, possesses a dynamic coun-terpart. Assuming transversality of the stable and unstable manifolds of the target andsource stationary cluster solutions, respectively, along heteroclinic orbits, there also ex-ists a direct heteroclinic connection between any two equilibria connected by a directed16 igure 5.1: Trajectories obtained from numerical simulations of Eq. (1.8) for c = − . λ = 0 .
18 and N = 32 units. We consider 2-cluster solutions x = · · · = x N = ξ , x N +1 = · · · = x N = ξ with ξ and ξ as in section 3. Starting from N = 4 and n = N + 1, we apply a small random perturbationto x n = ξ at a time indicated by the dashed vertical lines. We then integrate the system with rebel x n (cid:54) = ξ , ξ , until the dynamics settles again. See the colored rebel transients of x n from ξ (top) to ξ (bottom), along which the rebel x n changes its cluster affiliation. We repeat this process until thesystem enters a blocking region because a stationary 3-cluster state is initiated by the rebel at x n = ξ (blue) between ξ and ξ ; see the last state shown in the figure. Figure 5.2:
The rebel transients of figure 5.1 are inserted into the diagram of λ -levels from figure4.4, within the level curve of λ = 0 .
18. The color coding of the rebel transients is the same as above.A magenta arrow indicates the drift direction of N /N , induced by the heteroclinic rebel transients.For further discussion of the yellow curves, and of the dark shaded blocking region where stationaryrebel 3-clusters bifurcate and persist, we refer to figure 5.3 below. λ . This dynamic transitivity isa consequence of the so-called λ -Lemma; see for example [PadM82]. The useful prop-erty of transversality of invariant manifolds, often called the Morse or Morse-Smaleproperty, is generic for general vector fields, by the Kupka-Smale theorem. For ourmuch more restrictive class of equivariant vector fields (1.8), however, transversality isa much more delicate assumption – somewhat beyond the scope of our present paper.Heteroclinic orbits between y = 0 and y = ( α + 1) s are blocked when the third rebelequilibrium y = ¯ y ( s ) of (5.2) is located strictly between y = 0 and y = ( α + 1) s .Therefore we call the equilibrium ¯ y ( s ) in (5.2), (5.3) blocking , if 0 < y ( s ) < ( α + 1) s .The blocking regions , in contour plots 5.2–5.5, consist of those ( N /N, s ) for whichthe equilibrium ¯ y ( s ) blocks rebel heteroclinic orbits between the two large competingclusters. Instead the rebels are ready to form a tiny third cluster between the largeones, which may turn out stable, destabilizing the larger competitors, or unstable,stabilizing the 2-cluster status quo.The blocking boundaries of the blocking region are characterized by those values of( α, s ) for which ¯ y ( s ) = 0 or ¯ y ( s ) = ( α + 1) s , respectively. For the blocking boundary¯ y ( s ) = 0 we obtain the graphs s = s ( α ) := 12 − α > , (5.4) λ = λ ( α ) := 1 − ( c + 2) α (2 − α ) . (5.5)Indeed (5.5) follows from (5.4) and (4.3). The blocking boundary ¯ y ( s ) = ( α + 1) s isanalogously characterized by s = s ( α ) := 11 − α > , (5.6) λ = λ ( α ) := α α − ( c + 2)(1 − α ) . (5.7)See figures 5.2–5.5, where we have added the two blocking boundaries as solid blackcurves to the corresponding previous plots 4.4, 4.7 for c = − . , − .
75. The blockingboundaries are easily distinguished by their values at α = 0 : s = 1 / , λ = 1 / s = 1 , λ = 0. Also note the poles at α = 2 , N /N = 2 / α =1 / , N /N = 1 /
3, respectively.It is remarkable that the rebel dynamics (5.2) does not depend on the cubic coefficient c ,at all. In particular the blocking regions in the ( α, s )-plane, and their black boundaries(5.4), (5.6), coincide in figures 5.2–5.5. Any differences arise from the configuration oflevel curves λ = λ ( α, s ), which certainly depend on c via (4.3); see also (5.5) and (5.7).We determine the blocking regions and the direction of heteroclinic rebel dynamics in(5.2) next. Off the black blocking boundaries (5.4), (5.6), we sort the three stationarysolutions y = 0 , ( α + 1) s, ¯ y ( s ) as η < η < η , i.e.(5.8) { , ( α + 1) s, ¯ y ( s ) } = { η , η , η } . igure 5.3: Level curves of λ = λ ( α, s ) for nontrivial stationary solutions s = s ∗ > c = − . < −
1, in coordinates ( N /N, arctan s ). See figure 4.4 for axes and colorcodings. The two solid black curves mark the boundaries of the blocking region. The new dotdashedyellow curve between C and E marks the level λ = λ ( E ) = − ( c + 1). See figure 5.4 for a zoom intothat region and a discussion of the tangency point T . The dashed yellow curve λ = 1 / λ -level where one solid black blocking boundary terminates at N /N = 0. The other solid blackblocking boundary left terminates at the level λ = 0 indicated by the previous dotted yellow curve.The blocking region is located between the two solid black boundaries and is indicated by a darkershading. Outside the shaded blocking region, magenta arrows along the level curves of λ = λ ( α, s )to the right, i.e. towards larger cluster fractions N /N , indicate heteroclinic rebel orbits from thecluster N to the cluster N . Similarly, magenta arrows to the left, i.e. towards smaller fractions N /N , indicate heteroclinic rebel orbits in the opposite direction, favoring the cluster N . Note howdirections change across the blocking region and across the red saddle-node curves. Magenta arrowsare drawn solid, in the s -stable region, and are drawn dashed in the s -unstable region; see also figures4.4 and 4.7. In the s -stable region, for example, rebellions from N to any N < N will cause N to grow beyond equal parity N = N , across the white line N /N = 1 / E and F : fromminority to majority. Growth of N only terminates at the solid black blocking boundary, between A and B . See text for further details. Figure 5.4:
Zoom of figure 5.3. Note the tangency point T , further enlarged in the insert. Levelcurves of λ are tangent to the blocking boundary at T , from inside the dark shaded blocking region.In particular, level curves which emanate from the blocking boundary, between C and T , terminateon the blocking boundary, between the break-even point E and the tangency T . igure 5.5: Level curves of λ = λ ( α, s ) as in figure 5.3, but for c = − . > −
1. The basic locationsof the s -stable and the shaded y -blocking regions look similar, at first sight, but there are subtledifferences in detail. See text. Then ˙ y = ( y − η )( y − η )( y − η ) implies instability of the smallest and largest equilibria y = η , η , and stability of the intermediate equilibrium y = η . The two heteroclinicorbits run from η and η to η , respectively.For s >
0, i.e. 0 < ( α + 1) s , this leaves us with the following three cases for the thirdequilibrium ¯ y ( s ). Region 1: ¯ y ( s ) = η .This case is equivalent to (2 − α ) s − y ( s ) = η < η < η = ( α + 1) s ,i.e. s > s of (5.4). Then blocking does not occur, and heteroclinic rebel migration y = ξ − ξ runs from y = ( α + 1) s = η down to y = 0 = η , i.e. from the cluster( N , ξ ) towards the cluster ( N , ξ ) . We indicate this migration by a magentaarrow towards larger α and N /N , in figures 5.3– 5.5. Region 2 (blocking): ¯ y ( s ) = η .Then η = 0 < ¯ y ( s ) = (2 − α ) s − η < η = ( α + 1) s , i.e. s is betweenthe two black blocking curves. Blocking occurs, and heteroclinic rebel migrationfrom either large cluster gets stuck at the intermediate equilibrium y = ¯ y ( s ) = η .The resulting tiny new stationary rebel cluster at that 3-cluster equilibrium mayin fact grow, at the expense of both large clusters, and with indefinite effectson their proportion α . Figures 5.2–5.5 indicate this blocking region by a darkershading. Region 3: ¯ y ( s ) = η .Then (2 − α ) s − y ( s ) = η > η = ( α + 1) s > η = 0, i.e. s is above theupper red curve, and hence 0 ≤ α < / , ≤ N /N < /
3. Blocking doesnot occur, and heteroclinic rebel migration runs from y = 0 = η upwards to y = ( α + 1) s = η , i.e. from the smaller cluster ( N , ξ ) towards the larger cluster20 N , ξ ): from minority to majority. We indicate this migration by a magentaarrow towards smaller α , in figures 5.3 and 5.5.For example consider the s-stable region in figure 5.3, i.e. for c = − .
3. The region islocated in the wedge between the lower black blocking boundary s and the right redsaddle-node curve. All level curves of λ = λ ( α, s ) in that region are oriented, along thesolid magenta arrows, towards their termination at the black blocking boundary s tothe left of D . Rebel heteroclinic migration towards the cluster N erodes the cluster N , until eventual termination of the 2-cluster regime at the blocking boundary s .In fact, consider the s -stable 2-cluster states, which start out below the dashed yellowlevel curve λ = 1 / N < N , i.e. from the left of the whiteline N /N = of equal parity N = N . All these initial conditions will be prone toheteroclinic rebellion from the cluster N to N , across the white line and well into theregion N > N : from minority to majority, across equal cluster size.In figure 5.5 in contrast, at c = − .
75, the red saddle-node boundary confines the s -stable subregion of region 1 to the left of the white line N /N = , i.e. to N < N .Therefore heteroclinic rebel orbits starting in the s -stable region cannot achieve equalparity, anymore. Instead, they face one of two possibilities:a) termination by blocking at the black blocking curve s , orb) termination at the red saddle-node curve.Migration from the larger cluster N to N gets stuck by an emerging tiny third rebelcluster, in case (a). The blocking 3-cluster equilibrium y = ¯ y ( s ) (cid:38) y = 0, across the black blocking boundary s = s ( α ). In case (b), anyfurther increase of N stops at some status quo uneven 2-cluster with N < N . Thecause is the saddle-node termination of the 2-cluster. Indeed, the value of α = N /N at the saddle-node intersection is the maximal available value of α for any stationary2-cluster, at that particular level of λ .For later reference we also determine the regions of the cubic parameter c for whichthe black blocking boundaries s = s ( α ) and s = s ( α ), respectively, intersect withspecific relevant dotted or dashed level curves of λ , or with the red saddle-node curves s = s minmax ( α ).Specifically we claim the following four intersection points A – D of the lower blackblocking boundary s ( α ) = 1 / (2 − α ): A := s ∩ { λ = 0 } , < α < ⇐⇒ − < c < + ∞ ;(5.9) B := s ∩ { λ = } , < α < ⇐⇒ − < c < − C := s ∩ { λ = − ( c + 1) } , < α < ⇐⇒ − < c < − ;(5.11) D := s ∩ s minmax , < α < ⇐⇒ − < c < − . (5.12)We also claim the following intersection D (cid:48) of the upper black blocking boundary s ( α ) = 1 / (1 − α ) > D (cid:48) := s ∩ s minmax , < α < ⇐⇒ −∞ < c < − .
21n addition, we mark the following two intersections with the white line N /N = α/ ( α + 1) = 1 / α = N /N = 1:(5.14) E = s ∩ { α = 1 } , λ = − ( c + 1); F = ( α = 1 , s = 0) , λ = 0 . The elementary proofs all follow the same pattern. We first insert s = 1 / (2 − α ) > λ in (4.3) or the expression (4.10) for s minmax , as required.For the specified λ -values, we may alternatively invoke (5.5), (5.7). The resulting linearequation for c provides the following explicit expressions: A = s ∩ { λ = 0 } : c = (1 − α ) /α , α = 1 / ( c + 2) ;(5.15) B = s ∩ { λ = } : c = − α − , α = − c + 1) ;(5.16) C = s ∩ { λ = − ( c + 1) } : c = − ( α − / ( α − , α = (4 c + 5) / ( c + 1) ;(5.17) D = s ∩ s minmax : c = − ( α + 1) / , α = − c − ,λ = (1 + c ) / (3 + 2 c ) . (5.18)This proves the four claims (5.9)–(5.12) on s . For s = 1 / (1 − α ) from (5.6), weobtain analogously(5.19) D (cid:48) = s ∩ s minmax : c = − ( α + 1) /α , α = − / (2 c + 1) ,λ = (1 + c ) / (3 + 2 c ) . This proves the remaining claim (5.13). We have omitted variants A (cid:48) , B (cid:48) , C (cid:48) ∈ s whichwill be irrelevant for our subsequent discussion.It remains to address possible tangencies between level curves λ = λ ( α, s ) and theblack boundaries of the blocking regions, in the ( N /N, arctan s )-plane. At such tan-gencies, the emanation/termination behavior of the formal rebel dynamics changes, aswe will illustrate in the next section. For now, we note that such tangencies T , T (cid:48) are characterized by unique extrema of λ ι ( α ) := λ ( α, s ι ( α )) along the black blockingboundaries s ι ( α ) , ι = 0 ,
1. Elementary calculations of high school type for the rationalexpressions (5.5), (5.7) of λ ι ( α ) provide the explicit expressions T : c = − α + 1) / ( α + 2) ∈ ( − , − , α = − c + 1) / ( c + 2) ,λ = ( c + 2) / (2 c + 3) ;(5.20) T (cid:48) : c = − α + 1) / (2 α + 1) ∈ ( − , − , α = − ( c + 2) / ( c + 1) ,λ = ( c + 2) / (2 c + 3) . (5.21)See figure 5.4 for an illustration of the tangency point T .Since we are interested in minority/majority transitions across the white line N /N = , below the black blocking curve s , we also determine the values of c where A , . . . , D ,and T cross α = 1: α = 1 A B C D T c − − / − / − − / c = − , − , − , − , − , − , as announced in (1.19) and asexemplified in the next section. As announced in (1.19), we illustrate the global heteroclinic dynamics of the 3-clustersystem (2.4), in the limit of large dimension N → + ∞ . See the skew product system(3.7), (3.8), and the scaled version (5.2). To represent the seven parameter intervalswhich are separated by the six critical cubic coefficients c = − , − , − , − , − , − of (1.19), we successively illustrate the global dynamics for the seven coefficients(6.1) c = − , − . , − . , − . , − . , − . , +1 ;see figures 6.1–6.8.We also address those non-blocking regions where, in addition, the driving 2-clusterdynamics s > s = s ∗ > λ = ( λ, s ) indicate heteroclinic rebellions in s -stable regions. Dashedmagenta arrows indicate s -unstable regions. This leaves two dashed magenta regionsin each of the figures 6.1–6.8. To enforce s -stability, in regions which are not s -stable originally, according to section4, we may reverse time in all ODEs . For the coefficients A, B, C in (1.10) this amountsto a reversal of all signs. In (1.8) and the following sections, we just replace ˙ x n = . . . , ˙ s = . . . , ˙ y = . . . by − ˙ x n = . . . , − ˙ s = . . . , − ˙ y = . . . .In all seven figures we have shaded the regions in the plane ( N /N, arctan( s )) whereheteroclinic rebel orbits between the two large clusters are blocked, according to section5. −∞ < c < − −∞ < c = − < − s -stablenon-blocked regions, indicated by solid magenta arrows. The dashed magenta arrowsindicate the two s -unstable regions.The lower s -stable region of solid magenta arrows is located between the lower blackblocking boundary s and the right red saddle-node curve. It is split in four by threeseparating non-solid yellow level curves. In all four subregions, the cluster N wins atthe expense of N . The successive heteroclinic dynamics leads to infinite growth of the2-cluster asynchrony(6.2) s = ( ξ − ξ ) / ( α + 1) −→ + ∞ , via a size ratio α which increases asymptotically to N /N = α (cid:37) /α c , given by (4.8).23 igure 6.1: Global dynamics in the plane ( N /N, arctan( s )) for case 6.1, −∞ < c = − < − D (cid:48) , E , F ,see (5.19), (5.14). The shaded region marks blocking of rebel heteroclinic dynamics between the twolarge clusters of size ratio α = N /N . Magenta arrows indicate the formal flow on the level setsof λ = λ ( α, s ). Solid magenta arrows are used in the s -stable region of the asynchronous 2-clusterequilibrium s = s ∗ >
0. Dashed magenta arrows account for the two s -unstable regions. See text forfurther details. Figure 6.2:
Zoom into the upper left s -unstable and s -stable regions of figure 6.1. < λ < /
4, all directedlevel curves of λ emanate from the left boundary N /N = 0 and terminate at α =1 /α c , s = + ∞ . This means that the heteroclinic rebel dynamics favors the growthof arbitrarily small clusters ( N , ξ ), “out of the blue”, over the cluster ( N , ξ ), untilthe cluster asynchrony blows up, s (cid:37) + ∞ , at the maximal sustainable size ratio α = N /N = 1 /α c >
1: from minority to majority.Between the dashed and the dotdashed yellow separatrixes, i.e. for 1 / < λ < λ ( E ) = − ( c + 1), minority N can still become majority, until s blows up. This time, however,at least a critical minimal size N of the smaller cluster is required, which dependson the value of λ . Indeed that critical size is determined by the realizable value of α = N /N at the intersection of the level curve of λ with the black blocking boundary s .Above the dotdashed separatrix, i.e. for λ > − ( c + 1) = λ ( E ), the rebel growth of N does not cross the white line N /N = . The cluster size N , initiating to the rightof E on the blocking boundary s , must exceed N from the start. To the right of thedotted separatrix from F , i.e. for given λ <
0, the minimally required cluster size of N is determined by the value α of the cluster ratio N /N on the right red saddle nodecurve s minmax corresponding to λ minmax = λ .The upper s -stable region of solid magenta arrows is located in the triangular wedgeabove D (cid:48) , between the left red saddle-node curve s minmax and the upper black blockingboundary s . Rebellions there originate from s and decrease α = N /N <
1, untilthey terminate at s minmax >
0, where 2-cluster solutions disappear into saddle-nodebifurcations. See the zoom 6.2 of figure 6.1.Similar remarks apply to the remaining two non-blocking regions which are s -unstable.The dashed magenta arrows indicate the resulting formal rebel dynamics. The upperleft s -unstable region is bounded below by the black blocking boundary s = s ( α )and the upper red saddle-node curve s = s minmax ( α ); see (5.6) and (4.10). In (5.13)and (5.19) we have denoted their intersection by D (cid:48) . The two yellow separatrix levels λ ( α, s ) = 0, dotted, and λ ( α, s ) = λ ( D (cid:48) ), solid, define three subregions, which aredistinguished by the eventual fate of the heteroclinic rebel dynamics. The rebellionmay terminate at the left boundary α = 0, at the black blocking boundary s ( α ) to theleft of D (cid:48) , or at the left red saddle-node cluster configuration s minmax ( α ) to the rightof D (cid:48) . In all three cases, the ongoing decay of α = N /N originates from asynchrony s = + ∞ , at finite size ratio α = α c < λ . All rebellions favor N over N , this time, and terminate at α = N /N = ∞ alias N = 0 , N = N . Heteroclinic rebels defect from N to the largercluster N . Defection originates from the red saddle-node boundary s minmax ( α ) > F , for some λ -dependent minimal α = N /N >
1. Note that majority N > N prevails, because the white line N /N = is not crossed.25 igure 6.3: Global dynamics in the plane ( N /N, arctan( s )) for case 6.2, − < c = − . < − / T (cid:48) between the upper solid black blocking boundary and the level set λ ( α, s ) = λ ( T (cid:48) ); see (5.21). − < c < − / − < c = − . < − / c = −
3, in the original s -stable regions with solid magentaarrow, and in the lower right s -unstable region with dashed arrows. Note however theintersection point D (cid:48) and the tangency point T (cid:48) on the upper black blocking boundary s . These points only affect level sets in the upper left s -unstable region of dashedmagenta rebel arrows. The corner point D (cid:48) and its level set λ ( α, s ) = λ ( D (cid:48) ) retaintheir previous significance. See in particular the previous zoom in figure 6.2. However,the new tangency point T (cid:48) comes with a level set λ ( α, s ) = λ ( T (cid:48) ) which consists of twobranches. Only above λ = λ ( T (cid:48) ) do we still observe termination at N = 0, originatingfrom asynchrony s = + ∞ at α = α c . Below the left branch of λ = λ ( T (cid:48) ), such rebellionoriginates from the blocking boundary, instead. Below the right branch, the rebellionstill originates from s = + ∞ , α = α c as before, but terminates at a minimal clusterratio α = α ( λ ) > − / < c < − / − / < c = − . < − / s -stableregion of solid magenta arrows, and two s -unstable time-reversed regions of dashedmagenta arrows. The only s -stable region, lower triangular between the lower blackblocking boundary s and the right red saddle-node curve, has now detached from thesingular tip s = + ∞ at α = 1 /α c . The new tip is at D ; see (5.12) and (5.18). Along s , the intersection points A , B , C have appeared, with the yellow λ -levels λ = 0 = λ ( F )(dotted), λ = 1 / λ = − ( c + 1) = λ ( E ) (dotdashed), respectively.See (5.9)–(5.10) and (5.15)–(5.17). The three yellow separatrices define four subregions.26 igure 6.4: Global dynamics in the plane ( N /N, arctan( s )) for case 6.3, − / < c = − . < − / Figure 6.5:
Global dynamics in the plane ( N /N, arctan( s )) for case 6.4, − / < c = − . < − / λ <
0, i.e. in the triangular subregion
ADF , rebellions start from the red saddle-node curve DF and terminate at the black blocking boundary segment AD . In thepentagonal subregion 0 < λ < , all rebellions start from “blue sky”, at α = N /N =0, with tiny N . They gain majority as they cross the dashed white break-even line N = N , and terminate at the black blocking boundary segment AB . For < λ < − ( c + 1) = λ ( E ), rebellions still gain majority across the white line, but they start onthe black blocking boundary s to the left of E and terminate on the segment BC of s . The black blocking segment CE to the right of E , finally, exhibits a new tangency T with the level curves of λ . For − ( c + 1) = λ ( E ) < λ < λ ( T ) this leads to rebellions,from N to increasing N > N already in majority, which start and terminate at theblack blocking boundary: from ET to CT . Except for the location to the right of thewhite break-even line, the dynamics follows the zoom in figure 5.4.The upper region of dashed magenta arrows has simplified: all rebellions now originatefrom the black blocking boundary s , with finite α and s , and terminate at α = 0.The lower dashed magenta arrow region, likewise, terminates at α = + ∞ , N = 0.For λ < λ ( D ), the rebellions originate from black blocking s and, for λ ( D ) < λ < − / < c < − / − / < c = − . < − /
4, of figure 6.5 has been prepared in section 5; seefigure 5.3. The two regions of dashed magenta arrows correspond to the previous case,verbatim.The s -stable triangular region of solid magenta arrows looks quite similar to figure 6.4,except for the position of the dashed white line EF of equal parity N = N . Thesegment CT on the black blocking boundary has in fact moved from the right of E to the left of E , i.e. from size ratios α > α <
1. Of the four s -stable regionsseparated by the three yellow level curves λ = 0 , , − ( c + 1), this only effects the region − c + 1 < λ < λ ( T ) which now features a minority N , still growing, rather than amajority. Rebellions lead from CT to ET , this time. − / < c < − − / < c = − . < −
1, as in figure 6.6, the situation in the two regions ofdashed magenta arrows remains the same, qualitatively, as in the two previous figures6.4 and 6.5. In the remaining unique s -stable region of solid magenta arrows, thedotdashed yellow level λ = − ( c + 1) = λ ( E ) has dropped below the dashed yellow level λ = , as c increased through − /
4. The region of rebellion from “blue sky” minority N = 0 to majority N > N , across the dashed white line EF , therefore requires0 < λ < − ( c + 1), now. Termination occurs at the black blocking segment EA of s .The second intersection point C of s with the yellow level λ = − ( c + 1) = λ ( E ) hasdisappeared. The region < λ < λ ( T ) now features growth of the minority N fromthe black blocking segment of s on the left of T to TB .28 igure 6.6: Global dynamics in the plane ( N /N, arctan( s )) for case 6.5, − / < c = − . < − Figure 6.7:
Global dynamics in the plane ( N /N α, arctan( s )) for case 6.6, − < c = − . < − / igure 6.8: Global dynamics in the plane ( N /N, arctan( s )) for case 6.7, − / < c = +1 < + ∞ . − < c < − / − < c = − . < − / s -stable region of solidmagenta arrows is now contained in the region N /N to the left of the dashed whiteline N = N of equal parity. Thus N is, and remains, in minority N < N . Theonly yellow separatrix λ = λ ( D ) > D of the lowerblack blocking boundary s with the only remaining red saddle-node curve s minmax ;see (5.12) and (5.18). All rebellions start from “blue sky” i.e. at vanishing N . In thesubregion 0 < λ < λ ( D ) they terminate at the red saddle-nodes. In the complementarysubregion λ ( D ) < λ < , they terminate at the black blocking boundary.The second time-reversed region of dashed magenta arrows, on the right, is nowsubdivided into three subregions by the two yellow separatrices AF of λ = 0 (dot-ted) and λ = − ( c + 1) = λ ( E ) (dotdashed). For λ ( D ) > λ > λ ( A ) = 0, thelonger cluster N decays by heteroclinic rebellions which proceed from the segment AD of the black blocking boundary s to the red saddle-nodes, where N is still inmajority. For λ < λ ( A ) = 0, rebellions terminate at N = 0. In the subregion λ ( A ) = 0 > c > − ( c + 1) = λ ( E ) the majority cluster N from the black blockingsegment AE of s crosses the dashed white line of equal parity until it terminates asminority. For λ ( E ) = − ( c + 1) > λ , the cluster N remains a minority, originatingfrom the black blocking segment of s , to the right of E , at a finite value of s . − / < c < + ∞ The final case is − / < c = 1 < + ∞ , as in figure 6.8. As for all c > − / s -stable region, with solid magenta arrows, and two time-reversed s -unstable regions with dashed magenta arrows; see figures 6.4–6.7. The upper left30ashed magenta region remains unchanged. The solid magenta region has lost D from its boundary: all rebellions originate from red saddle-nodes and terminate at N = 0 , α = 0, with N remaining in minority.The wedge of the second dashed magenta region, on the right, now reaches left all theway to the tip at α = 0 , s = where λ = . The two yellow level curves λ = 0 = λ ( A )and λ = − ( c + 1) = λ ( E ) < − divide the region into three subregions, just as in theprevious case − < c < − / < λ < originate from anywhere on the black blocking boundary, to the leftof A , i.e. at any size ratio 0 < α = N /N < α ( A ) = 1 / ( c + 2) < /
3; not just at0 < − c − α ( D ) < α < α ( A ) = 1 / ( c + 2) < α = 0, asthey did in the previous case. The two other subregions of λ <
0, as before, show howthe cluster N can decay to N = 0 from the maximal value of N /N = 1 − α/ ( α + 1)on the black blocking boundary, which is sustainable at the given level of λ <
0. Ifwe reverse time, to make this s -unstable region s -stable, then the growth of N = 0to the maximally sustainable N reveals the limitations of rebel dynamics defecting tominority. In this section, we study N globally coupled, identical Stuart-Landau oscillators(7.1) ˙ W n = (1 − (1 + i γ ) | W n | ) W n + β · ( (cid:104) W (cid:105) − W n ) . Here W n ∈ C indicate phase and amplitude of the n -th oscillator, n = 1 , . . . , N . Weconsider real amplitude dependence γ of individual periods, complex coupling β ∈ C ,and we abbreviate the average (cid:104) W (cid:105) := N (cid:80) W n , as before. Note S N -equivariance of(7.1) under the action analogous to (1.1).For a background and motivation we recall how (7.1) often serves, in physics, as a“normal form” for oscillatory systems close to the onset of oscillation and under theinfluence of a linear coupling through the mean field [GMK08]. This normal form hasbeen established to be a good approximation in a multitude of contexts from variousdisciplines, whether it be in physics, chemistry, biology, neuroscience, social dynam-ics, or engineering. For an overview see, e.g. [PRK03, PR15] or references 1-15 in[KGO15]. Our motivation to study (7.1) is to gain a deeper understanding of thedynamics of oscillatory electrochemical systems. Indeed global, linear coupling oftencontrols the evolution of the electrostatic potential of the working electrode, a cru-cial dynamic variable in electrochemical systems [WKH00, Kri01, PLK04, VBBK05,MGMK09, KK&al14, SZHK14, SK15, PH&al17, LSMK18, NKV19, HGK19]. Theglobal coupling originates from the electric control of the device: any potential dropin the electrolyte or the external electric circuit is fed back to the evolution of theelectrode potential at any location. Yet, there are many other situations where thedynamics of the electric potential is governed in almost the same manner as in electro-chemical systems. Examples include semiconductor devices [Sch01], gas discharge tubes31PBA10], or arrays of Josephson junctions [BVB97]. Along with these numerous appli-cations go various theoretical studies of the globally coupled Stuart-Landau ensemble[NK93, NK94, NK95, HR92, SF89, MS90, MMS91, DN04, DN06, KGO15, KHK19].Specifically, we consider bifurcations from the globally synchronous periodic solution(7.2) W n ( t ) = exp − (i γt )of amplitude 1 and minimal period 2 π/γ . Somewhat unconventionally, we rewrite (7.1)in “polar coordinates” Z n = R n + iΨ n via W n = exp( Z n ) as(7.3) ˙ Z n = ˙ W n /W n = 1 − (1 + i γ ) | W n | + β ( − N N (cid:88) k =1 W k /W n ) . We now invoke the notation (1.9) and define(7.4) R := N (cid:88) R n , r n := (cid:102) R n = R n − R, Φ := N (cid:88) Φ n , ϕ n := (cid:102) Φ n = Φ n − Φ ,Z := R + iΦ , z n := r n + i ϕ n , z = ( z n ) Nn =1 , to derive ˙ ϕ n = − γ e R (cid:103) e r n + Im( β (cid:104) e z (cid:105) (cid:103) e − z n ) , (7.5) ˙ r n = − e R (cid:103) e r n + Re( β (cid:104) e z (cid:105) (cid:103) e − z n ) , (7.6) ˙ R = 1 − e R (cid:104) e r (cid:105) + Re( β ( (cid:104) e z (cid:105)(cid:104) e − z (cid:105) − , (7.7)and the average phase(7.8) ˙Φ = − γe R (cid:104) e r (cid:105) + Im( β ( (cid:104) e z (cid:105)(cid:104) e − z (cid:105) − . Here we have slightly extended the notation (1.9) to include(7.9) (cid:104) e z (cid:105) := N N (cid:88) n =1 e z n = N N (cid:88) n =1 ∞ (cid:88) m =0 1 m ! z mn = ∞ (cid:88) m =0 1 m ! (cid:104) z m (cid:105) , (cid:102) e z n := e z n − (cid:104) e z (cid:105) = ∞ (cid:88) m =0 1 m ! ( z mn − (cid:104) z m (cid:105) ) = ∞ (cid:88) m =0 1 m ! (cid:102) z mn . The globally synchronous solution (7.2) becomes the trivial equilibrium z = 0 , R = 0 of(7.5)–(7.7), in this notation. The average phase Φ( t ) does not appear in these ODEs,due to S -equivariance of the original Stuart-Landau system (7.1) under uniform phaseshifts. We will therefore ignore the average phase Φ( t ), henceforth. We only keep inmind how equilibria of z , R , and heteroclinic orbits between them, actually indicateperiodic orbits and their heteroclinic connections, via the skew product structure of˙Φ = . . . , driven by the Φ independent dynamics of z , R , only.Our task, in the present section, is the derivation of the reduced flow (1.10), i.e.(7.10) ˙ x n = µ + x n + A (cid:102) x n + B (cid:102) x n + C (cid:104) x (cid:105) x n + . . .
32n a center manifold of the trivial equilibrium z = 0 , R = 0 of the system (7.5)– (7.7),at a zero eigenvalue µ + of the linearization. See for example [Carr, ChHa82, Van89]for a background on center manifolds.An outline of this standard procedure is as follows. We replace z n = r n + i ϕ n ∈ C by suitable linear real coordinates ( x n , y n ) such that the eigenspace of the mandatoryeigenvalue µ + = 0 is given by y = 0 , R = 0. The remaining eigenvalues will be µ − < x = 0 , R = 0, and µ = −
2, for x = y = 0. Since (cid:104) z (cid:105) = 0, by construction of z n = (cid:102) R n + i (cid:102) Φ n , we will inherit (cid:104) x (cid:105) = 0 = (cid:104) y (cid:105) , i.e. x , y ∈ X will realize the standardrepresentation of S N ; see (1.1), (1.7). Since the S N -invariant center manifold can bewritten as a graph of ( y , R ) over x , tangent to the eigenspace of µ + = 0 at the trivialequilibrium, truncation to second order yields y n = a (cid:102) x n + . . . , (7.11) R = b (cid:104) x (cid:105) + . . . , (7.12)with suitable real coefficients a, b calculated below. Substitution of (7.11), (7.12) intothe ODE ˙ x n = . . . with vanishing linear part then allows us to determine the coefficients A, B, C of the reduced flow (7.10) in the center manifold, up to third order in x , asrequired for our analysis of (1.10), (1.8). We can then invoke the results of sections1–6 to detect rebel heteroclinic dynamics between periodic 2-cluster solutions of theglobally coupled Stuart-Landau system (7.1). See [SEC03] for another example in aDarwinian setting.To substantiate the above outline we start from the following linear change of coordi-nates:(7.13) 2 dx n : = − r n + d +1 γ (cid:48) ϕ n , r n = (1 − d ) x n + (1 + d ) y n , dy n : = + r n + d − γ (cid:48) ϕ n , ϕ n = γ (cid:48) x n + γ (cid:48) y n . for n = 1 , . . . , N . The system on the right defines the inverse of the system on the left.Here d abbreviates the discriminant root(7.14) d := (cid:113) − β I − γβ I > , writing the real and imaginary parts of the complex linear coupling as β = β R + i β I .Of course we assume positive discriminant , i.e.(7.15) β I + 2 γβ I < . The coefficient γ (cid:48) in (7.13) is defined as(7.16) γ (cid:48) := β I + 2 γ . The two real eigenvalues of the linearization of (7.5), (7.6) at the trivial equilibrium z = 0 , R = 0 are(7.17) µ ± = − ( β R + 1) ± d . µ − < µ + is of algebraic and geometric multiplicity N −
1. Indeed the eigenspaces x = 0 , R = 0 and y = 0 , R = 0 are each isomorphic tothe standard irreducible representation X of S N . The requisite eigenvalue µ + = 0, atbifurcation, is picked such that µ − < µ + and the algebraically simple eigenvalue µ = −
2, in the synchrony direction of R , ensure exponential stability of the reducedflow on the center manifold of µ + . We collect some relations among the availablecoefficients:(7.18) β R = d − ,γ (cid:48) β I = ( β I + 2 γ ) β I = 1 − d = − ( β R + 2) β R ,β = ( d − − i( d + 1) /γ (cid:48) ) . Indeed, the first line follows from µ + = 0 and (7.17). The second line uses definition(7.16) of γ (cid:48) , the definition (7.14) of d , and the first line. The third line follows fromthe first and the second. In summary, (7.16) and (7.18) allow us to express the threefree real parameters γ, β R , β I of (7.1) by the two real parameters γ (cid:48) and d , at µ + = 0, with the only remaining constraint d > (cid:54) = γ (cid:48) . We will therefore express the remainingcoefficients a, b of (7.11), (7.12), and A, B, C of (7.10) in terms of γ (cid:48) and d .To calculate a, b we use existence and C k differentiability of the center manifold, forany k >
0. See [Van89]. We first expand the transformed ODE(7.19)0 + . . . = 2 dy (cid:48) n ( x ) ˙ x = 2 d ˙ y n = ˙ r n + d − γ (cid:48) ˙ ϕ n == − (1 + ( λ − γγ (cid:48) ) e R (cid:103) e r n + Re (cid:16) (1 − i d − γ (cid:48) ) β ( (cid:104) e z (cid:105) (cid:103) e − z n ) (cid:17) == µ − y n + . . . . Here we have substituted (7.6), (7.5) on the right, after the transformation (7.13). Onthe left, we have inserted the quadratic expansion (7.11). Note that ˙ x = µ + x + . . . with µ + = 0 is at least quadratic. Moreover, tangency of the center manifold to theeigenspace y n = R = 0 implies y (cid:48) n (0) = 0. Therefore, the left hand side of (7.19) startsat (ommited) cubic order. Substitution of (7.13), (7.18), and the expansion (7.11) onthe right side of (7.19), yield the result(7.20) a = − d γ (cid:48) d (cid:0) γ (cid:48) + ( d − (cid:1) (cid:0) γ (cid:48) + 3( d − (cid:1) , by comparison of quadratic coefficients. For R , we analogously obtain(7.21) 0 + . . . = R (cid:48) ( x ) ˙ x = ˙ R = 1 − e R (cid:104) e r (cid:105) + Re (cid:0) β ( (cid:104) e z (cid:105)(cid:104) e − z (cid:105) − (cid:1) , with a left hand side of at least cubic order. Substitutions and comparison of secondorder coefficients yield(7.22) b = (1 − d ) (cid:0) γ (cid:48) + ( d − d + 5) (cid:1) . To calculate the reduced flow ˙ x n = f n ( x ) in the center manifold, to order k ≥
2, it isalways sufficient to expand the center manifold itself to order k −
1. To determine the34uadratic coefficient A and the cubic coefficients B, C in (1.10), we expand(7.23) 2 d ˙ x n = − ˙ r n + d +1 γ (cid:48) ˙ ϕ n == − (cid:16) − d + 1) γγ (cid:48) (cid:17) e R (cid:103) e r k + Re (cid:16) − − i d +1 γ (cid:48) β ( (cid:104) e z (cid:105) (cid:103) e − z k ) (cid:17) == µ + x n + A (cid:102) x n + B (cid:102) x n + C (cid:104) x (cid:105) x n + . . . to cubic order. We use the substitutions (7.13) and (7.18) and insert the quadraticexpansions (7.11), (7.12) to finally obtain, with the prerequisite stamina, A = d − γ (cid:48) d (cid:0) γ (cid:48) + ( d + 1) (cid:1) (cid:0) γ (cid:48) − d − (cid:1) ;(7.24) B = − d (cid:16) d − γ (cid:48) d (cid:17) (cid:0) γ (cid:48) + ( d + 1) (cid:1) (cid:0) γ (cid:48) + ( d − (cid:1) ·· (cid:0) ( γ (cid:48) + d ) + 2 d − (cid:1) (cid:0) ( γ (cid:48) − d ) + 2 d − (cid:1) ;(7.25) C = d (cid:16) d − γ (cid:48) d (cid:17) (cid:18) γ (cid:48) − d − d + 1) γ (cid:48) − d + d − d + 22 d + 1) γ (cid:48) −− d + 1) ( d − (2 d + 3 d − γ (cid:48) + 9( d − (cid:19) . (7.26)In particular, scaling (1.11) for nonzero A, B and truncation to cubic order lead to thecubic normal form (1.8) studied in the previous sections. The remaining cubic coeffi-cient c = C/B , according to (1.12), is then given by the long but explicit expression(7.27) c = γ (cid:48) − d − d +1) γ (cid:48) − d + d − d +22 d +1) γ (cid:48) − d +1) ( d − (2 d +3 d − γ (cid:48) +9( d − − ( γ (cid:48) +( d +1) )( γ (cid:48) +( d − )(( γ (cid:48) + d ) +2 d − γ (cid:48) − d ) +2 d − . Our results are summarized in the contour plot of figure 7.1. First we note that therational function c = c ( γ (cid:48) , d ) of (7.27) is even in γ (cid:48) . We can therefore omit negative γ (cid:48) and only consider d, γ (cid:48) >
0. We recall the expressions (7.14) and (7.18) for d and γ (cid:48) , in terms of the original coefficients γ ∈ R and β ∈ C of the coupled Stuart-Landau system (7.1). The coefficient γ regulates the soft-/hard-spring characteristicof the individual Stuart-Landau oscillator, i.e. the monotone dependence of period onamplitude. Complex linear all-to-all coupling is regulated by β . Colors in figure 7.1indicate the seven intervals of c ∈ R which are complementary to the six critical levels(7.28) c = − , − , − , − , − , − , as identified in section 5. We have subdivided the intervals c < − c > − /
2, forclarity.For further illustration we briefly relate our present results to the discussion of the in [KHK19]. By definition, the 2-cluster singularity refers to thethe bifurcation point λ = 0 of an odd nonlinearity A = 0 in the dynamics (1.10) on thecenter manifold. The very value A = 0, however, is conspicuously absent in our scaledasymmetric version (1.8), due to the singular scaling (1.11) with τ = B/A . Fromthe outset, we note that any analysis of 2-cluster equilibria is subsumed as N = 0in our present setting. Therefore such results hold for all N , and are not restricted35 igure 7.1: Level sets of the cubic coefficient c = c ( γ (cid:48) , d ) in the cubic S N normal form (1.8), as afunction of the positive parameters γ (cid:48) and d . See (7.27). Since c ( γ (cid:48) , d ) = c ( − γ (cid:48) , d ) is quadratic in γ (cid:48) ,we only plot positive γ (cid:48) , d . See (7.14) and (7.18) for expressions of d and γ (cid:48) in terms of the originalcoefficients γ ∈ R , of period-amplitude dependence, and β ∈ C , of complex linear coupling, in theStuart-Landau setting (7.1). The singular set c = ±∞ , alias B = 0, is indicated by the white crescent.For resulting dynamics in the colored intervals of c see the representative figures 6.1– 6.8 of section 6.The white dot at d = 3 , γ (cid:48) = 2 √ c = 1 of fig. 6.8. to any asymptotics of large N . This extends to the bifurcation curves of rebel 3-cluster stationary solutions, at the blocking curves. Indeed, the defining kernels of thelinearization are independent of the size of the bifurcating cluster; see (4.39) and (4.40)in [Elm01].We can easily determine the 2-cluster singularities in the parameters γ (cid:48) , d of figure 7.1.Indeed, A = 0 in our derivation (7.24) is equivalent to the pair of straight lines(7.29) γ (cid:48) = 3( d − . Quite remarkably, insertion of (7.29), to eliminate γ (cid:48) , collapses the formidable expres-sion (7.27) of the cubic coefficient c in the scaled center manifold dynamics (1.8), alongthese lines, to become(7.30) c = d − . Conversely, for given c > −
2, we can now invoke (7.29), (7.18), and (7.16), successively,to determine the parameters of the 2-cluster singularity as(7.31) d = c + 2 , γ (cid:48) = √ c + 1) . Since (7.29) is in fact quadratic, we may in fact replace any occurrence of √
3, here andbelow, by −√
3. For brevity, we will only address the positive sign.At λ = 0, relations (7.18) then determine the original parameters β, γ as β = ( c + 1) − i √ ( c + 3)(7.32) γ = √ (2 c + 3) . (7.33) 36nsertion of (7.31) in (7.25) and (7.26), respectively, determines the modest expressions B = − ( c + 1) ( c + 3 c + 3) / (2 + c ) , (7.34) C = Bc . (7.35)Of course we may just as well invoke (7.33), anytime, to alternatively express allother parameters in terms of the soft/hard spring constant γ of (7.1), at the 2-clustersingularity.In the language of section 6, each size ratio α = N : N gives rise to up to threeparticular nonzero bifurcation values of the parameter λ in the scaled center manifolddynamics (1.8): the red saddle-node value λ minmax of (4.11) and the two blockingvalues λ ι , ι = 0 , λ, A, B, C in the general, unscaled center manifold setting (1.10), we just have torevert the scaling (1.11). The parameter values λ in (1.10), which correspond to eachof the above reference values λ ι , ι ∈ { minmax , , } , for fixed α , are then given by theasymptotic parabolas(7.36) λ = ( λ ι ( α ) /B ) A + . . . . Higher order terms in A go beyond our third order truncation of the flow (1.10) inthe center manifold, and also account for dependencies of the coefficients A, B, C on λ . This shows how all bifurcation curves emanate from the 2-cluster singularity at A = 0 , λ = 0, with horizontal tangent and curvatures given by the one remainingcoefficient c and the size ratios α .See [KFHK20] for numerical illustrations of the 2-cluster singularity, in the specialcase of N = 16 Stuart-Landau oscillators (7.1) with γ = 2. Specifically, size ratios α = N / ( N − N ) , N = 1 , . . . , γ = 2corresponds to the simplest case c = √ − / > − / c = 1 in fig. 6.8. The complex value of the coupling constant β at the 2-clustersingularity follows from (7.32).In conclusion, we have gone beyond the discussion of 2-cluster equilibria and theirstability. In fact, we have indicated rebel heteroclinic dynamics between them, in thelimit of large N . For each of the seven complementary intervals of the cubic coefficient c in the center manifold dynamics (1.8), we have represented the resulting heteroclinicrebel dynamics of section 5, between the two large clusters ( N , ξ ) and ( N , ξ ), infigures 6.1–6.8 of section 6, respectively. Since N → + ∞ is finite, in practice, we haveto interpret these figures on the grid of rational values α/ ( α + 1) = N /N , of course,for cluster sizes N = 0 , . . . , N . See figures 5.1, 5.2 for the appropiate interpretationof heteroclinic rebel transitions. All seven cases admit 2-cluster singularities. Indeed,even case 6.1, c < −
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