Uni-directional synchronization and frequency difference locking induced by a heteroclinic ratchet
aa r X i v : . [ n li n . AO ] M a y UNI-DIRECTIONAL SYNCHRONIZATION ANDFREQUENCY DIFFERENCE LOCKING INDUCED BY AHETEROCLINIC RATCHET ¨Ozkan Karabacak
Electronics and Communication Engineering DepartmentIstanbul Technical [email protected]@gmail.com
Abstract
A system of four coupled oscillators that exhibitsunusual synchronization phenomena has been ana-lyzed. Existence of a one-way heteroclinic network,called heteroclinic ratchet, gives rise to uni-directional(de)synchronization between certain groups of cells.Moreover, we show that locking in frequency differ-ences occur when a small white noise is added to thedynamics of oscillators.
Key words
Heteroclinic ratchets, synchronization, coupled oscil-lators, uni-directional desynchronization.
Phase oscillators are used as approxima-tions for the phase dynamics of coupled limitcycle oscillators in the case of weak cou-pling [Kuramoto, 1984; Pikovsky et al., 2001;Hoppensteadt and Izhikevich, 1997]. They ex-hibit synchronization and clustering phenomena[Kuramoto, 1984; Sakaguchi and Kuramoto, 1986],even if coupling function consist of the first har-monic only. If the second harmonic of the couplingfunction is considered, it is possible to observeswitchings between different clusterings as a resultof an asymptotically stable robust heteroclinic cycle[Hansel et al., 1993]. It is known that heterocliniccycles are not structurally stable but they may existrobustly for coupled systems. This is due to theexistence of robust invariant subspaces for certain cou-pling structures that may support robust heteroclinicconnections that are saddle-to-sink on the invariantsubspaces and form a heteroclinic cycles [Krupa, 1997;Ashwin and Field, 1999; Aguiar et al., 2011]. Exis-tence of robust heteroclinic cycles or more generallyheteroclinic networks in a system of three and four globally coupled phase oscillators have been analyzedin [Ashwin et al., 2008] and in [Ashwin et al., 2006],where an extreme sensitivity phenomenon to detuningof natural frequencies has been observed. Namely, os-cillators loose synchrony even for very small detuningof natural frequencies. [Karabacak and Ashwin, 2010]have considered the third harmonic of the couplingfunction and observed one-way heteroclinic networks,which are called heteroclinic ratchets. A heteroclinicratchet is a heteroclinic network that, for some axis,contains trajectories winding in one direction only.Heteroclinic ratchets give rise to extreme sensitivityto detuning of certain sign. Namely, synchronizationof a pair of oscillators is possible only when thenatural frequency of a certain oscillator is larger thanthe other. We call this phenomenon uni-directional(de)synchronization.In the sequel, we identify a heteroclinic ratchetfor a system of four coupled oscillators in Sec-tion 2. Although the system is less complicatedthan the original ratcheting system considered in[Karabacak and Ashwin, 2010], it exhibits more com-plicated dynamics: uni-directional synchronization be-tween groups of oscillators, explained in Sections 3 and4.
Consider the following well-known model of coupledphase oscillators: ˙ θ i = ω i + KN N X j =1 c ij g ( θ i − θ j ) . (1)Here, θ i ∈ T = [0 , π ) denotes the phase of oscillator i and ω i is its natural frequency. The connection matrix c ij } represents the coupling. c ij = 1 if oscillator i re-ceives an input from oscillator j and c ij = 0 otherwise.Since g ( · ) is a π -periodic function, it can bewritten as a sum of Fourier harmonics: g ( x ) = P ∞ k =1 r k sin( kx + α k ) . Without loss of generality, wemay set K = N and r = − by a scaling of time.Let us choose the following coupling function with twoharmonics only: g ( x ) = − sin( x + α ) + r sin(2 x ) (2)The model (1) is used as the approximate phase dy-namics of weakly coupled limit cycle oscillators, andthe weak coupling gives rise to a T phase-shift sym-metry in the phase model (1). Hence, the dynamics of(1) is invariant under the phase shift ( θ , θ , . . . , θ N ) ( θ + ǫ, θ + ǫ, . . . , θ N + ǫ ) for any ǫ ∈ T . Below, this symmetry is used to reducethe dynamics to an ( N − )-dimensional phase differ-ence system.Let us consider the coupled phase oscillator system(1) with the coupling structure given in Figure 1. Thisgives rise to the following system: ˙ θ = ω + g ( θ − θ ) + g ( θ − θ )˙ θ = ω + g ( θ − θ ) + g ( θ − θ )˙ θ = ω + g ( θ − θ ) + g (0)˙ θ = ω + g ( θ − θ ) + g ( θ − θ ) . (3)Defining phase difference variables as φ := θ − θ , φ := θ − θ and φ = θ − θ , we obtain the followingdynamical system for phase differences: ˙ φ = ω + g ( φ + φ − φ ) + g ( φ ) − g ( φ − φ ) − g (0)˙ φ = ω + g ( − φ − φ + φ ) + g ( φ ) − g ( − φ − φ ) − g ( − φ )˙ φ = ω + g ( φ − φ ) + g (0) − g ( − φ − φ ) − g ( − φ ) . (4) ω ij denotes the detuning between oscillator i and os-cillator j , namely ω ij = ω i − ω j .We first assume identical oscillators, that is ω = · · · = ω = ω = ⇒ ω ij = 0 ∀ i, j. (5)Oscillators with different natural frequencies will beconsidered in Section 3. Figure 1. An asymmetric coupled cell system: this gives the cou-pled system of form (3).
431 2 431 2(a) (b)
Figure 2. Two balanced colorings of the coupled cell system inFigure 1. These give rise to 2-dimensional invariant subspaces X and X of the system (4). The assumption that the oscillators are identicalmakes it possible to use the balanced coloring method[Stewart et al., 2003] to obtain invariant subspaces ofthe system (3). A coloring of cells in a coupled cellsystem is called balanced if cells with identical colorreceives the same number of inputs from cells of anygiven color. A balanced coloring gives rise to an in-variant subspace obtained by assuming that the cellsof same color have identical states. The converse ofthis statement is also true. Namely, for a given cou-pling structure, a coloring is balanced if the corre-sponding subspace is invariant under any system hav-ing that coupling structure. Therefore, the invariantsubspaces obtained by the balanced coloring methodare robust under the perturbations that preserve the cou-pling structure. For an introduction to this theory, see[Golubitsky and Stewart, 2006].Using the balanced coloring method the invariant sub-spaces of the coupled cell system given in Figure 1 canbe found as in Table 1. Using the above-mentionedphase difference reduction, the corresponding invari-ant subspace in T for the system (4) are also listedin Table 1. Note that for the system (4), there are onlytwo 2-dimensional invariant subspaces, namely X and X . Balanced colorings for these invariant subspacesare given in Figure 2. The invariant subspaces X , X and their intersection X can support a robust hete-roclinic cycle (see [Ashwin et al., 2011] for robustnesscriteria of heteroclinic cycles). alanced Invariant subspaces of the system (3)colorings on T and system (4) and on T { | | | } X = T ¯ X = T (whole space) { | | } X = { θ ∈ T | θ = θ } ¯ X = { φ ∈ T | φ = 0 } ( φ − φ plane) { | | } X = { θ ∈ T | θ = θ } ¯ X = { φ ∈ T φ = 0 } ( φ − φ plane) { | } X = { θ ∈ T | θ = θ , θ = θ } ¯ X = { φ ∈ T φ = φ = 0 } ( φ axis) { | } X = { θ ∈ T | θ = θ = θ } ¯ X = { φ ∈ T φ = φ = 0 } ( φ -axis) { } X = { θ ∈ T | θ = θ = θ = θ } ¯ X = { (0 , , } (origin) Table 1. Balanced colorings of the coupled system given in Fig-ure 1. The corresponding invariant subspaces for the system (3) andthe corresponding reduced invariant subspaces for the system (4) aregiven.
A heteroclinic ratchet (first defined in[Karabacak and Ashwin, 2010]) is a heteroclinicnetwork that contains a heteroclinic cycle winding insome direction and does not contain another hetero-clinic cycle winding in the opposite direction. To beprecise, a heteroclinic cycle C ⊂ T N parametrizedby x ( s ) ( x : [0 , → T N ) is winding in some di-rection if there is a projection map P : R N → R such that the parametrization ¯ x ( s ) (¯ x : [0 , → R N ) of the lifted heteroclinic cycle ¯ C ⊂ R N satisfies lim s → P (¯ x ( s )) − P (¯ x (0)) = 2 kπ for some positiveinteger k . A heteroclinic cycle winding in the oppositedirection would satisfy the same condition for anegative integer k (see [Ashwin and Karabacak, 2011]for general properties of heteroclinic ratchets).As discussed above, the system (4) may have a robustheteroclinic network on the invariant subspaces X and X . Such a heteroclinic network should be connectingsaddles on X = X ∩ X . Reducing the equations in(4) to X and considering identical natural frequencies,we get ˙ φ = g ( φ ) − g ( − φ ) . (6)This system is Z -equivariant, and therefore it can ad-mit a codimension-1 pitchfork bifurcation of the zerosolution under some nondegeneracy conditions. Thesaddles emanating from this bifurcation are on X and they are of the form p = (0 , , p ) and q =(0 , − p ) . The value p can be obtained by solving φ φ ( ) q qqq qq qqp p p pp φ φ (a) (b) π π φ φ φ π π p pp Figure 3. Phase portraits of the system (4) on invariant subspaces ¯ X (a) and ¯ X (b) are illustrated for parameters given in (8). Redlines indicate robust heteroclinic trajectories. Thick red lines arethe winding heteroclinic trajectories. The robust heteroclinic ratchetformed by these winding and non-winding heteroclinic trajectoriesand the saddles p and q is shown in (c). g ( p ) − g ( − p ) = 0 as p = cos − (cid:18) cos α r cos α (cid:19) . (7)In order to show that there exist heteroclinic connec-tions between p and q we use the simulation programXPPAUT [Ermentrout, 2002]. We identify a hetero-clinic ratchet for the parameter values α = − , r = 0 . (8)(see Figure 3). On X , the heteroclinic ratchet con-tains a non-winding trajectory and a trajectory windingalong + φ and − φ directions (see Figure 3a). On X ,it contains a non-winding trajectory and a trajectorywinding along + φ direction (see Figure 3b). Thesefour connections and the saddles p and q form a hetero-clinic ratchet (see Figure 3c). For the parameters givenin (8), p can be found as (0 , , . . Consideringthe Jakobien of (4) at p , we can find the eigenvalues ofthe saddle p as λ ( p )1 = − . , λ ( p )2 = 0 . and λ ( p )3 = − . . Similarly, the eigenvalues of q = − p = (0 , , . can be found as λ ( q )1 = 0 . and λ ( q )2 = − . and λ ( q )3 = − . . Theseeigenvalues of p and q correspond to the eigenvectors v = (1 , , ∈ X , v = (0 , , ∈ X and v =(0 , , ∈ X . A heteroclinic cycle is attracting if thesaddle quantity, defined as the absolute value of the ra-tio between the product of the eigenvalues correspond-ing to the expanding connections and the product of the
100 200 300 400 500t-6-4-20246 φ i φ φ φ Figure 4. A solution of the system (4) converging to the hetero-clinic ratchet. Initial states are chosen as (2 , , . . The solutionshows the peculiar property for heteroclinic cycles that the residencetime near equilibria increases as t → ∞ , before φ and φ getlocked at zero due to the precision errors. eigenvalues corresponding to the contracting connec-tions is smaller than 1 [Melbourne, 1989]. Hence, wecan conclude that the heteroclinic ratchet for the system(4) with parameters given in (8) is asymptotically sta-ble since the saddle quantity (cid:12)(cid:12)(cid:12)(cid:12) λ ( p )2 λ ( q )1 λ ( q )1 lambda ( p )2 (cid:12)(cid:12)(cid:12)(cid:12) = 0 . is less than one.A solution of (4) converging to the heteroclinic ratchetcan be seen in Figure 4. The increase in the residencetime near equilibria is typical for a solution convergingto a heteroclinic network. Winding of φ and φ occurat the same time, respectively in positive and negativedirections, due to the winding heteroclinic trajectory on X (see Figure 3a). Winding of φ occur in the positivedirection due to the winding heteroclinic trajectory on X (see Figure 3b). Since at each turn the solution getscloser to the equilibria p and q , after some time, φ and φ get locked at zero due to the precision errors. Notethat, the invariant subspaces X and X serve as bar-riers, and therefore no solution can pass through them.For this reason the solution winds in φ and φ direc-tions only one time. Since winding in − φ directionoccurs together with the winding in + φ direction, thisalso happens only one time. However, these barrierscan be broken by noise and/or detuning of natural fre-quencies leading to the uni-directional synchronizationphenomenon. We say that oscillators i and j are (frequency) syn-chronized if the observed frequency differences Ω ij = lim t →∞ | θ Li − θ Lj | t , (9)is equal to zero. Here θ Li ∈ R is the lifted phase vari-able for θ i ∈ T . It is know that coupled oscillators canget frequency synchronized when the distance between -0.03 -0.02 -0.01 0 0.01 0.02 0.03 ω -0.4-0.200.20.4 Ω ij Ω Ω Ω t -200-1000100200 φ i φ φ φ Figure 5. Uni-directional synchronization with respect to ω . Themain graph shows the frequency differences Ω , Ω and Ω for(3) with parameters given in (8) as a function of detuning ω when ω = ω = 0 . Oscillators are frequency synchronized when ω ≤ and the synchronization fails for oscillator pairs (13) and (24) whenever ω > . The insets show time evolution of thephase differences φ i for a positive value of ω . -0.03 -0.02 -0.01 0 0.01 0.02 0.03 ω Ω ij Ω Ω Ω Figure 6. Uni-directional synchronization with respect to ω . Thefrequency differences Ω , Ω and Ω for (3) with parametersgiven in (8) are shown as a function of detuning ω when ω = ω = 0 . Oscillators are frequency synchronized when ω ≤ and the synchronization fails for the oscillator pair (24) whenever ω > . their natural frequencies, namely | ω ij | := | ω i − ω j | issmall enough. If frequency synchronization of oscil-lators i and j occurs only when a specific one of theoscillators has greater natural frequency, namely fora certain sign of ω ij , we say that synchronization isuni-directional. Uni-directional synchronization phe-nomenon has been shown to occur for oscillator pairswhen an asymptotically stable heteroclinic ratchet ex-ists in the phase space [Karabacak and Ashwin, 2010;Ashwin and Karabacak, 2011].For the system (4), we investigate the effect of detun-ings ω , ω and ω on the synchronization of oscil-lators, respectively in Figure 5, 6 and 7. Due to thewinding connections in the heteroclinic ratchet, uni-directional synchronization occurs for detunings ω and ω . However, because of the connection windingboth in + φ and − φ directions, a positive detuning ω -0.1-0.500.50.1 Ω ij Ω Ω Ω Figure 7. Bi-directional synchronization (the usual case) with re-spect to ω . The frequency differences Ω , Ω and Ω for(3) with parameters given in (8) are shown as a function of detuning ω when ω = ω = 0 . Oscillators are frequency synchro-nized when | ω | is small enough and the synchronization fails forthe oscillator pair (34) for large values of | ω | .Detuning Natural Winding ObservedDirection Frequencies Direction Frequencies + ω ω + , ω, ω, ω + φ , − φ Ω + , Ω + , Ω , Ω + + ω ω, ω + , ω, ω + φ Ω , Ω + , Ω , Ω+ ω ω + , ω, ω + , ω Ω , Ω , Ω , Ω Table 2. The effect of detunings on the synchronization of oscil-lators for the system (4). These can be obtained from Figures 5, 6and 7. Negative detunings have not been considered as they have noeffect. ω + ( Ω + ) represents a number slightly larger than ω ( Ω ). ω leads to synchronization of oscillators , and .This is because θ − θ ∼ = − ( θ − θ ) = ⇒ θ ∼ = θ .The synchronized groups of oscillators for each detun-ing case are given in Table 2. It is interesting that theoscillators , and get synchronized for a positivedetuning of ω , although the space { θ ∈ T | θ = θ = θ } is not one of the synchronization spaces ob-tained by the balanced coloring method in Section 2.1.Hence, it is not an invariant subspace. Noise induced uni-directional desynchro-nization of oscillators has been observed in[Karabacak and Ashwin, 2010] for a coupled sys-tem admitting a heteroclinic ratchet. Here, we showthat existence of a heteroclinic ratchet for the system(4) leads to a locking in frequency differences whena small noise is applied. Figure 8 shows a solutionof the system (4) under white noise with amplitude − . The noisy solution exhibits approximatelyequal number of windings in φ and φ directions.This is because the noise is homogeneous and theinvariant subspace X (resp. X ) divides any ǫ -ballaround the equilibrium q (resp. p ) into two regionsof attractions of equal volume for the winding and φ i φ φ φ Figure 8. A solution of the system (4) for parameters given in(8) with no detuning and with additive white noise (amplitude =10 − ). non-winding trajectories. On the other hand, the num-ber of windings in φ and − φ directions are exactlythe same because of the structure of the heteroclinicratchet in Figure 3.As a result, the solution gives rise to the followingfrequency locking between frequency differences: Ω = Ω = − Ω . (10)This is in agreement with the simulation results givenin Figure 8. Therefore, the observed oscillator frequen-cies Ω i := lim t | θ Li ( t ) | /t are in the following form: Ω Ω Ω Ω = Ω + δ ΩΩ + δ Ω + 2 δ , (11)where δ is a positive number. This type of a result can-not be seen directly from the connection structure ofthe coupled system, and is a consequence of the par-ticular heteroclinic ratchet that the system admits. Al-though the noise induces synchronization of oscillators and , the balanced coloring method explained inSection 2.1 does not give { θ ∈ T | θ = θ } as aninvariant subspace. We have analyzed a system of four coupled phase os-cillators. The existence of an asymptotically stable het-eroclinic ratchet gives rise to uni-directional synchro-nization of certain groups of oscillators and induce aparticular locking in the frequency differences of oscil-lators when small amplitude white noise is introducedto the system. These phenomena also lead to frequencysynchronization of some oscillators, that can not befound by using the balanced coloring method, thereforedoes not correspond to any synchrony subspace.For future works, the relation between the connec-tion structure and possible synchronization groups cane studied. Although the synchronization groups cannot be inferred from the coupling structure directly, thecoupling structure serves to create invariant subspaceson which heteroclinic ratchets can be supported. Forthis reason, the coupling structure plays an indirect roleon the existence of possible synchronization groups.Another direction could be to study bifurcations of het-eroclinic ratchets that result in winding periodic orbitson torus. This can explain the effect of small detuningsof natural frequencies on the observed frequencies in acomplete way.
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