Synchronization of oscillators with hyperbolic chaotic phases

Synchronization in a population of oscillators with hyperbolic chaotic phases is studied for two models. One is based on the Kuramoto dynamics of the phase oscillators and on the Bernoulli map applied to these phases. This system possesses an Ott-Antonsen invariant manifold, allowing for a derivation of a map for the evolution of the complex order parameter. Beyond a critical coupling strength, this model demonstrates bistability synchrony-disorder. Another model is based on the coupled autonomous oscillators with hyperbolic chaotic strange attractors of Smale-Williams type. Here a disordered asynchronous state at small coupling strengths, and a completely synchronous state at large couplings are observed. Intermediate regimes are characterized by different levels of complexity of the global order parameter dynamics.


I. INTRODUCTION
Synchronization of chaotic oscillators has many aspects [1], one generally distinguishes complete, generalized, and phase synchronization. The latter property is related to chaotic oscillators with well-defined phases. Many chaotic oscillators, like the Rössler system, possess chaotic amplitudes, while the phase in such systems is not chaotic and corresponds to a zero Lyapunov exponent. The dynamics of the phase is of diffusion type, and correspondingly the phase synchronization phenomena for such oscillators are close to those for periodic oscillators with a certain level of noise in the phase dynamics.
In a seminal paper [2] S. P. Kuznetsov constructed a physical model of an oscillator with a chaotic phase. In this construction, the process has amplitude modulation, and at each period of modulation the phase experience a doubling map. The overall attractor is hyperbolic and belongs to a Smale-Williams solenoid class. In a series of subsequent publications, summarized in the book [3], S. P. Kuznetsov and co-authors provided many examples of systems with hyperbolic phase chaos.
In this paper we study synchronization properties of the oscillators with chaotic phases.
First, we construct a rather abstract model, where phase chaos and synchronizing interactions are separated in time (Section II). Namely, the process consists of two epochs: in one epoch phase oscillators interact according to the Kuramoto global coupling scheme, and in another epoch the phases undergo a chaotic Bernoulli map. This model demonstrates, for certain values of parameters, a bistability between a desynchronized and a synchronized states. In Section III we consider coupled autonomous oscillators with chaotic phases, constructed by S. P. Kuznetsov and the author in [4]. This system demonstrates a rather reach behavior with asynchronous, completely synchronous, and complex partially synchronous states.

II. KURAMOTO-BERNOULLI MODEL
In this section we construct a model of interacting phase oscillators, which combines features of the Kuramoto model [5] (global attractive coupling of the phases) with the hyperbolic chaotic dynamics of the phases described by a Bernoulli map.

A. Kuramoto ensemble and OA evolution
Consider N phase oscillators ϕ k interacting via Kuramoto mean-field couplinġ Here Z is the complex mean field, and µ is the coupling constant. Quantity R is called Kuramoto order, it characterizes asynchronous (R = 1) and synchronous (R = 1) regimes.
We assume that all the oscillators have the same frequency, and write equations in the reference frame where this frequency vanishes, so it does not enter in (1). For µ > 0 the coupling is attractive, and in this situation all the oscillators eventually synchronize: R → 1, and a state where ϕ 1 = ϕ 2 = . . . ϕ N establishes.
Synchronization transition is monotonous (in fact, there exists a Lyapunov function that governs it), but it can be generally hardly expressed analytically. An analytic solution is, however, possible, if the Ott-Antonsen (OA) ansatz [6], which applies to the thermodynamic limit N → ∞, is peformed. In the OA ansatz it is assumed that the distribution of the phases is a wrapped Cauchy distribution, and the complex circular moments can all be expressed via the complex mean field Z k = Z k . Then the equation for the order Evolution of the complex mean field during a time interval T is One can see that the only parameter in this transformation is γ = exp(−µT ). Evidently, R → 1 as n → inf ty, and the rate of this convergence is larger for smaller γ.

B. Bernoulli map of phases
Consider a Bernoulli map acting on the phases with an integer parameter K. For an ensemble of Bernoulli maps (4), it is easy to express the evolution of the probability density of phases through the complex circular moments (2): One can see that the OA ansatz is invariant under Bernoilli maps.
Thus, the evolution of the complex mean field under the Bernoulli map is

C. Kuramoto ensemble and Bernoilli map
We construct a Kuramoto-Bernoulli (KB) model as a sequence of applications of the Kuramoto dynamics (3) and of the Bernoulli dynamics (5). Application of the expressions (3), (5) leads to the following map for the order parameter This map has always a stable asynchronous fixed point R as = 0, and a synchronous fixed in this case also an unstable partially synchronous fixed point with 0 < R ps < 1 exists, so there is a bistability asynchrony-synchrony.
The threshold for synchrony stability (6)   We illustrate the dynamics of the KB model in Fig. 1. There we show the evolution of the oder parameter R for different values of parameter γ and different initial states. The fully synchronous state is absorbing (exactly the same phases remain the same) for all system sizes N , while there are finite-size fluctuations around the disordered state. For small γ, one observes a finite-size induced transition to the synchronous state.

III. GLOBALLY COUPLED KUZNETSOV-PIKOVSKY (KP) OSCILLATORS
Here we study globally coupled chaotic phase oscillators introduced by S. P. Kuznetsov and the author in Ref. [4].
Below we fix the internal coupling parameter ε = 0.075. For ε = 0, system (7) has a stable homoclinic cycle, where the modes are excited consequentially w → v → u → w → . . ., with increasing periods of the cycle. The effect of coupling ε > 0 is twofold: first, the cycle period is limited from above (see Fig. 2(b)), and second, at each stage where a mode amplitude passes close to zero, its phase attains the doubled value of the exciting mode. The latter property is described in Ref. [4] in details; here we illustrate it with figure 3. Thus, the KP oscillator (7) has a chaotic phase obeying a Bernoulli map.

B. Globally coupled KP oscillators
Here we introduce a global coupling of N oscillators (numbered by index k), such that complete synchrony is possible: The coupling term is proportional to the parameter µ, it contains three complex mean fields U, V, W , corresponding to three modes of each oscillator. I calculated the time average |Z| t and its fluctuations (|Z| − |Z| t ) 2 t , these quantities are shown in Fig. 4 with red (fluctuations as error bars). Additionally, for each moment of time, I calculated the spread in the ensemble and then average this quantity over time. This quantity is shown with blue.
Below I describe different states on the bifurcation diagram.
1. Complete synchronization. This regime is observed for µ > 0.95. Here u k = u j , v k = v j , w k = w j for all k, j. In this state D = 0.
2. Asynchronous state. This regime is observed for µ < ∼ 0.22. Here the mean field vanishes, and one has effectively a set of non-interacting oscillators.
3. Periodic mean field. This regime is observed in the range 0.22 < ∼ µ < ∼ 0.31. Here the complex mean field Z(t) is nearly periodic. We illustrate this in Fig. 5(a,b). There are visible fluctuations for µ = 0.23, but for µ = 0.3 periodicity is nearly perfect. The transition to a periodic mean field at µ ≈ 0.22 is very much similar to one described in Ref. [7].  Fig. 5(c). At µ = 0.35 the mean field is close to periodic one, but has a seemingly nearly quasiperiodic modulation.

Irregular mean field
This state is observed for 0.4 < ∼ µ < ∼ 0.95, we illustrate it in Fig. 6 (a-c). Fluctuations of the mean field are essential, eventually for large µ they become close to the fluctuations of the field z(t) in one chaotic oscillator.

IV. DISCUSSION
In this paper I studied effects of coupling on oscillators with hyperbolic chaotic dynamics of the phases. In the simplest, rather artificial Kuramoto-Bernoulli model, an exact mapping for the order parameter has been derived in the Ott-Antonsen approximation in the thermodynamic limit. The dynamics here, beyond a certain level of coupling, is bistable: syn- chronous and asynchronous states coexist. In relatively small ensembles, for strong enough coupling, only synchronous states survives as it is a truly absorbing one. A more realistic model of coupled autonomous continuous-time oscillators with hyperbolic dynamics of the phases demonstrated much more rich dynamics. Together with a fully asynchronous state at small coupling strengths, and a completely synchronous at strong coupling strengths, it demonstrates different states with partial synchrony. Close to the asynchronous state, the mean field is nearly periodic; and with increase of coupling strength it becomes irregular through presumably a quasi-periodic state. Detailed study of partially synchronous states in this model will be a subject of a separate study.