Three Types of Transitions to Phase Synchronization in Coupled Chaotic Oscillators

We study the effect of noncoherence on the onset of phase synchronization of two coupled chaotic oscillators. Depending on the coherence properties of oscillations characterized by the phase diffusion, three types of transitions to phase synchronization are found. For phase-coherent attractors this transition occurs shortly after one of the zero Lyapunov exponents becomes negative. At rather strong phase diffusion, phase locking manifests a strong degree of generalized synchronization, and occurs only after one positive Lyapunov exponent becomes negative. For intermediate phase diffusion, phase synchronization sets in via an interior crises of the hyperchaotic set.

Figure 1

Upper panel (a)–(c): projections of the attractors of the Rössler systems (1) onto the plane $(x,y)$; middle panel (d)–(f): projections onto $(\dot{x},\dot{y})$; lower panel (g)–(i): distribution of the return times $T$. The parameters are $\omega =0.98$ and $a=0.16$ (a),(d),(g), $a=0.22$ (b),(e),(h), and $a=0.28$ (c),(f),(i).

Figure 2

Phase diffusion coefficient $D_{\varphi }$ (3) vs $a$. $\omega =0.98$.

Figure 3

Critical coupling curves. $l_1$ corresponds to the onset of CPS, i.e., below this line the oscillations are not synchronized, and above this line the phase and frequency locking conditions are fulfilled; $l_2$ corresponds to the transition of one of the zero LEs negative values; and $l_3$ corresponds to zero crossing of one of the positive LEs. Note that in this figure we do not separate the cases where the synchronization occurs between regular and chaotic oscillations.

Figure 4

Projections of trajectories of the Rössler systems for $a=0.22$ on the plane $({\varphi }_1,{\varphi }_2)$ for the coupling strength $d$ outside (a) ($d=0.055$) and within (b) ($d=0.075$) the synchronization region.

Figure 5

(a) Time evolution of phase difference. (b) Variables ${\dot{y}}_{1,2}$ in system (1) for $a=0.2925$ and $d=0.179\in [d_{l_3},d_{l_1}]$. Solid and dotted lines correspond to the first and second oscillators, respectively. In the time interval between dashed lines the first oscillator produces four rotations in the $({\dot{x}}_1,{\dot{y}}_1)$ plane around the origin, but the second oscillator generates only three rotations, which leads to a phase slip in (a).

Figure 6

Mean frequency ratio ${\Omega }_2/{\Omega }_1$ versus coupling. The fitting curve ($\sim |d-d_{l_1}{|}^{0.5}$) for $a=0.16$ (dashed line). The fitting straight lines for $a=0.22$ (solid line) and $a=0.28$ (dotted line) are presented.