Phase synchronization of chaotic systems with small phase diffusion

Abstract

The geometric theory of phase locking between periodic oscillators is extended to phase coherent chaotic systems. This approach explains the qualitative features of phase locked chaotic systems and provides an analytical tool for a quantitative description of the phase locked states. Moreover, this geometric viewpoint allows us to identify obstructions to phase locking even in systems with negligible phase diffusion, and to provide sufficient conditions for phase locking to occur. We apply these techniques to the Rössler system and a phase coherent electronic circuit and find that numerical results and experiments agree well with theoretical predictions.

Phase synchronization and topological defects in inhomogeneous media

“…Recently, attention has turned to the study of coupled chaotic oscillators and, in particular, to the phenomenon of phase synchronization. Provided that the phase can be defined [4,5], two coupled nonidentical chaotic oscillators are said to be phase synchronized if their frequencies are locked [1,6]. This appears to be a general phenomenon and it has been observed in such diverse systems as electrically coupled neurons [7,8], biomedical systems [9], chemical systems [10], and spatially extended ecological systems [11], to name only a few.…”

Rapid convergence of time-averaged frequency in phase synchronized systems

“…The definition is ambiguous because it depends on the choice of the Poincaré surface. Yet, any choice of a phase variable for chaotic oscillators investigated in this paper leads to the same macroscopic behavior [5]. With this phase definition, the phase dynamics can be described by…”

“…Recently, a considerable amount of research has been devoted to the study of coupled chaotic oscillators and, in particular, to the phenomenon of phase synchronization. Provided that the phase can be defined [4,5], two coupled nonidentical chaotic oscillators are said to be phase synchronized if their frequencies are locked but their amplitudes are not [1,6]. This appears to be a general phenomenon and it has been observed in systems as diverse as electrically coupled neurons [7,8], biomedical systems [9], chemical systems [10], and spatially extended ecological systems [11].…”

Data-driven optimal filtering for phase and frequency of noisy oscillations: Application to vortex flow metering

“…The problem of defining a steady, local phase when the internal dynamics of a deterministic, chaotic oscillator are known was treated rigorously in Ref. [20]. But as internal-and measurement noise become stronger, some temporal averaging is required and locality in time has to traded for accuracy and/or unambiguity.…”