Geometric and dynamic perspectives on phase-coherent and noncoherent chaos

Abstract

Statistically distinguishing between phase-coherent and noncoherent chaotic dynamics from time series is a contemporary problem in nonlinear sciences. In this work, we propose different measures based on recurrence properties of recorded trajectories, which characterize the underlying systems from both geometric and dynamic viewpoints. The potentials of the individual measures for discriminating phase-coherent and noncoherent chaotic oscillations are discussed. A detailed numerical analysis is performed for the chaotic Rössler system, which displays both types of chaos as one control parameter is varied, and the Mackey-Glass system as an example of a time-delay system with noncoherent chaos. Our results demonstrate that especially geometric measures from recurrence network analysis are well suited for tracing transitions between spiral-and screw-type chaos, a common route from phase-coherent to noncoherent chaos also found in other nonlinear oscillators. A detailed explanation of the observed behavior in terms of attractor geometry is given. PACS numbers:Oscillatory processes can be frequently observed in natural and technological systems. Often, the corresponding dynamics is not strictly periodic, but shows more complex temporal variability patterns characterized by a fast divergence of trajectories with arbitrarily close initial conditions [1-3]. There are numerous examples of such chaotic oscillators for which long-term predictions of amplitudes and phases are not possible. Therefore, studying their phase dynamics has recently attracted particular interest, e.g., regarding the process of phase synchronization between different coupled systems [4,5]. However, most existing methods suitable for this purpose require the explicit definition of an appropriate phase variable, which can become a non-trivial problem in the case of noncoherent chaotic oscillations. Therefore, studying the phase coherence properties of chaotic systems has become an important problem in both theoretical and experimental studies [6]. In this work, we propose some methods based on the concept of recurrences in phase space, which allow studying complementary aspects of chaotic oscillators relating to the geometric structure of, and the dynamics on the attractor. Specifically, we derive a detailed characterization of changes of the geometric structure of complex systems in phase space with varying control parameter, which accompany transitions from phase-coherent to noncoherent dynamics.

Phase coherence and attractor geometry of chaotic electrochemical oscillators

“…The considered definition provides a generic way for studying geometric properties of chaotic attractors by means of complex network methods. 29,58,64 In turn, the properties of RNs do not capture the dynamics on the attractor. It is important to point out that a RN is an approximation of an underlying continuous graph with uncountably many vertices and edges associated to the attractor in the corresponding phase space.…”

“…Here, the chaotic attractor undergoes fundamental structural changes accompanying a transition between phase-coherent and noncoherent dynamics due to a collision of the growing chaotic attractor with some homoclinic orbit of the system, which is reflected by a sharp transition in the associated dynamical as well as geometric properties. 13,29 As a result, there is no simple linear transformation (or projection) of the system's coordinates that leads to rotations with a well-defined center. We conjecture that such an abrupt change is typical for a chaotic system with missing structural phase coherence.…”

“…In turn, for the Hilbert phase and related approaches, the correct estimation of measures such as the phase diffusion coefficient or coherence factor-particularly from experimental time series-can also be a challenging task. 29 This suggests that in the case of experimental applications, one should compare various definitions of phase coherence and carefully discuss which measure is the most appropriate in practice.…”

“…Specifically, the two recurrence time distributions closely resemble those obtained for the standard (phase-coherent) R€ ossler system and the associated (non-phase-coherent) funnel attractor. 29 Finally, we study the 2nd-order Renyi entropy K 2 computed from the RPs for different values of RR (see Ref. 46, for details on the used estimator).…”

“…[54][55][56] Finally, e-recurrence networks (RNs) provide a graphtheoretical framework for quantifying various aspects of the underlying attractor's geometry. 29,[57][58][59][60][61][62][63][64][65][66][67] In the following, we will use pðsÞ and K 2 for further characterizing the dynamical complexity of the observed electrochemical oscillations. For this purpose, we will restrict our attention to the first N ¼ 10 000 points of the embedded time series in order to keep the computational efforts at an acceptable level.…”