A Phase Description of Mutually Coupled Chaotic Oscillators

Abstract

The synchronization of rhythms is ubiquitous in both natural and engineered systems, and the demand for data-driven analysis is growing. When rhythms arise from limit cycles, phase reduction theory shows that their dynamics are universally modeled as coupled phase oscillators under weak coupling. This simple representation enables direct inference of inter-rhythm coupling functions from measured time-series data. However, strongly rhythmic chaos can masquerade as noisy limit cycles. In such cases, standard estimators still return plausible coupling functions even though a phase-oscillator model lacks a priori justification. We therefore extend the phase description to the chaotic oscillators. Specifically, we derive a closed equation for the phase difference by defining the phase on a Poincaré section and averaging the phase dynamics over invariant measures of the induced return maps. Numerically, the derived theoretical functions are in close agreement with those inferred from time-series data. Consequently, our results justify the applicability of phase description to coupled chaotic oscillators and show that data-driven coupling functions retain clear dynamical meaning in the absence of limit cycles.

Figures (6)

• Figure 1: Rhythmic time series are often analyzed under the assumption of an underlying limit cycle, but may be generated by rhythmic chaos. (a) Nearly periodic signals with sharp spectral peaks appear consistent with the phase description. (b) Under this assumption, a standard data-driven procedure infers an effective interaction function from the observed time series (green arrow). (c) However, similar rhythmic time series can be generated by a pair of mutually coupled chaotic oscillators (purple arrow). In this case, the limit-cycle hypothesis underlying the inference is violated, making the validity of the inferred coupling function questionable.
• Figure 2: Schematic of the proposed theory. (a) We consider two weakly coupled chaotic oscillators that can exhibit phase synchronization. (b) For each oscillator, we define a cross section $\mathcal{S}$ and decompose the dynamics into an amplitude $R$ and a phase $\phi$. Assuming a time-scale separation where the amplitude dynamics evolve much faster than the phase difference $\psi$, we treat $\psi$ as effectively constant over a single period. To capture the chaotic nature of the amplitude fluctuations, we construct two invariant measures, $\mu$ and $\nu$, for the induced return maps. (c) Averaging the phase dynamics with respect to these measures yields the effective phase-difference dynamics.
• Figure 3: (a) A comparison of the theoretically derived interaction function (red line) and the statistically inferred coupling function (blue line) for the coupled Lorenz system [Eq. \ref{['eq:double lorenz']}]. The parameters are set to $(\alpha,\varepsilon)=(1.0002,0.02)$. The inferred function corresponds to the one shown in Fig. \ref{['fig:intro']}(b). (b) The grayscale map illustrates the frequency difference on the $(\alpha,\varepsilon)$ plane, where the white region indicates the Arnold tongue. The insets confirm that our theory accurately predicts the coupling functions even near the chaos-specific distortion of the Arnold tongue.
• Figure 4: Comparison results for the heterogeneous system composed of Rössler and Sprott--N oscillators [Eq. \ref{['eq:rossler sprottn']}]. Same as Fig. \ref{['fig:result_lorenz']}, but for the heterogeneous case. Theoretically derived (red) and statistically inferred (blue) coupling functions. (b) The grayscale map representing the frequency difference. The insets demonstrate the agreement between theory and inference for representative parameter sets both inside and outside the synchronization region.
• Figure 5: The trajectories of the oscillators (gray), the selected cross sections (green), and the phases determined by the cross sections (the 2D projection shown below each oscillator). The lower panels display the return time histograms for the selected cross sections. For each histogram, the scale $T$ of the horizontal axis represents the average return time to the cross section. The vertical axis shows the density. From left to right: the Lorenz oscillator, the Rössler oscillator, and the Sprott-N oscillator.