On the origin of chaotic attractors with two zero Lyapunov exponents in a system of five biharmonically coupled phase oscillators

TL;DR

This paper investigates the chaotic behavior of five coupled phase oscillators with biharmonic coupling, revealing the existence of strange attractors with two zero Lyapunov exponents near a specific bifurcation point, explained through bifurcation analysis.

Contribution

It demonstrates the emergence of strange spiral attractors with two zero Lyapunov exponents in a system of five coupled oscillators and links this phenomenon to a codimension three bifurcation and the ACST system.

Findings

Chaotic attractors with two zero Lyapunov exponents are found in the system.

Bifurcation analysis links the attractors to a codimension three bifurcation.

The observed dynamics relate to the ACST system's behavior.

Abstract

We study chaotic dynamics in a system of four differential equations describing the dynamics of five identical globally coupled phase oscillators with biharmonic coupling. We show that this system exhibits strange spiral attractors (Shilnikov attractors) with two zero (indistinguishable from zero in numerics) Lyapunov exponents in a wide region of the parameter space. We explain this phenomenon by means of bifurcation analysis of the three-dimensional Poincar\'e map for the system under consideration. We show that the chaotic dynamics develop here near a codimension three bifurcation, when a periodic orbit (fixed point in the Poincar\'e map) has the triplet $(1, 1, 1)$ of multipliers. As it is known, the asymptotic flow normal form for this bifurcation coincides with the three-dimensional Arneodo-Coullet-Spiegel-Tresser (ACST) system in which spiral attractors exist. Based on this, we conclude that the additional near-zero Lyapunov exponent for orbits in the observed attractors appear due to the fact that the corresponding three-dimensional Poincar\'e map is close to the time-shift map of the three-dimensional ACST-system.