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

Abstract

We study chaotic dynamics in a system of four differential equations describing the interaction of five identical phase oscillators coupled via biharmonic function. 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 a three-dimensional Poincaré map for the system under consideration. We show that chaotic dynamics develop here near a codimension three bifurcation, when a periodic orbit (fixed point of the Poincaré map) has the triplet of multipliers ( 1 , 1 , 1 ). As it is known, the flow normal form for such bifurcation is the well-known three-dimensional Arneodó-Coullet-Spiegel-Tresser (ACST) system, which exhibits spiral attractors. According to this, we conclude that the additional zero Lyapunov exponent for orbits in the observed attractors appears due to the fact that the corresponding three-dimensional Poincaré map is very close to the time-shift map of the ACST-system.