Torus Doublings in Symmetrically Coupled Period-Doubling Systems

Abstract

As a representative model for Poincaré maps of coupled period-doubling oscillators, we consider two symmetrically coupled Hénon maps. Each invertible Hénon map has a constant Jacobian b (0 < b < 1) controlling the "degree" of dissipation. For the singular case of infinite dissipation (b = 0), it reduces to the non-invertible logistic map. Instead of period-doubling bifurcations, antiphase periodic orbits (with a time shift of half a period) lose their stability via Hopf bifurcations, and then smooth tori, encircling the anti-phase mother orbits, appear. We study the fate of these tori by varying b. For large b, doubled tori are found to appear via torus doubling bifurcations. This is in contrast to the case of coupled logistic maps without torus doublings. With decreasing b, mechanisms for disappearance of torus doublings are investigated, and doubled tori are found to be replaced with simple tori, periodic attractors, or chaotic attractors for small b. These torus doublings are also observed in two symmetrically coupled pendula that individually display a period-doubling transition to chaos.

Lag Synchronization in Coupled Multistable van der Pol-Duffing Oscillators

“…For = 13.0125 subsystems are on one attractor and for = 13.013 they group equally on two tori. This continues for ∈ [ 2 , 3 ) and for 3 we obtain another bifurcation known as the doubling of torus [34,35]. Previously, subsystems are grouped equally on the attractors and for ∈ [ 3 , 4 ) there are no qualitative changes in the dynamics.…”