Torus Doublings in Symmetrically Coupled Period-doubling Systems

As a representative model for Poincare maps of coupled period-doubling oscillators, we consider two symmetrically Coupled Henon maps. Each invertible Henon 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, anti-phase 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.