Formation and destruction of multilayered tori in coupled map systems.

The paper first illustrates how multilayered tori can arise through one or more pitchfork bifurcations of the saddle cycle on an ordinary resonance torus. The paper hereafter describes three different scenarios by which a multilayered torus can be destructed. One scenario involves a saddle-node bifurcation in which the middle layer of a three-layered torus disappears in an abrupt transition to chaos while the outer-layer manifolds and their associated saddle and unstable-focus cycles continue to exist and to control the transient dynamics. In a second scenario, the unstable focus cycles of the intermediate layers in a five-layered torus turn into unstable nodes, and closed loop connections are established between the unstable nodes and the points of the stable resonance node on the torus. Finally, a third scenario describes a transition in which homoclinic bifurcations destroy first the outer layers and thereafter also the inner layer. The paper also illustrates how the formation and destruction of multilayered tori can occur in the cluster dynamics of an ensemble of globally coupled maps. This leads to three additional scenarios for the destruction of multilayered tori.