Smooth and Non-Smooth Dependence of Lyapunov Vectors upon the Angle Variable on a Torus in the Context of Torus-Doubling Transitions in the Quasiperiodically Forced Henon Map

摘要

A transition from a smooth torus to a chaotic attractor in quasiperiodicallyforced dissipative systems may occur after a finite number of torus-doublingbifurcations. In this paper we investigate the underlying bifurcationalmechanism which seems to be responsible for the termination of thetorus-doubling cascades on the routes to chaos in invertible maps underexternal quasiperiodic forcing. We consider the structure of a vicinity of asmooth attracting invariant curve (torus) in the quasiperiodically forced Henonmap and characterize it in terms of Lyapunov vectors, which determinedirections of contraction for an element of phase space in a vicinity of thetorus. When the dependence of the Lyapunov vectors upon the angle variable onthe torus is smooth, regular torus-doubling bifurcation takes place. On theother hand, the onset of non-smooth dependence leads to a new phenomenonterminating the torus-doubling bifurcation line in the parameter space with thetorus transforming directly into a strange nonchaotic attractor. We argue thatthe new phenomenon plays a key role in mechanisms of transition to chaos inquasiperiodically forced invertible dynamical systems.