DYNAMICAL SYSTEMS EXCITED BY TEMPORAL INPUTS: FRACTAL TRANSITION BETWEEN EXCITED ATTRACTORS

Abstract

This paper presents a framework for dissipative dynamical systems excited by external temporal inputs. We introduce a set {Il} of temporal inputs with finite intervals. The set {Il} defines two other sets of dynamical systems. The first is the set of continuous dynamical systems that are defined by a set {fl} of vector fields on the hyper-cylindrical phase space ℳ. The second is the set of discrete dynamical systems that are defined by a set {gl} of iterated functions on the global Poincaré section Σ. When the inputs are switched stochastically, a trajectory in the space ℳ converges to an attractive invariant set with fractal-like structure. We can analytically prove this result when all of the iterated functions satisfy a contraction property. Even without this property, we can numerically show that an attractive invariant set with fractal-like structure exists.