Dynamical systems excited by temporal inputs: Fractal transition between excited attractors

This paper presents a framework for dissipative dynamical systems excited by external temporal nputs. We introduce a set {I-l} of temporal inputs with finite intervals. The set {I-l} defines two other sets of dynamical systems. The first is the set of continuous dynamical systems that are defined by a set {f(l)} 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 {g(l)} of iterated functions on the global Poincare section Sigma. 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.