Dynamical Systems Excited by Temporal Inputs: Fractal Transition Between Excited Attractors

This paper presents a framework for dissipative dynamical systems excited by external temporal inputs. 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 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.