FRACTALS AND CLOSURES OF LINEAR DYNAMICAL SYSTEMS EXCITED STOCHASTICALLY BY TEMPORAL INPUTS

Fractals and closures of two-dimensional linear dynamical systems excited by temporal inputs are investigated. The continuous dynamics defined by the set of vector fields in the cylindrical phase space is reduced to the discrete dynamics defined by the set of iterated functions on the Poincaré section. When all iterated functions are contractions, it has already been shown theoretically that a trajectory in the cylindrical phase space converges into an attractive invariant set with a fractal-like structure. Calculating analytically the Lipschitz constants of iterated functions, we show that, under some conditions, noncontractions often appear. However, we numerically show that, even for noncontractions, an attractive invariant set with a fractal-like structure exists. By introducing the interpolating system, we can also show that the set of trajectories in the cylindrical phase space is enclosed by the tube structure whose initial set is the closure of the fractal set on the Poincaré section.