Torus-doubling bifurcations and strange nonchaotic attractors in a vibro-impact system

Abstract

Whether strange nonchaotic attractors (SNAs) can occur typically in dynamical systems other than quasiperiodically driven systems has been an open problem. Here we show that the SNAs can be induced by the interruption of torus-doubling bifurcation in a periodically driven vibro-impact system. It is found that the creation of SNAs occurs due to the collision of doubled torus with some unstable periodic orbits. Based on the Poincaré map technique, two types of codimension-3 bifurcations are represented and typical torus-doubling bifurcations can occur in the neighborhood of these bifurcation points. We use the singular continuous Fourier spectrum and its scaling to verify the strangeness (nondifferentiability) of the attractors. The implicitly transcendental map is used to calculate the largest Lyapunov exponent by QR-based algorithm, which ensures that the underlying dynamics (with nonpositive Lyapunov exponents) is nonchaotic. The associated mechanism is described for the creation of SNAs through detecting unstable periodic orbits from Poincaré map series.