Bifurcations of the symmetric quasi-periodic motion and Lyapunov dimension of a vibro-impact system

Abstract

Under the suitable parameter combination, the vibro-impact system with symmetry exhibits symmetric quasi-periodic motions in the vicinity of Neimark–Sacker bifurcation (i.e., NS bifurcation) point satisfying 1:2 resonant conditions and double Neimark–Sacker bifurcation (i.e., NS–NS bifurcation) point. Based on the fact that the Poincare map P is the second iteration of another virtual implicit map Q, the coexistence and symmetry of the attractor of the Poincare map is discussed by means of the limit set theory. The map Q can capture two conjugate limit sets and hence can capture two conjugate quasi-periodic motions. Based on the Poincare map, QR method is used to compute Lyapunov dimension, which can be used to characterize various quasi-periodic motions. Numerical simulation shows that the symmetric quasi-periodic motion can lose the stability and bifurcate into various subharmonic quasi-periodic motions via period-doubling bifurcation. Torus bifurcation of the symmetric quasi-periodic motion can also take place, which induces to a symmetric torus in the Poincare section. Bifurcation of quasi-periodic motion will give birth to the coexistence of quasi-periodic motion. It is shown that all quasi-periodic attractors have at least a zero Lyapunov exponent, but various quasi-periodic attractors have different Lyapunov dimensions.