The dynamics of a symmetric impact oscillator between two rigid stops

A single-degree-of-freedom symmetric impact oscillator between two rigid stops is considered. The system is strongly nonlinear due to the existence of impacts. The symmetric period n−2n−2 motion of the system is obtained analytically, and the Poincaré map is established. The dynamics of the system can be investigated by studying the symmetric fixed point, because symmetric period motion corresponds to a symmetric fixed point of the Poincaré map. Because of the symmetry of the Poincaré map, a symmetric fixed point of the Poincaré map only has pitchfork bifurcation. The stability of the symmetric fixed point is determined by the eigenvalues of the Jacobian matrix of the Poincaré map. For an attractor in the Poincaré section, all the Lyapunov exponents can be computed via the Jacobian matrix of the Poincaré map. Numerical simulations show that the symmetric period motion is stable for larger values of excitation frequency, smaller values of excitation amplitude and smaller values of mass. When the control parameter changes continuously, the symmetric fixed point loses its stability, and generates a pair of antisymmetric period-doubling sequences via pitchfork bifurcation, which subsequently lead to symmetric chaos. The top Lyapunov exponent can be used to distinguish long periodic motion from chaotic motion.