Low-dimensional maps for piecewise smooth oscillators

Abstract

Dynamics of the piecewise smooth nonlinear oscillators is considered, for which, general methodology of reducing multidimensional flows to low-dimensional maps is proposed. This includes a definition of piecewise smooth oscillator and creation of a global iterative map providing an exact solution. The global map is comprised of local maps, which are constructed in the smooth sub-regions of phase space. To construct this low-dimensional map, it is proposed to monitor the points of intersections of a chosen boundary between smooth subspaces by a trajectory. The dimension reduction is directly related to the dimension of the chosen boundary, and the lower its dimension is, the larger dimension reduction can be achieved. Full details are given for a drifting impact oscillator, where the five-dimensional flow is reduced to one-dimensional (1D) approximate analytical map. First an exact two-dimensional map has been formulated and analysed. A further reduction to 1D approximate map is introduced and discussed. Standard nonlinear dynamic analysis reveals a complex behaviour ranging from periodic oscillations to chaos, and co-existence of multiple attractors. Accuracy of the constructed maps is examined by comparing with the exact solutions for a wide range of the system parameters.