Switching-induced bifurcation analysis for piecewise nonlinear dynamical systems: A semi-analytical approach

In piecewise nonlinear dynamical systems, the flow is governed by different vector fields across discontinuous boundaries. When switching occurs at these boundaries, the governing vector field changes, inevitably causing the steady-state response to retain transient dynamics from individual subsystems. However, obtaining a complete analytical solution-including the transient component-for each nonlinear subsystem remains an unresolved challenge. This presents significant obstacles to finding unstable hidden bifurcation routes in such systems. The main objective of this paper is to obtain the complete bifurcation trees for piecewise nonlinear dynamical systems, enabling a comprehensive analysis of unconventional bifurcations induced by flow switching. A semi-analytical framework is proposed which integrates generalized mapping structures with local singularity theory to systematically characterize flow-switching dynamics at discontinuous boundaries. A generalized mapping formalism with closed-form constraint conditions at switching points is developed to parameterize periodic motions with higher-order singularities. To demonstrate the effectiveness of the proposed method, we analyze a piecewise nonlinear memristor circuit system exhibiting complex bifurcation and chaotic behaviors. This approach can be readily extended to study other piecewise nonlinear systems.