Existence of homoclinic orbits in three-dimensional piecewise linear forced oscillator systems

It is a challenging task to prove mathematically the existence of homoclinic orbits in a high dimensional piecewise-smooth dynamical system. Motivated by [Llibre et al., IJBC, 2007], this paper focuses on the existence of Shilnikov type homoclinic orbits connecting the saddle-focus equilibrium in three-dimensional continuous piecewise linear forced oscillator systems. Based on the half Poincaré map and invariant manifold theory, we can establish the general conditions on the existence of homoclinic orbits, and give the exact parameter analysis for the existence of such homoclinic orbits by rigorously mathematical analysis and symbolic calculations. Finally, some exact numerical examples are presented to illustrate our results.