Homoclinic trajectories and chaotic behaviour in a piecewise linear oscillator

In this paper we consider the equation ẍ + x = sin(√2t) + s(x), where s(x) is a piece-wise linear map given by min{5x, 1} if x ≥ 0 and by max{-1, 5x} if x < 0. The existence of chaotic behaviour in the Smale sense inside the instability area is proven. In particular transversal homoclinic fixed point is found. The results follow from the application of topological degree theory the computer-assisted verification of a set of inequalities. Usually such proofs can not be verified by hands due to vast amount of computations, but the simplicity of our system leads to a small set of inequalities that can be verified by hand.