Chaotic thresholds for the piecewise linear discontinuous system with multiple well potentials

• We investigate the bifurcations and the chaos of a piecewise linear discontinuous (PWLD) system.
• All of the solutions of equilibria, periodic orbits and homoclinic-like and heteroclinic-like orbits are obtained for the unperturbed system.
• The chaotic solutions are obtained analytically for the perturbed PWLD system.
• The Melnikov method to detect the chaotic behavior analytically from the broken of the homoclinic-like and heteroclinic-like orbits.
• The numerical results presented herein this paper show the complicated non-linear dynamics.

In this paper we investigate the bifurcations and the chaos of a piecewise linear discontinuous (PWLD) system based upon a rig-coupled SD oscillator, which can be smooth or discontinuous (SD) depending on the value of a system parameter, proposed in [18], showing the equilibrium bifurcations and the transitions between single, double and triple well dynamics for smooth regions. All solutions of the perturbed PWLD system, including equilibria, periodic orbits and homoclinic-like and heteroclinic-like orbits, are obtained and also the chaotic solutions are given analytically for this system. This allows us to employ the Melnikov method to detect the chaotic criterion analytically from the breaking of the homoclinic-like and heteroclinic-like orbits in the presence of viscous damping and an external harmonic driving force. The results presented here in this paper show the complicated dynamics for PWLD system of the subharmonic solutions, chaotic solutions and the coexistence of multiple solutions for the single well system, double well system and the triple well dynamics.