Chaotic threshold for the smooth-and-discontinuous oscillator under constant excitations
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The smooth-and-discontinuous oscillator under constant excitations (CSD) is an unsymmetrical system with an irrational restoring force, which leads to a barrier to detect the chaotic threshold using conventional nonlinear techniques. The goal of the present work is to overcome the trouble raised by constant excitations and irrational nonlinearity to investigate the necessary and insufficient condition for chaos. A smooth approximate system, whose behaviors are topologically equivalent to the ones of the original system, is proposed to investigate the chaotic boundary. The Hamiltonian system of the approximate system is analyzed and the homoclinic orbits in analytical form are obtained. The Melnikov method is employed to determine the distance between the stable and unstable manifolds under the perturbation of damping and external forcing. Bifurcation diagrams and numerical simulations are used to reveal the motion of chaos and the period for approximate system which is in good agreement with the original system. It is worth recalling that the approach for constructing the approximate function of smoothness is presented, which puts forward a step toward the solution of the irrational nonlinearity system.