Vibration of Generalized Double Well Oscillators

Abstract

We have applied the Melnikov criterion to examine a global homoclinic bifurcation and transition to chaos in a case of a double well dynamical system with a nonlinear fractional damping term and external excitation. The usual double well Duffing potential having a negative square term and positive quartic term has been generalized to a double well potential with a negative square term and a positive one with an arbitrary real exponent $q > 2$. We have also used a fractional damping term with an arbitrary power $p$ applied to velocity which enables one to cover a wide range of realistic damping factors: from dry friction $p \to 0$ to turbulent resistance phenomena $p=2$. Using perturbation methods we have found a critical forcing amplitude $\mu_c$ above which the system may behave chaotically. Our results show that the vibrating system is less stable in transition to chaos for smaller $p$ satisfying an exponential scaling low. The critical amplitude $\mu_c$ as an exponential function of $p$. The analytical results have been illustrated by numerical simulations using standard nonlinear tools such as Poincare maps and the maximal Lyapunov exponent. As usual for chosen system parameters we have identified a chaotic motion above the critical Melnikov amplitude $\mu_c$.