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Vibration of generalized double well oscillators. (English) Zbl 1136.34044
This paper gives an application of the well-known Melnikov method to a practical mechanical system. The authors consider a double well dynamical system with a nonlinear fractional damping term and external excitation, i.e., \[ \ddot x+\alpha \dot x| \dot x| ^{p-1}+\delta x+ \gamma \text{ sgn}(x)| x| ^{q-1}=\mu \cos\omega t, \] where \(p>0,q> 2\) are real numbers. They present the Melnikov function with infinite integrals of hyperbolic functions and authors calculate simple zeros of the Melnikov function examine homoclinic bifurcations leading to chaos in this system.

MSC:
34C28 Complex behavior and chaotic systems of ordinary differential equations
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
37G20 Hyperbolic singular points with homoclinic trajectories in dynamical systems
74H45 Vibrations in dynamical problems in solid mechanics
74H65 Chaotic behavior of solutions to dynamical problems in solid mechanics
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