Analytical proof on the existence of chaos in a generalized Duffing-type oscillator with fractional-order deflection

This paper gives a mathematically analytical proof on the existence of chaos in a generalized Duffing-type oscillator with fractional-order deflection. The analytical expressions of the two homoclinic orbits which connect a hyperbolic saddle point are solved. Then, Melnikov’s procedure for the definition of the criteria for transversal intersection of the stable and unstable manifolds is shown. In this process, we found that the expressions of the Melnikov functions could not be solved analytically, mainly because the homoclinic orbits are highly complicated. To this end, an effective numerical algorithm is proposed to compute the corresponding Melnikov functions, and, using the presented algorithms, the critical parameter curve for the existence of chaos in the Smale horse sense is shown. We also give the simulation graph depicting the intersection of stable and unstable manifolds, which can make the generation mechanism of chaotic dynamics more clear. Therefore, we provide a rigorous theoretical foundation to support studies and applications of this important class of dynamical systems.