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Complex dynamics in Josephson system with two external forcing terms. (English) Zbl 1149.34332
Summary: We consider the Josephson system $$\cases \dot x=y,\\ \dot y=-\sin x-k\sin 2x+\beta-\alpha(\cos x+2k\cos 2x)y+f_1\sin\omega_1t+f_2\sin\omega_2t.\endcases\tag{E_1}$$ By applying Melnikov’s method, we prove that criterion of existence of chaos under periodic perturbation. By second-order averaging method and the Melnikov method, we obtain the criterion of existence of chaos in averaged system under quasi-periodic perturbation for $\omega _{2} = \omega _{1} + \epsilon \nu$, and cannot prove the criterion of existence of chaos in averaged system under quasi-periodic perturbation for $\omega _{2} = n\omega _{1} + \epsilon \nu (n\geqslant 2$ and $n \in \Bbb N$), where $\nu$ is not rational to $\omega _{1}$. We also study the effects of the parameters of system on dynamical behaviors by using numerical simulation.

##### MSC:
 34C60 Qualitative investigation and simulation of models (ODE) 34D08 Characteristic and Lyapunov exponents 34C29 Averaging method 34C23 Bifurcation (ODE) 34C28 Complex behavior, chaotic systems (ODE) 37D45 Strange attractors, chaotic dynamics
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