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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.

34C60Qualitative investigation and simulation of models (ODE)
34D08Characteristic and Lyapunov exponents
34C29Averaging method
34C23Bifurcation (ODE)
34C28Complex behavior, chaotic systems (ODE)
37D45Strange attractors, chaotic dynamics
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