Complex dynamics in Josephson system with two external forcing terms

Josephson system with two external forcing terms is investigated. By applying Melnikov method, we prove that criterion of existence of chaos under periodic perturbation. By second-order averaging method and Melnikov method, we obtain the criterion of existence of chaos in averaged system under quasi-periodic perturbation for ω₂ = ω₁ + ε{lunate}ν, and cannot prove the criterion of existence of chaos in averaged system under quasi-periodic perturbation for ω₂ = nω₁ + ε{lunate}ν (n {greater than or slanted equal to} 2 and n ∈ N), where ν is not rational to ω₁. We also study the effects of the parameters of system on dynamical behaviors by using numerical simulation. The numerical simulations, including bifurcation diagram of fixed points, bifurcation diagram of system in three- and two-dimensional space, homoclinic and heteroclinic bifurcation surface, Maximum Lyapunov exponent, phase portraits, Poincaré map, are also plotted to illustrate theoretical analysis, and to expose the complex dynamical behaviors, including the period-n (n = 1, 2, 5, 7) orbits in different chaotic regions, cascades of period-doubling bifurcation from period-1, 2 and 5 orbits, reverse period-doubling bifurcation, onset of chaos which occurs more than once for two given external frequencies and chaos suddenly converting to periodic orbits, transient chaos with complex periodic windows and crisis, reverse period-5 bubble, non-attracting chaotic set and nice attracting chaotic set. In particular, we observe that the system can leave chaotic region to periodic motion by adjusting damping α, amplitude f₁ and frequency ω₂ of external forcing which can be considered as a control strategy.