Complex Dynamics In Josephson System With Parametric And External Excitations

Josephson system with parametric and external excitations is investigated in detail. The threshold values of existence of chaotic motion are obtained under the periodic perturbation. we prove the criterion of existence of chaos in averaged system under quasi-periodic perturbation forω2=ω1+∈v by applying the second-order averaging method and Melnikov's method, and prove that the criterion of existence of chaos in second-order averaged system under quasi-periodic perturba-tion forω2= nω1+∈v,n≥2 cannot be obtained by applying Melnikov's method. Numerical simulations including homoclinic bifurcation surfaces, bifurcation dia-grams, Lyapunov exponent, phase portraits and Poincaree map, not only show the consistence with the theoretical analysis but also exhibit some new complex dynamics, including the periodic doubling bifurcation and the reversed periodic doubling bifurcation leading to chaos, the chaos suddenly appearing and disap-pearing to periodical orbits, non-attractive chaotical sets, chaotical attractor, and the interleaving occurrences of chaotic behaviors and invariant torus, etc.The paper consists of three chapters. Chapter 1 is the preparation knowledge. A brief review of second-order averaging methods and Melnikov's methods,chaos and some routes to chaos for continual dynamical system is presented.In chapter 2, we briefly introduce the backgrounds, histories and known re-search results of Josephson system.In chapter 3, we study Josephson system with parametric and external exci-tations by using second-order averaging methods and Melnikov's methods. The threshold values of existence of chaotic motion are obtained under the periodic perturbation. We prove the criterion of existence of chaos in averaged system under quasi-periodic perturbation forω2=ω1+∈v by applying the second-order averaging method and Melnikov's method, and prove that the criterion of exis-tence of chaos in second-order averaged system under quasi-periodic perturbation forω2= nω1+∈v,n≥2 cannot be obtained by applying Melnikov's method. The theoretical results are verified and some new dynamics are demonstrated by numerical simulation.