Global dynamics of the Josephson equation in TS1

Abstract

The Josephson equation ϕ˙=y,y˙=−sin⁡ϕ+ϵ(a−(1+γcos⁡ϕ)y) was researched by Sanders and Cushman (1986) [12] for its phase portraits when ϵ>0 is small by applying the averaging method. The parameter ϵ can actually be large or even any real number in the practical application of this model. When |ϵ| is not small, we cannot apply the averaging method because the system is not near-Hamiltonian. For general ϵ∈R, we present complete dynamics and more complex bifurcations of the Josephson equation in TS1, including saddle-node bifurcation, Hopf bifurcation, Bogdanov-Takens bifurcation, homoclinic loop bifurcation, two-saddle heteroclinic loop bifurcation, upper saddle connection bifurcation and lower saddle connection bifurcation. Moreover, we prove the monotonicity of bifurcation functions with respect to parameters and the nonexistence of a two-saddle heteroclinic loop for all a≠0.