Global dynamics of the Josephson equation in T S1

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:269

Article

Chen, Hebai¹  Tang, Yilei² 

[1] Cent South Univ, Sch Math & Stat, Changsha 410083, Hunan, Peoples R China

[2] Shanghai Jiao Tong Univ, Sch Math Sci, MOE LSC, Shanghai 200240, Peoples R China

关键词: Limit cycle;    Josephson equation;    Homoclinic loop;    Two-saddle loop;    Saddle connection;   

DOI  :  10.1016/j.jde.2020.03.048

来源: Elsevier

【 摘 要 】

The Josephson equation (phi) over dot = y, (y) over dot = -sin phi + epsilon(a (1 + gamma cos phi)y) was researched by Sanders and Cushman (1986) [12] for its phase portraits when epsilon > 0 is small by applying the averaging method. The parameter E can actually be large or even any real number in the practical application of this model. When vertical bar epsilon vertical bar is not small, we cannot apply the averaging method because the system is not near-Hamiltonian. For general epsilon is an element of 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 not equal 0. (C) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

Free