Numerical evidence for global bifurcations leading to switching phenomena in long Josephson junctions

Abstract

Fluxons in long Josephson junctions are physical manifestations of travelling waves that connect rest states of the model partial differential equation (p.d.e.), which is a perturbed sine-Gordon equation. In the reduced traavelling wave ordinary differential equation (o.d.e.), fluxons correspond to heteroclinic connections between fixed points. In the absence of surface impedence effects, fluxons persist in parameter regimes until the fixed points disappear, after which the system ‘switches’ to another configuration. It is known that the presence of surface impedence produces a singular perturbation of the model equation, together with a new phenomenon: the fluxons switch in parameter regimes before the fixed points are lost. Why this occurs is unknown, and is the focus of this paper. Two disjoint possibilities are: (1) instability: fluxons still exist, but they become unstable in the p.d.e. due to surface impedance effects; (2) nonexistence: the fluxons fail to exist, even though the fixed points remain. Here, we provide compelling numerical evidence for the second scenario, characterized by a global bifurcation in the travelling wave phase space: a breakdown of heteroclinicorbits, undetected at the local linearized level. Moreover, this global o.d.e. bifurcation occurs at parameter values consistent with the p.d.e. switching phenomenon.