Prediction of chaos in Josephson junction by the Melnikov function technique

Abstract

 Users may download and print one copy of any publication from the public portal for the purpose of private study or research.  You may not further distribute the material or use it for any profit-making activity or commercial gain  You may freely distribute the URL identifying the publication in the public portal If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.

Bifurcations of periodic solutions and chaos in Josephson system with parametric excitation

“…System (33) always has a trivial equilibrium (u, v) = (0, 0), which corresponds to the harmonic x(t) of System (2). The eigenvalue equation at the trivial point is…”

“…Each additional pair of nontrivial fixed points (r ± , θ ± ), (r ± , θ ± + π) of System (34) correspond to a single subharmonic of period two of System (2) and are given approximately by…”

“…For example: the extension of the Josephson equation including a so-called cos ψ term has been performed by Schlup [16] by using perturbation theory and numerical calculation; Abidi and Chua [1] , and Odyniec and Chua [14] have supplied analytic treatment in the case when the Josephson junction possesses periodic orbits; Salam and Sastry [17] and Bartuccelli et al [2] proved the existence of chaos by the Melnikov method for certain parameter values. More dynamical information include heteroclinic chaos, subharmonic bifurcation and chaos are discussed by Jing [6,7] and Jing, et al [8,9] using singular perturbation method, second order average method and Melnikov method together with numerical simulation.…”

“…In region I, there is only a stable trivial equilibrium of System (33) corresponding to the stable nonresonant harmonic of System (2). In region II, there is a unstable harmonic and a resonant subharmonic corresponding to a pair of stable points of System (34).…”

“…(iii) Fig.3 (P1)-(P2) show the bifurcation curves for System (2) in the (Ω, f) space. In region I, there is only a stable trivial equilibrium of System (33) corresponding to the stable nonresonant harmonic of System (2).…”