Hopf Bifurcations and Chaos in the S-Shaped I-V Characteristic Systems : Condensed Matter and Statistical Physics

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Structures of chaos in semiconductor systems with the S-shaped I-V characteristic under an applied ac+bc bias are investigated from both the geometric and statistical viewpoint by using the dynamical systems approach. Their equations of motion are reduced to a three-dimensional non-hyperbolic map by taking the Poincare sections every period T=l/f_0 of the ac bias. As the dc component of applied bias is increased with the frequency f_0 fixed at some values, two different bifurcation sequences to chaos are found before entering the region of the S-shaped characteristic. Namely, for f_0<f_0^c(≃0.82), a subcritical (inverted) Hopf bifurcation of a limit cycle to a torus occurs, leading to torus wrinkling into chaos and crisis-induced intermittent chaos, whereas, for f_0>f_0^c, a supercritical (normal) Hopf bifurcation occurs, leading to torus doubling, torus fractalization into chaos and crisis-induced intermittent chaos. In the case caused by the supercritical Hopf bifurcation, statistical treatments of chaotic orbits are carried out for the crisis-induced intermittency, leading to P(τ)∝exp(-τ/<τ>) for the distribution of their lifetimes τ for staying within a chaotic region which was the chaotic attractor just before the crisis, where <τ> is the mean lifetime with <τ>∝ε^<-γ>, (γ≃1.2), ε being the deviation from the onset of crisis.
理論物理学刊行会の論文
1994-06-25