Hopf Bifurcations and Chaos in the S-Shaped I-V Characteristic Systems

Abstract

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 Poincaré sections every period T = 1/f0 of the ac bias. As the dc component of applied bias is increased with the frequency f0 fixed at some values, two different bifurcation sequences to chaos are found before entering the region of the S-shaped characteristic. Namely, for f0 ≪ f0c (≃0.82), a subcritical (inverted) Hopf bifucation of a limit cycle to a torus occurs, leading to torus wrinkling into chaos and crisis-induced intermittent chaos, whereas, for f0 > f0c, 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-r, (γ≃1.2), ε being the deviation from the onset of crisis.