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3(b) wanders from one stagnant region to another in an unpredictable way, whose overall feature looks like a living animal. As in the case (I), however, the positive Lyapunov exponent is vanishing. The power spectrum of Fig. 3(a) again shows the 1/f 2 law (Fig. 3(c)). Locally, a picture of the tilted band structure 38 Chapter 3 is meaningful. Laminar oscillations in the plateaus (see the inset of Fig. 3(a)) are caused by the increase of kn values within each band, whereas bursts are due to the Zener tunneling to adjacent bands at the zone boundaries.

General Remarks Despite the absence of genuine chaos in quantum systems, we have shown some examples of pseudo-chaos in 1-d quantum transport. This transport, induced by quantum tunneling, has no classical counterpart. The pseudo-chaos has no positive Lyapunov exponent but bears a positive component in the distribution of the local Lyapunov exponent. The pseudo-chaos of the Krönig-Penny model in the applied electric field breaks the law of large numbers, leading to 1/f fluctuations. Chaos resulting from double-barrier structures has proved to be of the virtual reality, since it is traced back to the artificial nonlinearity as a consequence of the mean-field approximation of the many-body interaction.

Typically, in chaotic systems without any bifurcation, there exist isolated and unstable periodic orbits bearing 20 Chapter 2 the positive Lyapunov exponents. Although the Lebesgue measure of periodic orbits is vanishing in the chaotic sea, their number is infinite. 17) Tr G0(E) comes from a contribution from zero-length orbits. 18) The stability exponent available from the eigenvalue of M depends on the type of fixed points. Corresponding to unstable and stable periodic orbits, one has =exp (± u ) and exp (iu ), respectively.