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We show that avoided crossings of energy bands may give rise to a variety of phenomena such as transitions from metal to insulator and vice versa, changes in localization lengths, and changes in the fractal dimension of energy spectra. We explain the occurrence of these phenomena in the kicked Harper model under classically chaotic conditions and predict them to occur in other systems.
As examples of quantum-"classical" coupling systems, multi-component systems are studied by semiclassical evaluations of the Feynman kernels in the coherent-state representation. From the observation of the phase space caustics due to the presence of the internal degree of freedom (IDF), two phenomena are explained in terms of the semiclassical theory: (1) The quantum oscillations of the IDF induce quantum interference patterns in the Hushimi representation; (2) Chaotic dynamics destroys the coherence of the quantum oscillations.
Conductance fluctuations have been studied in a soft wall stadium and a Sinai billiard defined by electrostatic gates on a high mobility semiconductor heterojunction. These reproducible magnetoconductance fluctuations are found to be fractal confirming recent theoretical predictions of quantum signatures in classically mixed (regular and chaotic) systems. The fractal character of the fluctuations provides direct evidence for a hierarchical phase space structure at the boundary between regular and chaotic motion.
A new method is introduced to create artificial time sequences that fulfil given constraints but are random otherwise. Constraints are usually derived from a measured signal for which surrogate data are to be generated. They are fulfilled by minimizing a suitable cost function using simulated annealing. A wide variety of structures can be imposed on the surrogate series, including multivariate, nonlinear, and nonstationary properties. When the linear correlation structure is to be preserved, the new approach avoids certain artifacts generated by Fourier-based randomization schemes.
We examine the effect of short unstable periodic orbits on wavefunction statistics in a classically chaotic system, and find that the tail of the wavefunction intensity distribution in phase space is dominated by scarring associated with the least unstable periodic orbits. In an ensemble average over systems with classical orbits of different instabilities, a power law tail is found, in sharp contrast to the exponential prediction of random matrix theory. The calculations are compared with numerical data and quantitative agreement is obtained
We study a resistively shunted semiconductor superlattice subject to a high-frequency electric field. Using a balance equation approach that incorporates the influence of the electric circuit, we determine numerically a range of amplitude and frequency of the ac field for which a dc bias and current are generated spontaneously and show that this region is likely accessible to current experiments. Our simulations reveal that the Bloch frequency corresponding to the spontaneous dc bias is approximately an integer multiple of the ac field frequency.
Although hydrogen in external fields is a paradigm for the application of periodic orbits and the Gutzwiller trace formula to a real system, the trace formula has never been applied successfully to other Rydberg atoms. We show that spectral fluctuations of general Rydberg atoms are given with remarkable precision by the addition of diffractive terms. Previously unknown features in atomic spectra are exposed: there are new modulations that are neither periodic orbits nor combinations of periodic orbits; `core-shadowing' generally decreases primitive periodic orbit amplitudes but can also lead to increases.
It has been proposed to make practical use of chaos in communication, in enhancing mixing in chemical processes and in spreading the spectrum of switch-mode power suppies to avoid electromagnetic interference. It is however known that for most smooth chaotic systems, there is a dense set of periodic windows for any range of parameter values. Therefore in practical systems working in chaotic mode, slight inadvertent fluctuation of a parameter may take the system out of chaos. We say a chaotic attractor is robust if, for its parameter values there exists a neighborhood in the parameter space with no periodic attractor and the chaotic attractor is unique in that neighborhood. In this paper we show that robust chaos can occur in piecewise smooth systems and obtain the conditions of its occurrence. We illustrate this phenomenon with a practical example from electrical engineering.
We present a new method for the computation of Lyapunov exponents utilizing representations of orthogonal matrices applied to decompositions of M or MM_trans where M is the tangent map. This method uses a minimal set of variables, does not require renormalization or reorthogonalization, can be used to efficiently compute partial Lyapunov spectra, and does not break down when the Lyapunov spectrum is degenerate.
Statistical properties of energy levels and eigenfunctions in a ballistic system with diffusive surface scattering are investigated. The two-level correlation function, the level number variance, the correlation function of wavefunction intensities, and the inverse participation ratio are calculated.
We investigate the dynamical behaviour of two limit cycle oscillators that interact with each other via time delayed coupling and find that time delay can lead to amplitude death of the oscillators even if they have the same frequency. We demonstrate that this novel regime of amplitude "death" also exists for large collections of coupled identical oscillators and provide quantitative measures of this death region in the parameter space of coupling strength and time delay. Its implication for certain biological and physical applications is also pointed out.
It is reported that the characteristic relations of type-II and type-III intermittencies, with respective local Poincar$\acute{e}$ maps of $x_{n+1}= (1+ε)x_n + a x^3_n$ and $x_{n+1}= - (1+ε)x_n - a x^3_n$, are both $- ln (ε)$ under the assumption of uniform reinjection probability. However, the intermittencies have various characteristic relations such as $ε^{-ν}$ ($1/2 \leq ν\leq 1$) depending on the reinjection probability. In this Letters the various characteristic relations are discussed, and the $ε^{-1/2}$ characteristic relation is obtained experimentally in an electronic circuit, with uniform reinjection probability.
We have found an analytic expression for the multivariate generating function governing all n-point statistics of random multiplicative cascade processes. The variable appropriate for this generating function is the logarithm of the energy density, ln epsilon, rather than epsilon itself. All cumulant statistics become sums over derivatives of ``branching generating functions'' which are Laplace transforms of the splitting functions and completely determine the cascade process. We show that the branching generating function is a generalization of the multifractal mass exponents. Two simple models from fully developed turbulence illustrate the new formalism.
A Lagrangian method for the numerical simulation of the Kraichnan passive scalar model is introduced. The method is based on Monte--Carlo simulations of tracer trajectories, supplemented by a point-splitting procedure for coinciding points. Clean scaling behavior for scalar structure functions is observed. The scheme is exploited to investigate the dependence of scalar anomalies on the scaling exponent $ξ$ of the advecting velocity field. The three-dimensional fourth-order structure function is specifically considered.
We study the behavior of vortex filaments subject to a uniform density of phase twist in oscillatory media described by the complex Ginzburg-Landau equation. The first instability is a supercritical Hopf bifurcation to stable propagating helical vortices. The secondary instability, also a forward Hopf bifurcation, leads to quasiperiodic supercoiled filaments. The structural changes undergone by these dynamical objects are akin to those of twisted elastic rods, in spite of the presence of the ribbon component of the twist, particular to phase singularities.
In this letter a new solvable model of synchronization dynamics is introduced. It consists of a system of long range interacting tops with random precession frequencies. The model allows for an explicit study of orientational effects in synchronized phenomena. A stability analysis of the incoherent solution is performed for different types of orientational disorder. A system with only orientational disorder always synchronizes in the absence of external noise.
Treating a many-body Fermi system in terms of a single particle in a deforming mean field. We relate adiabatic geometric phase to susceptibility for the noncyclic case, and to its derivative for the cyclic case. Employing the semiclassical expression of susceptibility, the expression for geometric phase for chaotic quantum system immediately follows. Exploiting the well-known association of the absorptive part of susceptibility with dissipation, our relations may provide a quantum mechanical origin of the damping of collective excitations in Fermi systems.
Intensities of LEED and PED are analyzed from a statistical point of view. The probability distribution is compared with a Porter-Thomas law, characteristic of a chaotic quantum system. The agreement obtained is understood in terms of analogies between simple models and Berry's conjecture for a typical wavefunction of a chaotic system. The consequences of this behaviour on surface structural analysis are qualitatively discussed by looking at the behaviour of standard correlation factors.