Chaotic Dynamics

We investigate the interplay of collective and chaotic motion in a classical self-bound N-body system with two-body interactions. This system displays a hierarchy of three well separated time scales that govern the onset of chaos, damping of collective motion and equilibration. Comparison with a mean-field problem shows that damping is mainly due to dephasing. The Lyapunov exponent, damping and equilibration rates depend mildly on the system size N.

Strange nonchaotic attractors (SNAs) can be created due to the collision of an invariant curve with itself. This novel ``homoclinic'' transition to SNAs occurs in quasiperiodically driven maps which derive from the discrete Schrödinger equation for a particle in a quasiperiodic potential. In the classical dynamics, there is a transition from torus attractors to SNAs, which, in the quantum system is manifest as the localization transition. This equivalence provides new insights into a variety of properties of SNAs, including its fractal measure. Further, there is a {\it symmetry breaking} associated with the creation of SNAs which rigorously shows that the Lyapunov exponent is nonpositive. By considering other related driven iterative mappings, we show that these characteristics associated with the the appearance of SNA are robust and occur in a large class of systems.

This brief note argues that, contrary to the claim of Ishioka and Fuchikami (chao-dyn/9902012), Landauer's principle is concerned a priori with entropy generation in computing processes. The concept of heat, in this principle, is only relevant when a connection with thermodynamics is established.

We critique the measure of complexity introduced by Shiner, Davison, and Landsberg in Ref. [1]. In particular, we point out that it is over-universal, in the sense that it has the same dependence on disorder for structurally distinct systems. We then give counterexamples to the claim that complexity is synonymous with being out of equilibrium: equilibrium systems can be structurally complex and nonequilibrium systems can be structurally simple. We also correct a misinterpretation of a result given by two of the present authors in Ref. [2]. [1] J. S. Shiner, M. Davison, and P. T. Landsberg, ``Simple Measure for Complexity'', Phys. Rev. E 59 (1999) 1459-1464. [2] J. P. Crutchfield and D. P. Feldman, ``Statistical Complexity of Simple One-Dimensional Spin Systems'', Phys. Rev. E 55:2 (1997) 1239R-1243R.

Short comment for a recent paper, suggesting for some directions related to previous studies in chaotic dynamics.

We first study birational mappings generated by the composition of the matrix inversion and of a permutation of the entries of 3×3 matrices. We introduce a semi-numerical analysis which enables to compute the Arnold complexities for all the 9! possible birational transformations. These complexities correspond to a spectrum of eighteen algebraic values. We then drastically generalize these results, replacing permutations of the entries by homogeneous polynomial transformations of the entries possibly depending on many parameters. Again it is shown that the associated birational, or even rational, transformations yield algebraic values for their complexities.

To ease analysis and simulation we make low-dimensional models of complicated dynamical systems. Centre manifold theory provides a systematic basis for the reduction of dimensionality from some detailed dynamical prescription down to a relatively simple model. An initial condition for the detailed dynamics also has to be projected onto the low-dimensional model, but has received scant attention. Herein, based upon the reduction algorithm in~\cite{Roberts96a}, I develop a straightforward algorithm for the computer algebra derivation of this projection. The method is systematic and is based upon the geometric picture underlying centre manifold theory. The method is applied to examples of a pitchfork and a Hopf bifurcation. There is a close relationship between this projection of initial conditions and the correct projection of forcing onto a model. I reaffirm this connection and show how the effects of forcing, both interior and from the boundary, should be properly included in a dynamical model.

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.

The slow passage through a Hopf bifurcation leads to the delayed appearance of large amplitude oscillations. We construct a smooth scalar feedback control which suppresses the delay and causes the system to follow a stable equilibrium branch. This feature can be used to detect in time the loss of stability of an ageing device. As a by-product, we obtain results on the slow passage through a bifurcation with double zero eigenvalue, described by a singularly perturbed cubic Lienard equation.

We apply the Ott, Grebogy and Yorke mechanism for the control of chaos to the analytical oscillator model of a leaky tap obtaining good results. We exhibit the robustness of the control against both dynamical noise and measurement noise.A possible way of controlling experimentally a leaky tap using magnetic-field-produced variations in the viscosity of a magnetorheological fluid is suggested.