Recent Developments in Chaotic Time Series Analysis

In this paper, two issues are addressed: (1) the applicability of the delay-coordinate embedding method to transient chaotic time series analysis, and (2) the Hilbert transform methodology for chaotic signal processing.A common practice in chaotic time series analysis has been to reconstruct the phase space by utilizing the delay-coordinate embedding technique, and then to compute dynamical invariant quantities of interest such as unstable periodic orbits, the fractal dimension of the underlying chaotic set, and its Lyapunov spectrum. As a large body of literature exists on applying the technique to time series from chaotic attractors, a relatively unexplored issue is its applicability to dynamical systems that exhibit transient chaos. Our focus will be on the analysis of transient chaotic time series. We will argue and provide numerical support that the current delay-coordinate embedding techniques for extracting unstable periodic orbits, for estimating the fractal dimension, and for computing the Lyapunov exponents can be readily adapted to transient chaotic time series.A technique that is gaining an increasing attention is the Hilbert transform method for signal processing in nonlinear systems. The general goal of the Hilbert method is to assess the spectrum of the instantaneous frequency associated with the underlying dynamical process. To obtain physically meaningful results, it is necessary for the signal to possess a proper rotational structure in the complex plane of the analytic signal constructed by the original signal and its Hilbert transform. We will describe a recent decomposition procedure for this task and apply the technique to chaotic signals. We will also provide an example to demonstrate that the methodology can be useful for addressing some fundamental problems in chaotic dynamics.

EXPERIMENTAL CONTROL OF CHAOS BY MODIFICATIONS OF DELAYED FEEDBACK

We present two modifications of the delayed feedback control technique for controlling chaotic dynamical systems. In these methods, control force is applied only when trajectories enter neighborhoods of the targets. So three shortcomings in the delayed feedback technique, the nonsmallness of control force, impossibility of targeting unstable periodic orbits and birth of undesirable stable orbits, are improved. The delay coordinate embedding technique is also used for specifying the target orbits and determining whether trajectories enter their neighborhood. We demonstrate the effectiveness of the two approaches for an experimental system, a feedback controlled pendulum.

Experimental Evidence of Quasiperiodicity and Its Breakdown in the Column-Pendulum Oscillator

A new vibration absorbing device is introduced for large flexible structures. The phase-space of the experimental system is reconstructed via delay-coordinate embedding technique. Experimental dynamics indicate that the motion is predominantly quasiperiodic, confirming the existence of invariant tori. Within the quasi-periodic region, there are windows containing intricate webs of phase-locked periodic responses. The quasiperiodic and the phase-locked responses are clearly visualized on the cover of the torus. Increase in the amplitude of excitation results in distortion of the invariant torus due to the resonance overlap. Due to the resonance overlap, the return map extracted from the experimental data becomes noninvertible. Furthermore, a burst of frequencies appears on the Fourier spectrum. This scenario is similar to many experimental observations of hydrodynamical instabilities; the breakup of the tori in these experiments is related to the onset of turbulence.

Explicit Realization of Chaos Control in an NMR-Laser Experiment

The usefulness of the Ott-Grebogi-Yorke control method is demonstrated by stabilizing a chaotic NMR-laser system around an unstable period-one orbit. We have used a six-dimensional delay-coordinate embedding technique in order to fully determine the stability properties of the orbit controlled. Our analysis yields small time-dependent perturbations of the system quality factor capable to perform real-time control.