E. R. Hunt

Keeping chaos at bay

The use of electronic circuits in studying chaotic dynamics and control are reviewed. Since all chaotic systems have several properties in common, simple circuits are analogous to much more complicated ones, such as lasers. Consequently, the methods developed to control chaos in electronic circuits are applicable to many diverse physical systems. The controlling device itself is a high-speed analog circuit. In applying perturbations, no calculations are made; instead, trial-and-error adjustments are used to locate the desired behavior. The initial observations of chaos in electronics, the development of the Ott-Grebogi-Yorke method for calculating the perturbations needed to stabilize a periodic orbit in a chaotic system and the occasional proportional feedback method, and their applications are discussed.<<ETX>>

Derivative control of the steady state in Chua's circuit driven in the chaotic region

Experimentally demonstrates that Chuas circuit, operating in the double-scroll chaotic regime, may be brought to either of the two unstable, stationary state fixed points by means of derivative control. >

CONTROLLING CHAOS IN CHUA'S CIRCUIT

The occasional proportional feedback (OPF) control technique has been successful in stabilizing periodic orbits in both periodically driven and autonomous systems undergoing chaotic behavior. By applying this technique to the well-known Chuas circuit, we are able to control a variety of periodic orbits including single-correction, low-period orbits and multiple-correction, high-period orbits. Also, by employing two control circuits, we are able to stabilize orbits that visit both regions of Chuas circuits double-scroll attractor, applying corrections in each of these regions during a single orbit.

Stochastic resonance in coupled nonlinear dynamic elements

We investigate the response of a linear chain of diffusively coupled diode resonators under the influence of thermal noise. We also examine the connection between spatiotemporal stochastic resonance and the presence of kink-antikink pairs in the array. The interplay of nucleation rates and kink speeds is briefly addressed. The experimental results are supplemented with simulations on a coupled map lattice. We furthermore present analytical results for the synchronization and signal processing properties of a Phi(4) field theory and explore the effects of various forms of nonlinear coupling. (c) 1998 American Institute of Physics.

CONTROLLING CHAOS IN A SIMPLE AUTONOMOUS SYSTEM: CHUA’S CIRCUIT

The occasional proportional feedback (OPF) control technique has been successful in stabilizing periodic orbits in both periodically driven and autonomous systems undergoing chaotic behavior. By applying this technique to the well-known Chua’s circuit, we are able to control a variety of periodic orbits including single-correction, low-period orbits and multiple-correction, high-period orbits. Also, by employing two control circuits, we are able to stabilize orbits that visit both regions of Chua’s circuit’s double-scroll attractor, applying corrections in each of these regions during a single orbit.

Stability analysis of fixed points via chaos control.

This paper reviews recent advances in the application of chaos control techniques to the stability analysis of two-dimensional dynamical systems. We demonstrate how the systems response to one or multiple feedback controllers can be utilized to calculate the characteristic multipliers associated with an unstable periodic orbit. The experimental results, obtained for a single and two coupled diode resonators, agree well with the presented theory. (c) 1997 American Institute of Physics.

Stable states and kink dynamics in a system of coupled diode resonators

Abstract We investigate the origin of stable spatially extended waveforms in an open flow system consisting of unidirectionally coupled chaotic oscillators. Results are obtained for an experimental system consisting of coupled diode resonators as well as for a coupled map lattice, a numerical model comprised of logistic maps. Under the conditions studied, spatially coherent or laminar states in both systems are convectively unstable, typically inducing high-dimensional, complex spatio-temporal dynamics. In each system stable spatial waveforms are shown to exist and are stabilized by either anchoring the first oscillator onto specific temporal orbits or by employing periodic boundary conditions. The latter case combined with a reduced drive amplitude also allows for travelling phase-kink solutions which move with constant velocity. This velocity is roughly inversely proportional to the coupling resistance and also depends on the number of kinks present in the system. We show evidence that the stable states of the system are associated with travelling kinks.

Stabilizing spatiotemporal patterns in a convectively unstable open flow system via kink-antikink pairs

Abstract A variety of spatial patterns embedded within an underlying spatiotemporally chaotic state is stabilized via various methods of boundary control. Results are presented for an experimental system consisting of unidirectionally coupled diode resonator circuits as well as for a coupled map lattice. Spatiotemporal chaos is eliminated either by locking the first resonator onto selected temporal periods or by utilizing periodic boundary conditions. We give experimental evidence that the resulting stable waveforms are related to moving phase-kink-antikink pairs.

Measurements of f(α) for Multifractal Attractors in Driven Diode Resonator Systems

A single diode resonator system is a nonlinear circuit consisting of a series combination of a resistance, inductance and a p-n junction diode. Driven single diode resonator systems experimentally show the period-doubling route to chaos, while driven coupled diode resonator systems show the quasi-periodic route to chaos. We report f(α) spectra1,2 calculated from experimental data3,4 and model calculations5 for driven diode resonator systems at transitions to chaos. The method used to obtain f(α) has no “free” adjustable parameters.