Scott Hayes

Control and applications of chaos

This review describes a procedure for stabilizing a desirable chaotic orbit embedded in a chaotic attractor of dissipative dynamical systems by using small feedback control. The key observation is that certain chaotic orbits may correspond to a desirable system performance. By carefully selecting such an orbit, and then applying small feedback control to stabilize a trajectory from a random initial condition around the target chaotic orbit, desirable system performance can be achieved. As applications, three examples are considered: (1) synchronization of chaotic systems; (2) conversion of transient chaos into sustained chaos; and (3) controlling symbolic dynamics for communication. The first and third problems are potentially relevant to communication in engineering, and the solution of the second problem can be applied to electrical power systems to avoid catastrophic events such as the voltage collapse.

Coding information in the natural complexity of chaos

The complexity inherent in the dynamics of chaotic oscillators can be utilized to produce digital communication signals. Symbolic (digital) information is encoded in large-scale features of the waveform by use of small perturbations to control the symbolic dynamics. A digital signal can thus be produced directly at the transmission stage, with no need for subsequent amplification. Because tiny perturbations control the dynamics, the circuitry for controlling the oscillator could often be entirely microelectronic. We first illustrate the main idea using the Lorenz system, which has a particularly simple symbolic dynamics description. We then describe other aspects of this mechanism for information transmission in the context of a previous paper.

Reverse-Time Chaos from a Randomly Driven Filter

A linear filter driven by a random signal is shown to generate a waveform that is chaotic under time reversal. That is, the waveform exhibits determinism and a positive Lyapunov exponent when viewed backward in time. The filter is realized in a passive electronic circuit, and the resulting waveform exhibits a Lorenz-like butterfly structure. This method for generating chaotic waveforms may be useful for a number of potential applications, including spread-spectrum communication and ultra-wideband (UWB) radar. The filter also demonstrates that chaos may be connected to physical theories beyond those described by deterministic nonlinear dynamical systems

Chaos: control and communication

This paper addresses two related issues: (1) control of chaos and, (2) controlling symbolic dynamics for communication. For control of chaos, we discuss the idea for realizing desirable periodic motion by applying small perturbations to an accessible parameter of the system. The key observations is that a chaotic attractor typically has embedded densely within it an infinite number of unstable periodic orbits. Since we wish to make only small controlling perturbations to the system, we do not envision creating new orbits with very different properties from the already existing orbits. Thus we seek to exploit the already existing unstable periodic orbits and unstable steady states. Our approach is as follows: We first determine some of the unstable low-period periodic orbits and unstable steady states that are embedded in the chaotic attractor. We then examine these orbits and choose one which yields improved system performance. Finally, we apply small controls so as to stabilize this already existing orbit. For the issue of communication, we describe an experiment verifying that the injection of small current pulses can be used to control the symbolic dynamics of a chaotic electrical oscillator to produce a digital communication waveform.