Stephen Lynch

Small-amplitude limit cycle bifurcations for Liénard systems with quadratic or cubic damping or restoring forces

We consider the second-order equation where f and g are polynomials with deg f,deg g n. Our interest is in the maximum number of isolated periodic solutions which can bifurcate from the steady state solution x = 0. Alternatively, this is equivalent to seeking the maximum number of limit cycles which can bifurcate from the origin for the Li?nard system, Assuming the origin is not a centre, we show that if either f or g are quadratic, then this number is . If f or g are cubic we show that this number is , for all 1n50. The results also hold for generalized Li?nard systems.

Small-amplitude limit cycles of certain Liénard systems

The paper is concerned with the number of limit cycles of systems of the form ẋ = y–F(x), ẏ = –g(x), where F and g are polynomials. For several classes of such systems, the maximum number of limit cycles that can bifurcate out of a critical point under perturbation of the coefficients in F and g is obtained (in terms of the degree of F and g).

Analysis of optical instabilities and bistability in a nonlinear optical fibre loop mirror with feedback

Abstract The dynamics of a nonlinear optical fibre loop mirror with feedback and a CW input is examined. The device, first investigated by Shi, Optics Comm. 107 (1994) 276, using a graphical technique, is re-examined using linear stability analysis and iterative methods. Results show that the known bistable behaviour of the device can be affected by Ikeda instabilities. We also report that more than one method of analysis is required to fully classify the dynamical behaviour of the device.

Generalized cubic Liénard equations

Abstract This article presents a new method for determining the Liapunov quantities of Lienard systems with either cubic damping or restoring terms. The first eleven quantities have been computed on a PC, whereas the algorithm used previously requires the use of high powered computers with lots of memory. The reduction part of the algorithm is simplified by expressing the Liapunov quantities in a special form. The maximum number of small-amplitude limit cycles which may be bifurcated from the origin is given for certain systems.

Generalized quadratic Liénard equations

Abstract In this article we give explicit formulae for the Liapunov quantities of generalized Lienard systems with either quadratic damping or restoring coefficients. These quantities provide necessary conditions in order for the origin to be a centre. Recent results are also presented for these systems.

Limit cycles of generalized Liénard equations

Abstract The generalized Lienard equations of the form: x = h(y) − F(x), y = −g(x) where F, g, and h are polynomials, are examined. It has been found that the results given by Blows, Lloyd and Lynch [1–5] for Lienard equations hold also for the generalized systems. A new result is also presented within this article.

FURTHER INVESTIGATION OF HYSTERESIS IN CHUA'S CIRCUIT

For a system to display bistable behavior (or hysteresis), it is well known that there needs to be a nonlinear component and a feedback mechanism. In the Chua circuit, nonlinearity is supplied by the Chua diode (nonlinear resistor) and in the physical medium, feedback would be inherently present, however, with standard computer models this feedback is omitted. Using Poincare first return maps, bifurcations for a varying parameter in the Chua circuit equations are investigated for both increasing and decreasing parameter values. Evidence for the existence of a small bistable region is shown and numerical methods are applied to determine the behavior of the solutions within this bistable region.

Oscillatory Threshold Logic

In the 1940s, the first generation of modern computers used vacuum tube oscillators as their principle components, however, with the development of the transistor, such oscillator based computers quickly became obsolete. As the demand for faster and lower power computers continues, transistors are themselves approaching their theoretical limit and emerging technologies must eventually supersede them. With the development of optical oscillators and Josephson junction technology, we are again presented with the possibility of using oscillators as the basic components of computers, and it is possible that the next generation of computers will be composed almost entirely of oscillatory devices. Here, we demonstrate how coupled threshold oscillators may be used to perform binary logic in a manner entirely consistent with modern computer architectures. We describe a variety of computational circuitry and demonstrate working oscillator models of both computation and memory.

Symbolic Computation of Lyapunov Quantities and the Second Part of Hilbert’s Sixteenth Problem

This tutorial survey presents a method for computing the Lyapunov quantities for Lienard systems of differential equations using symbolic manipulation packages. The theory is given in detail and simple working MATLAB and Maple programs are listed in this chapter. In recent years, the author has been contacted by many researchers requiring more detail on the algorithmic method used to compute focal values and Lyapunov quantities. It is hoped that this article will address the needs of those and other researchers. Research results are also given here.

Controlling chaos in nonlinear optical resonators

Abstract This article presents an analytical method for controlling the chaos in two nonlinear bistable optical resonators. The devices are a bulk cavity ring oscillator and a nonlinear simple fibre ring resonator. It is shown that there is a trade-off between flexibility and controllability in such devices. For the first time, as far as the authors are aware, theoretical studies have shown that by controlling the chaos within a bistable region it becomes possible to use a previously unstable device as a bistable resonator.