A HARMONIC BALANCE APPROACH FOR CHAOS PREDICTION: CHUA’S CIRCUIT

Abstract

The paper proposes a practical engineering approach for predicting chaotic dynamics in an important class of nonlinear systems. The aim of this approach is to provide a heuristic method of analysis which can give reasonably accurate answers but is far simpler to apply than other more rigorous methods based on nonlinear dynamics. Our approach is founded on the harmonic balance principle and uses standard describing function techniques well known to design engineers. Our method consists of a synergism of two independent techniques, each one constituting a possible mechanism for chaos. These two techniques are combined into a single algorithm which is highly efficient computationally, taking only a fraction of the time normally required by other more exact procedures. In order to present and illustrate our algorithm clearly, and in order to compare its predictions with readily available results obtained by rigorous methods, we have chosen Chua’s circuit as a vehicle to demonstrate the effectiveness of this approach. Chua’s circuit was chosen not only for the huge amount of results already published concerning the dynamics of this system, but also because it represents a real physical system easily built in the laboratory, and whose simple mathematical model has proven to be realistic and mathematically tractable. Although our algorithm does not guarantee its prediction is fully reliable, any more than the widely used describing function method does, its significance is based entirely on the empirical evidence that it yields qualitatively correct, though not exact, results for all of the chaotic phenomena that we have investigated in Chua’s circuit, as well as in many other chaotic systems. We hope therefore that our chaos prediction algorithm will find practical uses among engineers and scientists not familiar with more specialized mathematical approaches, in search of a simple and practical, although not rigorous, tool for analyzing systems with complex dynamics.