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Harmonic balance and the Hopf bifurcation

Published online by Cambridge University Press:  24 October 2008

D. J. Allwright
D. J. Allwright
Affiliation:
Peterhouse, Cambridge
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Summary

A form of the Hopf bifurcation theorem specially suited to systems in block-diagram form has been developed, which allows one to deal with high or infinite order linear elements solely in terms of their transfer functions. The results are proved by the method of harmonic balance, and, for general non-linear systems, lead to criteria for the existence and stability of bifurcated orbits generalizing those derived by various authors for systems of ordinary differential equations. In the particular case of control loops with a single non-linearity, a simple addition to the Nyquist diagram of the loop determines the amplitude and frequency of bifurcated orbits, and whether they occur when the equilibrium is stable or unstable. The analysis is independent of the central manifold theorem, and of Floquet theory.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 82 , Issue 3 , November 1977 , pp. 453 - 467
Copyright
Copyright © Cambridge Philosophical Society 1977

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References

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