The central point of this book, the celebrated Hopf bifurcation theorem, is revisited using the so-called frequency domain approach. Unlike the more traditional “time domain” approach, when the theory employs the state-space formulation for a system of ODEs, this technique deals with a closed loop system in terms of input-output relationships. Introducing a “state-feedback control”, one can so arrive at a linear system with a nonlinear feedback. The Laplace transform is then performed there on the time domain of a state-space system.
The frequency domain approach has been initiated by the series of papers due to D. J. Allwright, A. I. Mees and L. O. Chua in the late 1970’s. The present monograph aims to show its advantages for obtaining various types of bifurcation conditions to strongly nonlinear dynamical systems. It is rather technical, full of formulas and algebraic calculations, but it is self-consistent.
Formally, it is divided into the 7 following chapters and an appendix (involving higher-order Hopf bifurcation formulas in three parts):
1. Introduction. 2. The Hopf bifurcation theorem. 3. Continuation of bifurcation curves on the parameter plane. 4. Degenerate bifurcations in the space of system parameters. 5. High-order Hopf bifurcation formulas. 6. Hopf bifurcation in nonlinear systems with time delay. 7. Birth of multiple limit cycles.
Besides the existence and multiplicity of nonlinear oscillations, their Lyapunov stability is treated as well. Plenty of diagrams allow us to see the bifurcation scenarios in detail. Finally, many concrete examples suitably illustrate the whole exposition.