Frequency-Domain Approach to Hopf Bifurcation Analysis

Frequency-Domain Approach to Hopf Bifurcation Analysis

Continuous Time-Delayed Systems

Franco Sebastián Gentile, Jorge Luis Moiola;Guanrong Chen


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This book is devoted to the study of an effective frequency-domain approach, based on systems control theory, to compute and analyze several types of standard bifurcation conditions for general continuous-time nonlinear dynamical systems. A very rich pictorial gallery of local bifurcation diagrams for such nonlinear systems under simultaneous variations of several system parameters is presented. Some higher-order harmonic balance approximation formulas are derived for analyzing the oscillatory dynamics in small neighborhoods of certain types of Hopf and degenerate Hopf bifurcations.

The frequency-domain approach is then extended to the large class of delay-differential equations, where the time delays can be either discrete or distributed. For the case of discrete delays, two alternatives are presented, depending on the structure of the underlying dynamical system, where the more general setting is then extended to the case of distributed time-delayed systems. Some representative examples in engineering and biology are discussed.

  • Stability and Bifurcation Analysis
  • The Hopf Bifurcation Theorem in the Frequency Domain
  • Analysis of Static and Multiple Bifurcations
  • Degenerate Hopf Bifurcations
  • Higher-Order Hopf Bifurcation Formulas
  • Hopf Bifurcation in Continuous-Time Systems with Discrete-Time Delays
  • Hopf Bifurcation in Continuous-Time Systems with Distributed Time Delays
  • Degenerate Bifurcations in Time-Delayed Systems

Readership: Graduate students and researchers interested in nonlinear dynamics, control systems and applied mathematics for engineering systems.Hopf Bifurcation;Delayed Feedback Systems;Harmonic Balance;Control Systems0Key Features:
  • Offers an enlightening graphical representation for several types of Hopf and degenerate Hopf bifurcations
  • Enables the approximation of a periodic solution with higher-order precision in the vicinity of the singularity through the methodology
  • Presents a unified view for nonlinear control systems by using the harmonic balance method for continuous time-delayed feedback systems