The Hopf bifurcation theorem is one of the most important results for delay differential equations because it is essentially the only method for rigorously establishing the existence of periodic solutions. We begin with an example where the bifurcation can be easily calculated in order to show key features of a Hopf bifurcation.Then the theorem is stated without proof along with a discussion of the stability properties of bifurcating periodic solutions. The remainder of the chapter consists of numerous applications. A complete study of the Hopf bifurcation for the nonlinear negative feedback equation is carried out with particular application to the delayed logistic equation. Hopf bifurcation for second-order delayed negative feedback equations is thoroughly studied, including the inverted pendulum equation with delayed feedback. A gene regulatory network provides another example. The chapter concludes with a statement and elaboration on a beautiful Poincaré e-Bendixson theorem due to Mallet-Paret and Sell for a special class of delay differential systems,namely, monotone cyclic feedback systems.
KeywordsPeriodic Solution Hopf Bifurcation Implicit Function Theorem Homoclinic Orbit Delay Differential Equation
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