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**Introduction to dynamic bifurcation.**
*(English)*
Zbl 0544.58016

Bifurcation theory and applications, Lect. 2nd Sess. CIME, Montecatini/Italy 1983, Lect. Notes Math. 1057, 106-151 (1984).

[For the entire collection see Zbl 0527.00018.]

In these notes the author reviews a number of standard results in dynamic bifurcation theory together with some more recent results. The Lyapunov- Schmidt method is described and applied to prove existence of stable and unstable manifolds at hyperbolic equilibria and to obtain the bifurcation picture when a simple eigenvalue crosses zero. It is shown how one obtains dynamic behaviour from the bifurcation function, the saddle-node bifurcation is described, and applications are given to parabolic equations and to retarded functional differential equations. The next section is on the Hopf bifurcation, again with the emphasis on the information on the dynamic behaviour which one can obtain from the bifurcation function. Then the concept of exponential dichotomy is explained and used to obtain bifurcation of homoclinic orbits, again via a Lyapunov-Schmidt reduction. Exponential dichotomies are further explored to describe the flow near a transverse homoclinic point and to obtain a version of the ”shadowing lemma” as given in recent work of K. Palmer. Next the codimension one bifurcations in the plane are summarized, and two examples of codimension two bifurcations are discussed; in both these examples the results rely partly on a study of certain Abelian integrals.

In these notes the author reviews a number of standard results in dynamic bifurcation theory together with some more recent results. The Lyapunov- Schmidt method is described and applied to prove existence of stable and unstable manifolds at hyperbolic equilibria and to obtain the bifurcation picture when a simple eigenvalue crosses zero. It is shown how one obtains dynamic behaviour from the bifurcation function, the saddle-node bifurcation is described, and applications are given to parabolic equations and to retarded functional differential equations. The next section is on the Hopf bifurcation, again with the emphasis on the information on the dynamic behaviour which one can obtain from the bifurcation function. Then the concept of exponential dichotomy is explained and used to obtain bifurcation of homoclinic orbits, again via a Lyapunov-Schmidt reduction. Exponential dichotomies are further explored to describe the flow near a transverse homoclinic point and to obtain a version of the ”shadowing lemma” as given in recent work of K. Palmer. Next the codimension one bifurcations in the plane are summarized, and two examples of codimension two bifurcations are discussed; in both these examples the results rely partly on a study of certain Abelian integrals.

Reviewer: A.Vanderbauwhede

### MSC:

37G99 | Local and nonlocal bifurcation theory for dynamical systems |