Elements of Applied Bifurcation Theory

The favorable reaction to the ?rst edition of this book con?rmed that the publication of such an application-oriented text on bifurcation theory of dynamical systems was well timed. The selected topics indeed cover - jor practical issues of applying the bifurcation theory to ?nite-dimensional problems. This new edition preserves the structure of the ?rst edition while updating the context to incorporate recent theoretical developments, in particular, new and improved numerical methods for bifurcation analysis. The treatment of some topics has been clari?ed. Major additions can be summarized as follows: In Chapter 3, an e- mentary proof of the topological equivalence of the original and truncated normal forms for the fold bifurcation is given. This makes the analysis of codimension-one equilibrium bifurcations of ODEs in the book complete. This chapter also includes an example of the Hopf bifurcation analysis in a planar system using MAPLE, a symbolic manipulation software. Chapter 4 includes a detailed normal form analysis of the Neimark-Sacker bif- cation in the delayed logistic map. In Chapter 5, we derive explicit f- mulas for the critical normal form coe?cients of all codim 1 bifurcations of n-dimensional iterated maps (i. e. , fold, ?ip, and Neimark-Sacker bif- cations). The section on homoclinic bifurcations in n-dimensional ODEs in Chapter 6 is completely rewritten and introduces the Melnikov in- gral that allows us to verify the regularity of the manifold splitting under parameter variations.

to Dynamical Systems.- Topological Equivalence, Bifurcations, and Structural Stability of Dynamical Systems.- One-Parameter Bifurcations of Equilibria in Continuous-Time Dynamical Systems.- One-Parameter Bifurcations of Fixed Points in Discrete-Time Dynamical Systems.- Bifurcations of Equilibria and Periodic Orbits in n-Dimensional Dynamical Systems.- Bifurcations of Orbits Homoclinic and Heteroclinic to Hyperbolic Equilibria.- Other One-Parameter Bifurcations in Continuous-Time Dynamical Systems.- Two-Parameter Bifurcations of Equilibria in Continuous-Time Dynamical Systems.- Two-Parameter Bifurcations of Fixed Points in Discrete-Time Dynamical Systems.- Numerical Analysis of Bifurcations.