Elements of Applied Bifurcation Theory, Second Edition Yuri A. Kuznetsov Springer [...]... form of the fold bifurcation 4.3 Generic fold bifurcation 4.4 The normal form of the flip bifurcation 4.5 Generic flip bifurcation 4.6 The “normal form” of the Neimark-Sacker bifurcation 4.7 Generic Neimark-Sacker bifurcation 4.8 Exercises 4.9 Appendix 1: Feigenbaum’s universality 4.10 Appendix 2: Proof of. .. Detection and location of codim 1 bifurcations 10.2.3 Analysis of codim 1 bifurcations 10.2.4 Branching points 10.3 Two-parameter bifurcation analysis 10.3.1 Continuation of codim 1 bifurcations of equilibria and fixed points 10.3.2 Continuation of codim 1 limit cycle bifurcations 10.3.3 Continuation of codim 1 homoclinic orbits ... rather naive way as the number of conditions defining the bifurcation Topological normal forms (universal unfoldings of nondegenerate parameter-dependent systems) for bifurcations are defined, and an example of such a normal form is demonstrated for the Hopf bifurcation Chapter 3 One-parameter bifurcations of equilibria in continuous-time dynamical systems Two generic codim 1 bifurcations – tangent (fold)... Systems 8.1 List of codim 2 bifurcations of equilibria 8.1.1 Codim 1 bifurcation curves 8.1.2 Codim 2 bifurcation points 8.2 Cusp bifurcation 8.2.1 Normal form derivation 8.2.2 Bifurcation diagram of the normal form 8.2.3 Effect of higher-order terms 8.3 Bautin (generalized Hopf) bifurcation 8.3.1... 8.3.2 Bifurcation diagram of the normal form 8.3.3 Effect of higher-order terms 8.4 Bogdanov-Takens (double-zero) bifurcation 8.4.1 Normal form derivation 8.4.2 Bifurcation diagram of the normal form 8.4.3 Effect of higher-order terms 8.5 Fold-Hopf (zero-pair) bifurcation 8.5.1 Derivation of the normal form... 8.5.2 Bifurcation diagram of the truncated normal form 8.5.3 Effect of higher-order terms 8.6 Hopf-Hopf bifurcation 8.6.1 Derivation of the normal form 8.6.2 Bifurcation diagram of the truncated normal form 8.6.3 Effect of higher-order terms 8.7 Exercises 8.8 Appendix 1: Limit cycles and homoclinic orbits of Bogdanov... for a bifurcation appear naturally at this step An example of the Hopf bifurcation in a predator-prey system is analyzed Chapter 4 One-parameter bifurcations of fixed points in discrete-time dynamical systems The approach formulated in Chapter 3 is applied to study tangent (fold), flip (period-doubling), and Hopf (NeimarkSacker) bifurcations of discrete-time dynamical systems For the NeimarkSacker bifurcation, ... general approach: (1) formulation of the corresponding topological normal form and analysis of its bifurcations; (2) reduction of a generic parameterdependent system to the normal form up to terms of a certain order; and (3) demonstration that higher-order terms do not affect the local bifurcation diagram Step 2 (finite normalization) is performed by means of polynomial changes of variables with unknown coefficients... systems or systems theory Certain classical results, such as Andronov-Hopf and homoclinic bifurcation in two-dimensional systems, are presented in great detail, including self-contained proofs For more complex topics of the theory, such as homoclinic bifurcations in more than two dimensions and two-parameter local bifurcations, we try to make clear the relevant geometrical ideas behind the proofs but only... modifications of the technique to handle the Hopf bifurcation in some infinite-dimensional systems Chapter 6 Bifurcations of orbits homoclinic and heteroclinic to hyperbolic equilibria This chapter is devoted to the generation of periodic orbits via homoclinic bifurcations A theorem due to Andronov and Leontovich describing homoclinic bifurcation in planar continuous-time systems is formulated A simple proof is . 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 bifur- cation. linear algebra, analysis, and differential equations is required. A brief summary of general mathematical terms and results, which are assumed to be known in the main text, appears at the end of the. and iterated maps is given, and the geometry of the phase portrait near such points is studied. A bifurcation diagram of a parameter-dependent system is introduced as a partitioning of its parameter