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During the last few years several good textbooks on nonlinear dynamics have ap­ peared for graduate students in applied mathematics. It seems, however, that the majority of such books are still too theoretically oriented and leave many practi­ cal issues unclear for people intending to apply the theory to particular research problems. This book is designed for advanced undergraduate or graduate students in mathematics who will participate in applied research. It is also addressed to professional researchers in physics, biology, engineering, and economics who use dynamical systems as modeling tools in their studies. Therefore, only a moderate mathematical background in geometry, linear algebra, analysis, and differential equations is required. A brief summary of general mathematical terms and results that are assumed to be known in the main text appears at the end of the book. Whenever possible, only elementary mathematical tools are used. For example, we do not try to present normal form theory in full generality, instead developing only the portion of the technique sufficient for our purposes. The book aims to provide the student (or researcher) with both a solid basis in dynamical systems theory and the necessary understanding of the approaches, methods, results, and terminology used in the modem applied mathematics litera­ ture. A key theme is that of topological equivalence and codimension, or "what one may expect to occur in the dynamics with a given number of parameters allowed to vary.

### 1. Introduction to Dynamical Systems

Abstract
This chapter introduces some basic terminology. First, we define a dynamical system and give several examples, including symbolic dynamics. Then we introduce the notions of orbits, invariant sets, and their stability. As we shall see while analyzing the Smale horseshoe, invariant sets can have very complex structures. This is closely related to the fact discovered in the 1960s that rather simple dynamical systems may behave “randomly,” or “chaotically.” Finally, we discuss how differential equations can define dynamical systems of both finite and infinite dimensions.
Yuri A. Kuznetsov

### 2. Topological Equivalence, Bifurcations, and Structural Stability of Dynamical Systems

Abstract
In this chapter we introduce and discuss the following fundamental notions that will be used throughout the book: topological equivalence of dynamical systems and their classification, bifurcations and bifurcation diagrams, and topological normal forms for bifurcations. The last section is devoted to the more abstract notion of structural stability. In this chapter we will be dealing only with dynamical systems in the state space X = ℝ n .
Yuri A. Kuznetsov

### 3. One-Parameter Bifurcations of Equilibria in Continuous-Time Systems

Abstract
In this chapter we formulate conditions defining the simplest bifurcations of equilibria in n-dimensional continuous-time systems: the fold and the Hopf bifurcations. Then we study these bifurcations in the lowest possible dimensions: the fold bifurcation for scalar systems and the Hopf bifurcation for planar systems. Chapter 5 shows how to “lift” these results to n-dimensional situations.
Yuri A. Kuznetsov

### 4. One-Parameter Bifurcations of Fixed Points in Discrete-Time Systems

Abstract
In this chapter, which is organized very much like Chapter 3, we present bifurcation conditions defining the simplest bifurcations of fixed points in n-dimensional discrete-time dynamical systems: the fold, the flip, and the Neimark-Sacker bifurcations. Then we study these bifurcations in the lowest possible dimension in which they can occur: the fold and flip bifurcations for scalar systems and the Neimark-Sacker bifurcation for planar systems. In Chapter 5 it will be shown how to apply these results to n-dimensional systems when n is larger than one or two, respectively.
Yuri A. Kuznetsov

### 5. Bifurcations of Equilibria and Periodic Orbits in n-Dimensional Systems

Abstract
In the previous two chapters we studied bifurcations of equilibria and fixed points in generic one-parameter systems having the minimum possible phase dimensions. Indeed, the systems we analyzed were either one- or two-dimensional. This chapter shows that the corresponding bifurcations occur in “essentially” the same way for generic n-dimensional systems. As we shall see, there are certain parameter-dependent one- or two-dimensional invariant manifolds on which the system exhibits the corresponding bifurcations, while the behavior off the manifolds is somehow “trivial,” for example, the manifolds may be exponentially attractive. Moreover, such manifolds (called center manifolds) exist for many dissipative infinite-dimensional dynamical systems. Below we derive explicit formulas for the approximation of center manifolds in finite dimensions and for systems restricted to them at bifurcation parameter values. In Appendix 1 we consider a reaction-diffusion system on an interval to illustrate the necessary modifications of the technique to handle infinite-dimensional systems.
Yuri A. Kuznetsov

### 6. Bifurcations of Orbits Homoclinic and Heteroclinic to Hyperbolic Equilibria

Abstract
In this chapter we will study global bifurcations corresponding to the appearance of homoclinic or heteroclinic orbits connecting hyperbolic equilibria in continuous-time dynamical systems. We will consider in detail two- and three-dimensional cases for two reasons: first, because in these cases geometrical intuition can be fully exploited, and second, because all the main results can be generalized to higher dimensions.
Yuri A. Kuznetsov

### 7. Other One-Parameter Bifurcations in Continuous-Time Systems

Abstract
The list of possible bifurcations in multidimensional systems is not exhausted by those studied in the previous chapters. Actually, even the complete list of all generic one-parameter bifurcations is unknown.
Yuri A. Kuznetsov

### 8. Two-Parameter Bifurcations of Equilibria in Continuous-Time Dynamical Systems

Abstract
This chapter is devoted to generic bifurcations of equilibria in two-parameter systems of differential equations. First, we make a complete list of such bifurcations. Then, we derive a parameter-dependent normal form for each bifurcation in the minimal possible phase dimension and specify relevant nondegeneracy conditions. Next, we truncate higher-order terms and present the bifurcation diagrams of the resulting system. The analysis is completed by a discussion of the effect of the higher-order terms. In those cases where the higher-order terms do not qualitatively alter the bifurcation diagram, the truncated systems provide topological normal forms for the relevant bifurcations. The results of this chapter can be applied to n-dimensional systems by means of the parameter-dependent version of the Center Manifold Theorem (see Chapter 5).
Yuri A. Kuznetsov

### 9. Two-Parameter Bifurcations of Fixed Points in Discrete-Time Dynamical Systems

Abstract
This chapter is devoted to the study of generic bifurcations of fixed points of two-parameter maps. First we derive a list of such bifurcations. As for the final two bifurcations in the previous chapter, the description of the majority of these bifurcations is incomplete in principle. For all but two cases, only approximate normal forms can be constructed. Some of these normal forms will be presented in terms of associated planar continuous-time systems whose evolution operator φ 1 approximates the map in question (or an appropriate number of iterates of the map). We present bifurcation diagrams of the approximate normal forms and discuss their relationships with the original maps.
Yuri A. Kuznetsov

### 10. Numerical Analysis of Bifurcations

Abstract
In this chapter we shall describe some of the basic techniques used in the numerical analysis of dynamical systems. We assume that low-level numerical routines like those for solving linear systems, finding eigenvectors, and performing numerical differentiation of functions or integration of ODEs are known to the reader. Instead we focus on algorithms that are more specific to bifurcation analysis, specifically those for the location of equilibria (fixed points) and their continuation with respect to parameters, and for the detection, analysis, and continuation of bifurcations. Special attention is given to location and continuation of limit cycles and their associated bifurcations. We deal mainly with the continuous-time case and give only brief remarks on discrete-time systems. Appendix 1 summarizes estimates of convergence of Newton-like methods. Appendix 2 presents numerical methods for the continuation and analysis of homoclinic bifurcations. The bibliographical notes in Appendix 3 include references to standard noninteractive software packages and interactive programs available for continuation and bifurcation analysis of dynamical systems. Actually, the main goal of this chapter is to provide the reader with an understanding of the methods implemented in widely used software for dynamical systems analysis.
Yuri A. Kuznetsov