Discrete dynamical systems and chaos.

*(English)*Zbl 0819.34001
Pitman Monographs and Surveys in Pure and Applied Mathematics. 62. Harlow: Longman Scientific & Technical. New York, NY: John Wiley & Sons, Inc.. 282 p. (1992).

In this book, the author brings to undergraduate level the fundamental ideas, definitions and results on discrete dynamical systems and chaos. The only prerequisites are a few fundamental results from calculus of one variable. The book is divided into seven chapters. To further help readers understand the book, there are practice problems at the end of each chapter except the first.

In the first chapter, the author presents standard definitions and terminology. In particular, he defines a fixed point, a periodic point, a periodic orbit, limit point, limit sets and aperiodic orbits. The stability of stationary states is also introduced in the first chapter. The logistic map, the two species discrete predator-prey model, one- dimensional discrete models with delay, Lorenz model of atmospheric behavior and Hopfield model of neural networks are used as examples to motivate basic dynamical systems’ ideas.

An extensive analysis of one-dimensional dynamical systems is presented in the second chapter. The system considered depends on one parameter and is of the form \(X_{n+1} = F(x_ n,a)\) where \(F(.,a) : I_ a \to \mathbb{R}\) usually maps the interval \(I_ a\) into itself. The cobweb method, the idea of conjugacy, the stability and instability of stationary states and periodic orbits are all introduced and studied in the second chapter. Sarkovskii’s theorem and a sufficient condition for all orbits of a dynamical system in \(\mathbb{R}\) to converge to a stationary state are also presented. The author ends the chapter with a discussion of bifurcation points and bifurcation diagrams for one-parameter families of maps. The bifurcations studied include period doubling bifurcations route to chaos, pitchfork bifurcations, transcritical bifurcations and fold bifurcations.

Chapter three contains a thorough overview of the important topics of linear algebra and calculus of several variables. In particular, the author presented the standard definition of \(\mathbb{R}^ q\), and its structure. Matrices, with particular attention to eigenvalues and spectral radius, are also presented. In addition, the author discusses matrices as operators. The operator norm of a matrix, its relation to the spectral radius, the eigenvectors and the real canonical form of a matrix are all presented in the overview. The chapter ends with a summary on differentiability and first order approximation.

In the fourth chapter, the author uses three fundamental tools to analyze discrete linear dynamical systems governed by the action of a matrix. The three fundamental tools are (i) the spectrum, (ii) the real canonical form of a matrix, and (iii) the fundamental property of the spectral radius. First, the author uses the idea of conjugacy to study the relation between linear affine processes. Next, the case of matrices having spectral radius smaller one, or such that the inverse has spectral radius smaller than one is studied. In addition, the saddle case is studied. The case when at least an eigenvalue has modulus one is analyzed at the end of the chapter.

The fifth chapter is the more challenging part of the book. Here, the author studies the dynamics of nonlinear discrete systems. First, the following three types of maps are analyzed: (i) contractive, (ii) gradient, and (iii) triangular maps. Next, local asymptotic stabilities of equilibrium points and periodic orbits are analyzed. The author then studies repellers and saddles, with a brief excursion on stable and unstable manifolds. Dissipative and quasi-bounded maps, two families of maps for which the existence of a bounded and invariant region can be easily established, are also studied. Bifurcation theory that includes Hopf bifurcation is discussed at the end of the chapter.

Chaotic behavior is presented in the sixth chapter. Here, a notion of the core of a dynamical system is introduced. Attractors are particular subsets of the core of dynamical systems. In addition, a definition of chaotic dynamical system is presented. This definition is based on the presence of dense orbits and on the instability of all orbits. Other alternative definitions of chaos including the concept of sensitivity with respect to initial conditions are also discussed. The dimension of chaotic attractors is introduced and analyzed. The capacity (Hausdorff) dimension and the correlation dimension of Grassberger-Procaccia are the two types of dimensions discussed. The author ends the chapter with a study of Lyapunov exponents and their relation to stability and sensitivity with respect to initial conditions.

Chapter seven, the last chapter, contains an extensive analysis of four specific discrete models. The four models considered are: (i) blood cell population models, (ii) predator-prey models for competition between two species, (iii) Lorenz model for the dynamics of atmospheric change and (iv) Hopfield model of a neural network.

In the first chapter, the author presents standard definitions and terminology. In particular, he defines a fixed point, a periodic point, a periodic orbit, limit point, limit sets and aperiodic orbits. The stability of stationary states is also introduced in the first chapter. The logistic map, the two species discrete predator-prey model, one- dimensional discrete models with delay, Lorenz model of atmospheric behavior and Hopfield model of neural networks are used as examples to motivate basic dynamical systems’ ideas.

An extensive analysis of one-dimensional dynamical systems is presented in the second chapter. The system considered depends on one parameter and is of the form \(X_{n+1} = F(x_ n,a)\) where \(F(.,a) : I_ a \to \mathbb{R}\) usually maps the interval \(I_ a\) into itself. The cobweb method, the idea of conjugacy, the stability and instability of stationary states and periodic orbits are all introduced and studied in the second chapter. Sarkovskii’s theorem and a sufficient condition for all orbits of a dynamical system in \(\mathbb{R}\) to converge to a stationary state are also presented. The author ends the chapter with a discussion of bifurcation points and bifurcation diagrams for one-parameter families of maps. The bifurcations studied include period doubling bifurcations route to chaos, pitchfork bifurcations, transcritical bifurcations and fold bifurcations.

Chapter three contains a thorough overview of the important topics of linear algebra and calculus of several variables. In particular, the author presented the standard definition of \(\mathbb{R}^ q\), and its structure. Matrices, with particular attention to eigenvalues and spectral radius, are also presented. In addition, the author discusses matrices as operators. The operator norm of a matrix, its relation to the spectral radius, the eigenvectors and the real canonical form of a matrix are all presented in the overview. The chapter ends with a summary on differentiability and first order approximation.

In the fourth chapter, the author uses three fundamental tools to analyze discrete linear dynamical systems governed by the action of a matrix. The three fundamental tools are (i) the spectrum, (ii) the real canonical form of a matrix, and (iii) the fundamental property of the spectral radius. First, the author uses the idea of conjugacy to study the relation between linear affine processes. Next, the case of matrices having spectral radius smaller one, or such that the inverse has spectral radius smaller than one is studied. In addition, the saddle case is studied. The case when at least an eigenvalue has modulus one is analyzed at the end of the chapter.

The fifth chapter is the more challenging part of the book. Here, the author studies the dynamics of nonlinear discrete systems. First, the following three types of maps are analyzed: (i) contractive, (ii) gradient, and (iii) triangular maps. Next, local asymptotic stabilities of equilibrium points and periodic orbits are analyzed. The author then studies repellers and saddles, with a brief excursion on stable and unstable manifolds. Dissipative and quasi-bounded maps, two families of maps for which the existence of a bounded and invariant region can be easily established, are also studied. Bifurcation theory that includes Hopf bifurcation is discussed at the end of the chapter.

Chaotic behavior is presented in the sixth chapter. Here, a notion of the core of a dynamical system is introduced. Attractors are particular subsets of the core of dynamical systems. In addition, a definition of chaotic dynamical system is presented. This definition is based on the presence of dense orbits and on the instability of all orbits. Other alternative definitions of chaos including the concept of sensitivity with respect to initial conditions are also discussed. The dimension of chaotic attractors is introduced and analyzed. The capacity (Hausdorff) dimension and the correlation dimension of Grassberger-Procaccia are the two types of dimensions discussed. The author ends the chapter with a study of Lyapunov exponents and their relation to stability and sensitivity with respect to initial conditions.

Chapter seven, the last chapter, contains an extensive analysis of four specific discrete models. The four models considered are: (i) blood cell population models, (ii) predator-prey models for competition between two species, (iii) Lorenz model for the dynamics of atmospheric change and (iv) Hopfield model of a neural network.

Reviewer: A.-A.Yakubu (Washington)

##### MSC:

34-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordinary differential equations |

39-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to difference and functional equations |

37-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to dynamical systems and ergodic theory |

37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |