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**An introduction to dynamical systems and chaos.**
*(English)*
Zbl 1354.34001

New Delhi: Springer (ISBN 978-81-322-2555-3/hbk; 978-81-322-2556-0/ebook). xviii, 622 p. (2015).

The material of the book has been assembled from different papers and books published over the past 50 years on the strength of thinking and research experience of the author. The book contains 13 chapters covering all aspects of nonlinear dynamical continuous and discrete systems at the basic and advanced levels. The first chapter contains an introduction followed by a brief history of nonlinear science and discussion of one-dimensional continuous systems. Flows and their mathematical basis, qualitative approach, analysis of one-dimensional flows with examples, some important definitions, and conservative-dissipative systems are discussed in this chapter. Chapter 2 deals with linear systems of ordinary differential equations, both homogeneous and nonhomogeneous equations. The main emphasis is given for finding solutions of linear systems with constant coefficients so that the solution methods could be extended to higher dimensional systems easily. The well-known methods such as eigenvalue-eigenvector method and the fundamental matrix method have been described in detail. The properties of the fundamental matrix, the fundamental theorem, and important properties of exponential matrix function are given in this chapter. The general solution procedure for linear systems using fundamental matrix, the concept of generalized eigenvector, solutions of multiple eigenvalues, both real and complex, are discussed. Flows in \(\mathbb{R}^2\) that is, phase plane analysis, the equilibrium points and their stability characters, linearization of nonlinear systems, and its limitations are subject matters in Chapter 3. Mathematical pendulum problems and linear oscillators are also discussed in this chapter. Chapter 4 includes the theory of stability of linear and nonlinear systems. It also contains the notion of hyperbolicity, stable and unstable subspaces, Hartman-Grobman theorem, stable manifold theorem, and their applications. Chapter 5 is devoted to linear and nonlinear oscillations with some important theorems and physical applications. Chapter 6 presents the bifurcations in one-dimensional and two-dimensional systems depending on the parameters. Lorenz system and its properties are also given in this chapter. Chapter 7 discusses the basics of Lagrangian and Hamiltonian systems and their derivations. Hamiltonian flows, their properties, and a number of worked-out examples are presented in this chapter. Chapter 8 contains an introduction to the Lie symmetry under continuous group of transformations, invariance principle, and systematic calculation of symmetries for ordinary and partial differential equations. Chapter 9 discusses maps, their iterates, fixed points and their stabilities, periodic cycles, and some important theorems. In Chapter 10 some important maps, namely tent map, logistic map, shift map, Henon map, etc., are discussed elaborately. Chapter 11 deals with conjugacy and semi-conjugacy relations among maps, their properties, and proofs of some important theorems. Chapter 12 contains a brief history of chaos and its mathematical theory. Emphasis has been given to establish mathematical theories on chaotic systems, quantifying chaos and universality. Routes of chaos, chaotic maps, Sharkovskii ordering and other theory are discussed in this chapter. Chapter 13 is devoted to the study of fractals, their self-similarities, scaling, and dimensions of fractal objects with many worked-out examples.

A number of examples worked out in detail and exercises are included. The book can be useful for courses in dynamical systems and chaos, nonlinear dynamics, etc., for advanced undergraduate and postgraduate students in mathematics, physics and engineering.

A number of examples worked out in detail and exercises are included. The book can be useful for courses in dynamical systems and chaos, nonlinear dynamics, etc., for advanced undergraduate and postgraduate students in mathematics, physics and engineering.

Reviewer: Alexander Grin (Grodno)

### MSC:

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

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

37C75 | Stability theory for smooth dynamical systems |

34D20 | Stability of solutions to ordinary differential equations |

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

34D45 | Attractors of solutions to ordinary differential equations |

34C28 | Complex behavior and chaotic systems of ordinary differential equations |