Nonlinear dynamics and chaotic phenomena. An introduction (Q5906920)

This book has grown out of the author's lecture notes for a graduate level course on nonlinear dynamics for students in applied mathematics, physics and engineering. The emphasis is on a clear and comprehensive description of the basic concepts and results of dynamical system theory. Frequently new concepts are demonstrated via detailed examples, often further elaborated in extensive footnotes. Providing deeper insight into the underlying dynamical behavior is more important to the author than providing solution methods. Consequently to the Hamiltonian formulation of mechanics is given preference over the Newtonian approach. Although the author has tried to keep the mathematical formalism to a minimum, this text is definitely oriented towards the more theoretically inclined reader. The first three chapters introduce the reader to the background subjects of nonlinear differential equations, bifurcation theory and Hamiltonian dynamics. Phase space concepts are defined, and chapter 4 deals with integrable Hamiltonian systems. The celebrated Kolmogorov-Arnold-Moser theorem and criteria for the breakdown of integrability and transition to chaos are discussed together with topics such as Arnold diffusion and internal resonances. The appearance of chaotic motion in conservative systems via homoclinic intersections of unstable manifolds is treated in chapter 5. Quantitative measures for determining chaotic behavior, such as Lyapunov exponents, Kolmogorov entropy etc., are outlined, and the Baker's transformation is discussed as a paradigm of deterministic chaos. The subject of chaos in dissipative systems is covered in chapter 6 leading to the notion of a strange attractor with a fractal structure. The Lorenz attractor and the logistic map are discussed in detail as simple examples with a strange attractor. In chapter 7 fractals and multi-fractals are used to characterize fully-developed fluid turbulence. Though not rigorously derived from the governing equations, they seem to mimic the turbulence dynamics qualitatively. The last chapter is devoted to the still speculative topic of using the Painlevé property as test of integrability. The book concludes with a few exercises.