Attractor dimension estimates for dynamical systems: theory and computation. Dedicated to Gennady Leonov (Q2192973)

This book covers some special topics in the theory of dynamical systems, with emphasis on attractor dimension estimates and investigations on global attractors and invariant sets by means of Lyapunov functions and adapted metrics. It is interesting and very well written. Mostly, chapters are self-contained and rich of detailed explanations. Many powerful computational tools and algorithms provide a solid numerical background for the study of attractor dimensions. The authors present MATLAB programs for the computation of the dimensions and of Lyapunov exponents, and interestingly, they also visualize these quantities graphically. The effectiveness of introducing the tool of Lyapunov functions into the study of dimensional features is shown for several concrete dynamical systems, such as the Hénon map, Lorenz and Rössler systems, as well as their generalizations that one finds in various physical applications. This book is structured in three parts and each part consists of some chapters covering some special topics. In Part I, the authors provide the basic concepts of attractor theory, exterior products and dimension theory which are related to Lyapunov functions, dissipativity, homoclinic orbits, the Yakubovich-Kalman frequency theorem, the frequency-domain estimation of singular values, topological dimension, Hausdorff dimension, fractal dimension, topological entropy and dimension-like features. In Part II, the authors mention dimension properties of dynamical systems in Euclidean spaces. They provide dimension estimates for almost periodic flows. This part covers estimates of the topological dimension, of the Hausdorff dimension and of the fractal dimension for invariant sets of some concrete physical systems. Further, estimates of the Lyapunov dimension are given. Analytical formulas for the exact Lyapunov dimension of well-known dynamical systems are studied. Among these are: the Hénon map, the Lorenz system, the Glukhovsky-Dolzhansky system, the Yang-Tigan system and the Shimizu-Morioka system. Computation of attractors and Lyapunov dimension are also provided. In Part III, the authors explore dimension properties for dynamical systems on manifolds. This part provides basic concepts for dimension estimation on manifolds. The authors describe the Hausdorff dimension estimates for invariant sets of vector fields, the Lyapunov dimension as upper bound of the fractal dimension, the Hausdorff dimension estimates by use of a tubular Carathéodory structure and applications to stability theory. Dimension and entropy estimates for global attractors of cocycles on manifolds are discussed as well. Dimension estimates are discussed also for non-injective smooth maps, piecewise \(C^{1}\)-maps and maps with special singularity sets. In each part, the authors justify the results by providing theoretical and numerical examples and applications. To make the material self-contained, the authors present at the end of the book some basic concepts about differential manifolds, Riemannian manifolds, degree theory, homology theory, geometric measure theory and hyperbolicity in dynamical systems. Overall, this book contains advanced material on attractor dimension estimates for dynamical systems. This is definitely suitable for researchers in applied mathematics and computational theory of dynamical systems.