Attractor Dimension Estimates for Dynamical Systems: Theory and Computation

The main tool in estimating dimensions of invariant sets and entropies of dynamical systems developed in this book is based on Lyapunov functions. In this chapter we introduce the basic concept of global attractors. The existence of a global attractor for a dynamical system follows from the dissipativity of the system. In order to show the last property we use Lyapunov functions. In this chapter we also consider some applications of Lyapunov functions to stability problems of the Lorenz system. A central result is the existence of homoclinic orbits in the Lorenz system for certain parameters.

Global stability and dimension properties of nonlinear differential equations essentially depend on the contraction properties of k-parallelopipeds or k-ellipsoids under the flow of the associated variational equations. The goal of this second chapter is to develop some elements of multilinear algebra for the investigation of linear differential equations. This includes the discussion of singular value inequalities for linear operators in finite-dimensional spaces, the Fischer-Courant theorem as an extremal principle for eigenvalues of Hermitian matrices, exterior powers of operators and spaces, the logarithmic norm calculation and the use of the Kalman-Yakubovich frequency theorem for the effective estimation of time-dependent singular values of the solution operator to linear differential equations. The Kalman-Yakubovich frequency theorem is also used to get sufficient conditions for convergence in dynamical systems.

In Chap. 2 the dimension of a vector space was defined as the maximal number of linearly independent vectors existing in it. The simplest example of an n-dimensional space, whose dimension is understood in this sense, is the space \(\mathbb R^n\). The dimension theory, which was developed in the early 20th century, has extended this conception to more general classes of spaces and sets. In the following we give a short introduction into important notions of dimension for sets in general topological or metric spaces. We restrict ourselves to those dimensions and their properties which are especially useful in the investigation of ODE’s.

The first part (Sects. 4.2, 4.3, 4.5 and 4.6) of the present chapter contains several approaches to the investigation of the Fourier spectrum of almost periodic solutions to various differential equations. The core element here is the Cartwright theorem [6] that links the topological dimension of the orbit closure of an almost periodic flow and the algebraic dimension of its frequency module (Theorem 4.8). The next step is an extension of this theorem to non-autonomous differential equations (Theorem 4.11) originally presented in [7]. Applications of Cartwright’s theorems are given for almost periodic ODEs based on the approach due to R. A. Smith (Theorem 4.12) and for DDEs based on results of Mallet-Paret from [16] (Theorem 4.14). In Sect. 4.7 we develop a method for studying fractal dimensions of forced almost periodic oscillations using some kind of recurrence properties. This approach differs from the one due to Douady and Oesterlé and highly relies on almost periodicity. Some fundamental ideas firstly appeared in the works of Naito (see [17, 18]) and then were developed in [1, 2]. In Sect. 4.8 we study forced almost periodic oscillations in Chua’s circuit and compare the analytical upper estimates of the fractal dimension of their trajectory closures with numerical simulations given by the standard box-counting algorithm.

Nowadays there is a number of surveys and theoretical works devoted to Lyapunov exponents and Lyapunov dimension, however most of them are devoted to infinite dimensional systems or rely on special ergodic properties of a system. At the same time the provided illustrative examples are often finite dimensional systems and the rigorous proof of their ergodic properties can be a difficult task. Also the Lyapunov exponents and Lyapunov dimension have become so widespread and common that they are often used without references to the rigorous definitions or pioneering works. This chapter is devoted to the finite dimensional dynamical systems in Euclidean space and its aim is to explain, in a simple but rigorous way, the connection between the key works in the area: Kaplan and Yorke (the concept of Lyapunov dimension, 1979), Douady and Oesterlé (estimation of Hausdorff dimension via the Lyapunov dimension of maps, 1980), Constantin, Eden, Foias, and Temam (estimation of Hausdorff dimension via the Lyapunov exponents and Lyapunov dimension of dynamical systems, 1985–90), Leonov (estimation of the Lyapunov dimension via the direct Lyapunov method, 1991), and numerical methods for the computation of Lyapunov exponents and Lyapunov dimension. In this chapter a concise overview of the classical results is presented, various definitions of Lyapunov exponents and Lyapunov dimension are discussed. An effective analytical method for the estimation of the Lyapunov dimension is presented, its application to self-excited and hidden attractors of well-known dynamical systems is demonstrated, and analytical formulas of exact Lyapunov dimension are obtained.

In this chapter we present some auxiliary results from the linear operator theory and stability theory which are used in the sequel for dimension estimation. In Sect. 7.1 some elements of the exterior calculus of linear operators in linear spaces are introduced. Section 7.2 is concerned with orbital stability results for vector fields on Riemannian manifolds.

In this chapter generalizations of the Douady-Oesterlé  theorem (Theorem 5.​1, Chap. 5) are obtained for maps and vector fields on Riemannian manifolds. The proof of the generalized Douady-Oesterlé  theorem on manifolds is given in Sect. 8.1. In Sect. 8.2 it is shown that the Lyapunov dimension is an upper bound for the Hausdorff dimension. A tubular Carathéodory structure is used in Sect. 8.3 for the estimation of the Hausdorff dimension of invariant sets.

In this chapter we derive dimension and entropy estimates for invariant sets and global \(\mathcal{B}\)-attractors of cocycles in non-fibered and fibered spaces. A version of the Douady-Oesterlé theorem will be proven for local cocycles in an Euclidean space and for cocycles on Riemannian manifolds. As examples we consider cocycles, generated by the Rössler system with variable coefficients. We also introduce time-discrete cocycles on fibered spaces and define the topological entropy of such cocycles. Upper estimates of the topological entropy along an orbit of the base system are given which include the Lipschitz constants of the evolution system and the fractal dimension of the parameter dependent phase space.

In this chapter dimension estimates for maps and dynamical systems with specific properties are derived. In Sect. 10.1 a class of non-injective smooth maps is considered. Dimension estimates for piecewise non-injective maps are given in Sect. 10.2. For piecewise smooth maps with a special singularity set upper Hausdorff dimension estimates are shown in Sect. 10.3. Lower dimension estimates are shown in Sect. 10.4.