/pdf/introduction-to-the-modern-theory-of-dynamical-systems-1eced70mum.pdf

Introduction to the Modern Theory of Dynamical Systems

In this chapter we will embark upon the task of systematically identifying important specific phenomena associated with the asymptotic behavior of smooth dynamical systems We will build upon the results of our survey of specific examples in Chapter 1 as well as on the insights gained from the general structural approach outlined and illustrated in Chapter 2 Most of the properties discussed in the present chapter are in fact topological invariants and can be defined for broad classes of topological dynamical systems, including symbolic ones The predominance of topological invariants fits well with the picture that emerges from the considerations of Sections 21, 23, 24, and 26 The considerations of the previous chapter make it very plausible that smooth dynamical systems are virtually never differentiably stable and can only rarely be classified locally up to smooth conjugacy In contrast, structural and the related topological stability seem to be fairly widespread phenomena We will consider three broad classes of asymptotic invariants: (i) growth of the numbers of orbits of various kinds and of the complexity of orbit families, (ii) types of recurrence, and (iii) asymptotic distribution and statistical behavior of orbits The first two classes are of a purely topological nature; they are discussed in the present chapter The last class is naturally related to ergodic theory and hence we will provide an introduction to key aspects of that subject This will require some space so we put that material into a separate chapter The two chapters are intimately connected

Introduction to the Modern Theory of Dynamical Systems: PRINCIPAL CLASSES OF ASYMPTOTIC TOPOLOGICAL INVARIANTS

Introduction to the Modern Theory of Dynamical Systems: INTRODUCTION: WHAT IS LOW-DIMENSIONAL DYNAMICS?

Understanding the interplay of order and disorder in chaos is a central challenge in modern quantitative science. Approximate linear representations of nonlinear dynamics have long been sought, driving considerable interest in Koopman theory. We present a universal, data-driven decomposition of chaos as an intermittently forced linear system. This work combines delay embedding and Koopman theory to decompose chaotic dynamics into a linear model in the leading delay coordinates with forcing by low-energy delay coordinates; this is called the Hankel alternative view of Koopman (HAVOK) analysis. This analysis is applied to the Lorenz system and real-world examples including Earth’s magnetic field reversal and measles outbreaks. In each case, forcing statistics are non-Gaussian, with long tails corresponding to rare intermittent forcing that precedes switching and bursting phenomena. The forcing activity demarcates coherent phase space regions where the dynamics are approximately linear from those that are strongly nonlinear. The huge amount of data generated in fields like neuroscience or finance calls for effective strategies that mine data to reveal underlying dynamics. Here Brunton et al.develop a data-driven technique to analyze chaotic systems and predict their dynamics in terms of a forced linear model.

/pdf/chaos-as-an-intermittently-forced-linear-system-8yq9lxufzg.pdf

Chaos as an intermittently forced linear system.

We establish the convergence of a class of numerical algorithms, known as Dynamic Mode Decomposition (DMD), for computation of the eigenvalues and eigenfunctions of the infinite-dimensional Koopman operator. The algorithms act on data coming from observables on a state space, arranged in Hankel-type matrices. The proofs utilize the assumption that the underlying dynamical system is ergodic. This includes the classical measure-preserving systems, as well as systems whose attractors support a physical measure. Our approach relies on the observation that vector projections in DMD can be used to approximate the function projections by the virtue of Birkhoff's ergodic theorem. Using this fact, we show that applying DMD to Hankel data matrices in the limit of infinite-time observations yields the true Koopman eigenfunctions and eigenvalues. We also show that the Singular Value Decomposition, which is the central part of most DMD algorithms, converges to the Proper Orthogonal Decomposition of observables. We use this result to obtain a representation of the dynamics of systems with continuous spectrum based on the lifting of the coordinates to the space of observables. The numerical application of these methods is demonstrated using well-known dynamical systems and examples from computational fluid dynamics.

Ergodic theory, Dynamic Mode Decomposition and Computation of Spectral Properties of the Koopman operator

We explore some properties of Lyapunov exponents of measures preserved by smooth maps of the interval, and study the behaviour of the Lyapunov exponents under topological conjugacy.

/pdf/topological-invariance-of-the-sign-of-the-lyapunov-exponents-2wz7m40fyj.pdf

Topological invariance of the sign of the Lyapunov Exponents in one-dimensional maps

We show that for any C^1+alpha diffeomorphism of a compact Riemannian manifold, every non-atomic, ergodic, invariant probability measure with non-zero Lyapunov exponents is approximated by uniformly hyperbolic sets in the sense that there exists a sequence Omega_n of compact, topologically transitive, locally maximal, uniformly hyperbolic sets, such that for any sequence mu_n of f-invariant ergodic probability measures with supp (mu_n) in Omega_n we have mu_n -> mu in the weak-* topology.

Uniform hyperbolic approximations of measures with non zero Lyapunov exponents

We consider compact invariant sets \Lambda for C^{1} maps in arbitrary dimension. We prove that if \Lambda contains no critical points then there exists an invariant probability measure with a Lyapunov exponent \lambda which is the minimum of all Lyapunov exponents for all invariant measures supported on \Lambda. We apply this result to prove that \Lambda is uniformly expanding if every invariant probability measure supported on \Lambda is hyperbolic repelling. This generalizes a well known theorem of Mane' to the higher-dimensional setting.

A minimum principle for Lyapunov exponents and a higher-dimensional version of a Theorem of Mane'

We study the hyperbolicity of a class of horseshoes exhibiting an internal tangency, i.e. a point of homoclinic tangency accumulated by periodic points. We show that the periodic points are uniformly hyperbolic.

Hyperbolicity of periodic points for horseshoes with internal tangencies

We derive some new conditions for integrability of dynamically defined C 1 invariant splittings in arbitrary dimension and co-dimension. In particular we prove that every 2-dimensional C 1 invariant decomposition on a 3-dimensional manifold satisfying a volume domination condition is uniquely integrable. In the special case of volume preserving diffeomorphisms we show that standard dynamical domination is already sufficient to guarantee unique integrability.

/pdf/integrability-of-c-1-invariant-splittings-1rk5qfzldi.pdf

Stefano Luzzatto

Papers

Topological invariance of the sign of the Lyapunov Exponents in one-dimensional maps

Uniform hyperbolic approximations of measures with non zero Lyapunov exponents

A minimum principle for Lyapunov exponents and a higher-dimensional version of a Theorem of Mane'

Hyperbolicity of periodic points for horseshoes with internal tangencies

Integrability of C^1 invariant splittings