Extracting qualitative dynamics from experimental data

V. I. Arnold, on behalf of the International Mathematical Union has written to a number of mathematicians with a suggestion that they describe some great problems for the next century. This report is my response. Arnold's invitation is inspired in part by Hilbert's list of 1900 (see e.g. [Browder, 1976]) and I have used that list to help design this essay. I have listed 18 problems, chosen with these criteria:

Mathematical problems for the next century

In view of the minimization of a nonsmooth nonconvex function f, we prove an abstract convergence result for descent methods satisfying a sufficient-decrease assumption, and allowing a relative error tolerance. Our result guarantees the convergence of bounded sequences, under the assumption that the function f satisfies the Kurdyka–Łojasiewicz inequality. This assumption allows to cover a wide range of problems, including nonsmooth semi-algebraic (or more generally tame) minimization. The specialization of our result to different kinds of structured problems provides several new convergence results for inexact versions of the gradient method, the proximal method, the forward–backward splitting algorithm, the gradient projection and some proximal regularization of the Gauss–Seidel method in a nonconvex setting. Our results are illustrated through feasibility problems, or iterative thresholding procedures for compressive sensing.

/pdf/convergence-of-descent-methods-for-semi-algebraic-and-tame-sdj1f3ttk2.pdf

Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward-backward splitting, and regularized Gauss-Seidel methods

Probability and Measure

Introduction to Ordinary Differential Equations * Linear Systems and Stability of Nonlinear Systems * Applications * Hyperbolic Theory * Continuation of Periodic Solutions * Homoclinic Orbits, Melnikov?s Method, and Chaos * Averaging * Local Bifurcation * References * Index.

http://www.math.ecnu.edu.cn/~zmwang/teaching/Advanced_ODE-2014-2015/1-Textbook_and_Reference_book/2-Ordinary%20Differential%20Equations%20with%20Applications.pdf

Ordinary Differential Equations with Applications

0. Introduction.- I. Circle Diffeomorphisms.- 1. The Combinatorial Theory of Poincare.- 2. The Topological Theory of Denjoy.- 2.a The Denjoy Inequality.- 2.b Ergodicity.- 3. Smooth Conjugacy Results.- 4. Families of Circle Diffeomorphisms Arnol'd tongues.- 5. Counter-Examples to Smooth Linearizability.- 6. Frequency of Smooth Linearizability in Families.- 7. Some Historical Comments and Further Remarks.- II. The Combinatorics of One-Dimensional Endomorphisms.- 1. The Theorem of Sarkovskii.- 2. Covering Maps of the Circle as Dynamical Systems.- 3. The Kneading Theory and Combinatorial Equivalence.- 3.a Examples.- 3.b Hofbauer's Tower Construction.- 4. Full Families and Realization of Maps.- 5. Families of Maps and Renormalization.- 6. Piecewise Monotone Maps can be Modelled by Polynomial Maps.- 7. The Topological Entropy.- 8. The Piecewise Linear Model.- 9. Continuity of the Topological Entropy.- 10. Monotonicity of the Kneading Invariant for the Quadratic Family.- 11. Some Historical Comments and Further Remarks.- III. Structural Stability and Hyperbolicity.- 1. The Dynamics of Rational Mappings.- 2. Structural Stability and Hyperbolicity.- 3. Hyperbolicity in Maps with Negative Schwarzian Derivative.- 4. The Structure of the Non-Wandering Set.- 5. Hyperbolicity in Smooth Maps.- 6. Misiurewicz Maps are Almost Hyperbolic.- 7. Some Further Remarks and Open Questions.- IV. The Structure of Smooth Maps.- 1. The Cross-Ratio: the Minimum and Koebe Principle.- l.a Some Facts about the Schwarzian Derivative.- 2. Distortion of Cross-Ratios.- 2.a The Zygmund Conditions.- 3. Koebe Principles on Iterates.- 4. Some Simplifications and the Induction Assumption.- 5. The Pullback of Space: the Koebe/Contraction Principle.- 6. Disjointness of Orbits of Intervals.- 7. Wandering Intervals Accumulate on Turning Points.- 8. Topological Properties of a Unimodal Pullback.- 9. The Non-Existence of Wandering Intervals.- 10. Finiteness of Attractors.- 11. Some Further Remarks and Open Questions.- V. Ergodic Properties and Invariant Measures.- 1. Ergodicity, Attractors and Bowen-Ruelle-Sinai Measures.- 2. Invariant Measures for Markov Maps.- 3. Constructing Invariant Measures by Inducing.- 4. Constructing Invariant Measures by Pulling Back.- 5. Transitive Maps Without Finite Continuous Measures.- 6. Frequency of Maps with Positive Liapounov Exponents in Families and Jakobson's Theorem.- 7. Some Further Remarks and Open Questions.- VI. Renormalization.- 1. The Renormalization Operator.- 2. The Real Bounds.- 3. Bounded Geometry.- 4. The PullBack Argument.- 5. The Complex Bounds.- 6. Riemann Surface Laminations.- 7. The Almost Geodesic Principle.- 8. Renormalization is Contracting.- 9. Universality of the Attracting Cantor Set.- 10. Some Further Remarks and Open Questions.- VII. Appendix.- 1. Some Terminology in Dynamical Systems.- 2. Some Background in Topology.- 3. Some Results from Analysis and Measure Theory.- 4. Some Results from Ergodic Theory.- 5. Some Background in Complex Analysis.- 6. Some Results from Functional Analysis.

One-dimensional dynamics

Geometric theory of dynamical systems

1 Differentiable Manifolds and Vector Fields.- 0 Calculus in ?n and Differentiable Manifolds.- 1 Vector Fields on Manifolds.- 2 The Topology of the Space of Cr Maps.- 3 Transversality.- 4 Structural Stability.- 2 Local Stability.- 1 The Tubular Flow Theorem.- 2 Linear Vector Fields.- 3 Singularities and Hyperbolic Fixed Points.- 4 Local Stability.- 5 Local Classification.- 6 Invariant Manifolds.- 7 The ?-lemma (Inclination Lemma). Geometrical Proof of Local Stability.- 3 The Kupka-Smale Theorem.- 1 The Poincare Map.- 2 Genericity of Vector Fields Whose Closed Orbits Are Hyperbolic.- 3 Transversality of the Invariant Manifolds.- 4 Genericity and Stability of Morse-Smale Vector Fields.- 1 Morse-Smale Vector Fields Structural Stability.- 2 Density of Morse-Smale Vector Fields on Orientable Surfaces.- 3 Generalizations.- 4 General Comments on Structural Stability. Other Topics.- Appendix: Rotation Number and Cherry Flows.- References.

Geometric Theory of Dynamical Systems : An Introduction

Our aim is to show that the Julia-Fatou--Sullivan structure theory for the dynamics of rational maps is also valid for smooth endomorphisms of the circle (and of the interval) under extremely mild smoothness and non-flatness conditions. In order to stress the similarity between real and complex one-dimensional dynamics let us recall the main results from the Fatou-Julia-Sullivan theory. If f is a rational map then there is a dynamical decomposition of the Riemann sphere into the disjoint union of two totally invariant (i.e., both forward and backward invariant) sets J(f), F(f). Here, F(f) is the domain of normality of the family of iterates of f , and is called the Fatou set. Its complement, which is called the Julia set off , is a compact set, which contains all the complications of the dynamics off . The connected components of the open set F(f) are mapped onto each other by f. Hence the orbit of a component of F(f) is the union of some components of F(f). Julia proved in the beginning of the century that if a component of F(f) is periodic and contains an attracting periodic point then the orbit of this component must contain a critical point. Sullivan, in the remarkable paper [Su], showed via quasi-conformal deformations that the components are eventually periodic and fall into finitely many orbits. Let N be either the circle S J or a compact interval of the real line and f." N---~N be a smooth endomorphism. A critical point of f is a point where the derivative vanishes. A critical point is non-flat if some (higher) derivative is non-zero. A critical point is an inflection point if it has a neighbourhood where f is monotone. Otherwise it is called a turning point. Assume f is not a homeomorphism ( i f f is a homeomorphism then these maps correspond to degree + 1 rational maps of the Riemann sphere and the situation is

https://projecteuclid.org/download/pdf_1/euclid.acta/1485890671

Julia-Fatou-Sullivan theory for real one-dimensional dynamics

In this paper we prove that in any non-trivial real analytic family of quasiquadratic maps, almost any map is either regular (i.e., it has an attracting cycle) or stochastic (i.e., it has an absolutely continuous invariant measure). To this end we show that the space of analytic maps is foliated by codimension-one analytic submanifolds, “hybrid classes”. This allows us to transfer the regular or stochastic property of the quadratic family to any non-trivial real analytic family.

/pdf/regular-or-stochastic-dynamics-in-real-analytic-families-of-31yki9bbom.pdf

Welington de Melo

Papers

One-dimensional dynamics

Geometric theory of dynamical systems

Geometric Theory of Dynamical Systems : An Introduction

Julia-Fatou-Sullivan theory for real one-dimensional dynamics

Regular or stochastic dynamics in real analytic families of unimodal maps