Dynamical Systems 3
$C^r$ Stability Conjecture
Structural stability is a global topological property: a dynamical system is structurally stable if all nearby systems have the same orbit structure, up to continuous reparametrisation. Hyperbolicity is a local differential property: the tangent bundle over the recurrent set splits into uniformly contracting and expanding directions. That these two conditions should be equivalent is one of the deepest principles in smooth dynamics. Conjecture ($C^r$ Stability Conjecture, Palis–Smale, ~1970) Let $M$ be a closed smooth manifold and $r \geq 1$. If $f \in \mathrm{Diff}^r(M)$ is $C^r$-structurally stable, then $f$ is hyperbolic, i.e., it satisfies Axiom A and the Strong Transversality Condition.
Free Books on Dynamical Systems
Arxiv/ Free Books # 1. Lectures on Neural Dynamics - Francesco Bullo # Chapter 1: Neural circuit models based on firing rates and Hopfield networks: their dynamics, interconnections, and local Hebbian adaptation rules Chapter 2: Stability in dynamic neural networks using Lyapunov methods, multistability, and energy functions Chapter 3: Optimization in neural networks through biologically inspired gradient dynamics and sparse representations. Chapter 4: Unsupervised learning via neural dynamics, linking Hebbian rules to tasks like PCA, clustering, and similarity-based representation learning. 2. Linear Geometry and Algebra - Taras Banakh # Abstract: Linear Geometry studies geometric properties which can be expressed via the notion of a line. All information about lines is encoded in a ternary relation called a line relation. A set endowed with a line relation is called a liner. So, Linear Geometry studies liners. Imposing some additional axioms on a liner, we obtain some special classes of liners: regular, projective, affine, proaffine, etc. Linear Geometry includes Affine and Projective Geometries and is a part of Incidence Geometry. The aim of this book is to present a self-contained logical development of Linear Geometry, starting with some intuitive acceptable geometric axioms and ending with algebraic structures that necessarily arise from studying the structure of geometric objects that satisfy those simple and intuitive geometric axioms. We shall meet many quite exotic algebraic structures that arise this way: magmas, loops, ternary-ring, quasi-fields, alternative rings, procorps, profields, etc. We strongly prefer (synthetic) geometric proofs and use tools of analytic geometry only when no purely geometric proof is available. Liner Geometry has been developed by many great mathematicians since times of Antiquity (Thales, Euclides, Proclus, Pappus), through Renaissance (Descartes, Desargues), Early Modernity (Playfair, Gauss, Lobachevski, Bolyai, Poncelet, Steiner, Möbius), Late Modernity Times (Steinitz, Klein, Hilbert, Moufang, Hessenberg, Jordan, Beltrami, Fano, Gallucci, Veblen, Wedderburn, Lenz, Barlotti) till our contempories (Hartshorne, Hall, Buekenhout, Gleason, Kantor, Doyen, Hubault, Dembowski, Klingenberg, Grundhöfer).