Flows near compact invariant sets. Part I

• Autorzy: Pedro Teixeira
• Języki publikacji: EN

Abstrakty

EN

It is proved that near a compact, invariant, proper subset of a C⁰ flow on a locally compact, connected metric space, at least one, out of twenty eight relevant dynamical phenomena, will necessarily occur. Theorem 1 shows that the connectedness of the phase space implies the existence of a considerably deeper classification of topological flow behaviour in the vicinity of compact invariant sets than that described in the classical theorems of Ura-Kimura and Bhatia. The proposed classification brings to light, in a systematic way, the possibility of occurrence of orbits of infinite height arbitrarily near the compact invariant set in question, and this under relatively simple conditions. Singularities of $C^{∞}$ vector fields displaying this strange phenomenon occur in every dimension n ≥ 3 (in this paper, a $C^{∞}$ flow on 𝕊³ exhibiting such an equilibrium is constructed). Near periodic orbits, the same phenomenon is observable in every dimension n ≥ 4. As a corollary to the main result, an elegant characterization of the topological-dynamical Hausdorff structure of the set of all compact minimal sets of the flow is obtained (Theorem 2).

Kategorie tematyczne

• 37C27: Periodic orbits of vector fields and flows
• 58K45: Singularities of vector fields, topological aspects
• 37C70: Attractors and repellers, topological structure
• 37B35: Gradient-like and recurrent behavior; isolated (locally maximal) invariant sets
• 37B25: Lyapunov functions and stability; attractors, repellers
• 37C10: Vector fields, flows, ordinary differential equations

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2013
• Tom: 223
• Numer: 3
• Strony: 225-272

Daty

wydano

2013

Twórcy

autor

Pedro Teixeira

• Centro de Matemática da Universidade do Porto, Rua do Campo Alegre, 687, 4169-007 Porto, Portugal

Identyfikatory

DOI

10.4064/fm223-3-3