Invariant Sets, Limit Sets and the Attractor

  • Jack K. Hale
  • Luis T. Magalhães
  • Waldyr M. Oliva
Part of the Applied Mathematical Sciences book series (AMS, volume 47)


A function y(t) is said to be a global solution of an RFDE(F) on M, if it is defined for t ∈ (−∞,+ ∞) and, for every σ ∈ (−∞,+ ∞), xt(σ, yσ, F) = yt, t ≥ σ. The constant and the periodic solutions are particular cases of global solutions. The solutions with initial data in unstable manifolds of equilibrium points or periodic orbits are often global solutions, for example, when M is compact. An invariant set of an RFDE(F) on a manifold M, is a subset S of C0 = C0(I, M) such that for every φ ∈ S there exists a global solution x of the RFDE, satisfying x0 = φ and xt ∈ S for all t ∈ ℝ. The ω-limit set ω(φ) of an orbit γ+(φ) = { Фtφ, t ≥ 0} through φ is the set
$$\omega (\varphi ) = \mathop \cap \limits_{\tau 0} C\ell \mathop \cup \limits_{t\tau } {\Phi _t}\varphi .$$


Global Attractor Liapunov Function Infinite Dimensional Dynamical System Nonwandering Point Negative Orbit 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 1984

Authors and Affiliations

  • Jack K. Hale
    • 1
  • Luis T. Magalhães
    • 2
  • Waldyr M. Oliva
    • 3
  1. 1.Division of Applied MathematicsBrown UniversityProvidenceUSA
  2. 2.Universidade Tecnica de LisbõaLisbonPortugal
  3. 3.Departmento de Matemática Aplicada, Instituto de Matemática e EstatisticaUniversidade de São PauloSão PauloBrasil

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