Dynamical Systems: Basic Theory

  • George R. Sell
  • Yuncheng You
Part of the Applied Mathematical Sciences book series (AMS, volume 143)

Abstract

The basic concept underlying the study of dynamics in infinite dimensional spaces is that of a semiflow, or as it is sometimes called, a semigroup. This semiflow is a time-dependent action on the ambient space, which we assume to be a complete metric space W, for example, a Banach space or a Fréchet space. One should think of the semiflow as a mechanism for describing the solutions of an underlying evolutionary equation. This evolutionary equation is oftentimes the abstract formulation of a given partial differential equation or, sometimes, an ordinary differential equation with time delays. In this chapter we will examine some basic properties of semiflows. Our principal objective is to describe the longtime dynamics in terms of the invariant sets, the limit sets, and the attractors of the semiflow. A comprehensive theory of global attractors is included here. Later in this volume, we will develop the connections between the semiflow and the underlying evolutionary equation.

Keywords

Unstable Manifold Global Attractor Invariant Domain Nonlinear Semigroup Longtime Dynamic 
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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Additional Readings

  1. J M Ball (1997), Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations, J Nonlinear Sci 7, 475–502.MathSciNetMATHCrossRefGoogle Scholar
  2. N P Bhatia and G P Szegö (1970), Stability Theory of Dynamical Systems, Springer Verlag, New York.MATHCrossRefGoogle Scholar
  3. J E Billotti and J P LaSalle (1971), Dissipative periodic processes, Bull Am Math Soc 77, 1082–1088.MathSciNetMATHCrossRefGoogle Scholar
  4. J W Cholewa and T Dlotko (2000), Global Attractors in Abstract Parabolic Problems, London Math Soc Lecture Note Series, No 278, Cambridge Univ Press, Cambridge, UK.Google Scholar
  5. C C Conley (1978), Isolated Invariant Sets and the Morse Index, CBMS Regional Conference, vol 89, Am Math Soc, Providence.Google Scholar
  6. G Fusco and J K Hale (1989), Slow motion manifolds, dormant instability, and singular perturbations, J Dynamics Differential Equations 1, 75–94.MathSciNetMATHCrossRefGoogle Scholar
  7. J K Hale and H Kocak (1991), Dynamics and Bifurcations, Springer Verlag, New York.MATHCrossRefGoogle Scholar
  8. J K Hale, J P LaSalle, and M Slemrod (1972), Theory of a general class of dissipative processes, J Math Anal Appl 39, 177–191.MathSciNetMATHCrossRefGoogle Scholar
  9. E N Lorenz (1963), Deterministic nonperiodic flow, J Atmospheric Sci 20, 130–141.CrossRefGoogle Scholar
  10. V V Nemytskii and V V Stepanov (1960), Qualitative Theory of Differential Equations, Princeton Univ, Princeton NJ.MATHGoogle Scholar
  11. R J Sacker and G R Sell (1977), Lifting properties in skewproduct flows with applications to differential equations, Memoirs Am Math Soc, No 190.Google Scholar
  12. R J Sacker and G R Sell (1994), Dichotomies for linear evolutionary equations in Banach spaces, J Differential Equations 113, 17–67.MathSciNetMATHCrossRefGoogle Scholar
  13. G R Sell (1971), Topological Dynamics and Ordinary Differential Equations, Van Nostrand, New York.MATHGoogle Scholar
  14. G R Sell (1996), Global attractors for the three dimensional Navier-Stokes equations, J Dynamics Differential Equations 8, 1–33.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2002

Authors and Affiliations

  • George R. Sell
    • 1
  • Yuncheng You
    • 2
  1. 1.School of MathematicsUniversity of MinnesotaMinneapolisUSA
  2. 2.Department of MathematicsUniversity of South FloridaTampaUSA

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