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.
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Additional Readings
J M Ball (1997), Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations, J Nonlinear Sci 7, 475–502.
N P Bhatia and G P Szegö (1970), Stability Theory of Dynamical Systems, Springer Verlag, New York.
J E Billotti and J P LaSalle (1971), Dissipative periodic processes, Bull Am Math Soc 77, 1082–1088.
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.
C C Conley (1978), Isolated Invariant Sets and the Morse Index, CBMS Regional Conference, vol 89, Am Math Soc, Providence.
G Fusco and J K Hale (1989), Slow motion manifolds, dormant instability, and singular perturbations, J Dynamics Differential Equations 1, 75–94.
J K Hale and H Kocak (1991), Dynamics and Bifurcations, Springer Verlag, New York.
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.
E N Lorenz (1963), Deterministic nonperiodic flow, J Atmospheric Sci 20, 130–141.
V V Nemytskii and V V Stepanov (1960), Qualitative Theory of Differential Equations, Princeton Univ, Princeton NJ.
R J Sacker and G R Sell (1977), Lifting properties in skewproduct flows with applications to differential equations, Memoirs Am Math Soc, No 190.
R J Sacker and G R Sell (1994), Dichotomies for linear evolutionary equations in Banach spaces, J Differential Equations 113, 17–67.
G R Sell (1971), Topological Dynamics and Ordinary Differential Equations, Van Nostrand, New York.
G R Sell (1996), Global attractors for the three dimensional Navier-Stokes equations, J Dynamics Differential Equations 8, 1–33.
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© 2002 Springer Science+Business Media New York
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Sell, G.R., You, Y. (2002). Dynamical Systems: Basic Theory. In: Dynamics of Evolutionary Equations. Applied Mathematical Sciences, vol 143. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-5037-9_2
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DOI: https://doi.org/10.1007/978-1-4757-5037-9_2
Publisher Name: Springer, New York, NY
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Online ISBN: 978-1-4757-5037-9
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