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There is an extensive theory for the flow defined by dynamical systems generated by continuous semigroups T: ℝ+ X M → M, T(t,x): = T(t)x, where T(t) : M → M, ℝ+ = [0, ∞), and M is either a finite dimensional compact manifold without boundary or a compact manifold with boundary provided that the flow is differentiable and transversal to the boundary. The basic problem is to compare the flows defined by different dynamical systems. This comparison is made most often through the notion of topological equivalence. Two semigroups T and S defined on M are topologically equivalent if there is a homeomorphism from M to M which takes the orbits of T onto the orbits of S and preserves the sense of direction in time.
KeywordsFunctional Differential Equation Continuous Semigroup Parabolic Partial Differential Equation Morse Decomposition Large Initial Data
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