Generic Properties. The Theorem of Kupka-Smale
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The aim of the generic theory of differential equations is to study qualitative properties which are typical of the class of equations considered, in the sense that they hold for all equations defined by functions of a residual set of the function space being considered. More precisely, if X is a complete metric space, then a property P on the elements x ∈ X is said to be generic if there is a residual set Y ⊂ X such that each element of Y has property P. Recall that a residual set is a countable intersection of open dense sets. As for ordinary differential equations, the constant and the periodic solutions, and their stable and unstable manifolds, play an important role in the generic theory of RFDE’s.
KeywordsPeriodic Solution Periodic Orbit Null Space Unstable Manifold Implicit Function Theorem
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