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By Samuel Zaidman

ISBN-10: 0273084275

ISBN-13: 9780273084273

ISBN-10: 0822484277

ISBN-13: 9780822484271

ISBN-10: 1584880112

ISBN-13: 9781584880110

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1 and lim x(t, x0 ) − x(t, x0 ) = 0. 3(a). In order to tell whether x(t, x0 ) is stable we have to compare the values of x(t, x0 ) and x(t, x0 ) at the same values of t. Clearly, Lyapunov stability does not exhaust the whole spectrum of plausible asymptotic descriptions of solutions of a dynamical system with respect to each other. Consider an example. Let x(t, x0 ) and x(t, x0 ) be two solutions of the same system, and x0 = x0 . Then a possible characterization of their relative position in the state space could be ρ(t, x(t, x0 ), x(t, x0 )) = x(t, x0 ) A A = {p ∈ Rn |p = x(t, x0 ), t ∈ R}.

2 This led to the emergence of the new notion of a weakly attracting set, which was formally defined by J. 4 A set A is a weakly attracting, or Milnor attracting, set iff (1) it is closed, invariant, and (2) for some set V (not necessarily a neighborhood of A) with strictly positive measure and for all x0 ∈ V the following limiting relation holds: lim x(t, x0 ) = A ∀ x0 ∈ V (A). 3 is that the domain of attraction V is not necessarily a neighborhood of A. Despite the fact that this difference may look small and insignificant at first glance, it becomes very instrumental for successful statement and solution of particular problems of adaptation.

3). This figure demonstrates that for any neighborhood U (A) of the origin A there are points x ∈ U (A) such that solutions x(t, x ) escape the neighborhood U (A) and never come back. 3, A cannot be called an attracting set. On the other hand, there are points x ∈ U (A) such that limt→∞ x(t, x ) = 0. If U (A) is an open circle, then the number of such points is as large as the number of points corresponding to the solutions escaping U (A). Thus the set A bears an overall signature of attractivity.

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Abstract differential equations by Samuel Zaidman

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