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# Download Differential equations, chaos and variational problems by Vasile Staicu PDF

By Vasile Staicu

This choice of unique articles and surveys written by means of prime specialists of their fields is devoted to Arrigo Cellina and James A. Yorke at the celebration in their sixty fifth birthday. the amount brings the reader to the border of study in differential equations, a quick evolving department of arithmetic that, in addition to being a chief topic for mathematicians, is likely one of the mathematical instruments so much used either through scientists and engineers.

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1) x (t) = −f (x(t), u(t)) where u(t) ∈ U (x(t)) or x (t) ∈ −F (x(t)) There are several ways for describing continuity of the evolutionary system x ❀ S(x) with respect to the initial state, regarded as stability property. 2. The evolutionary system S is said to be upper semicompact from X to C(0, ∞; X) if for any xn ∈ X converging to x in X and for any evolution xn (·) ∈ S(xn ) starting at xn , there exists a subsequence of xn (·) converging to a evolution x(·) ∈ S(x) uniformly on compact intervals.

Indeed, on any ﬁxed small interval, say [s1 , s2 ] in [0, 1], when ε is small the values of sin( sε ) are distributed very closely to the distribution of the values of the sin function over one period; namely, the distribution is μ0 (dξ) = 1 π −1 (1 − ξ 2 )− 2 dξ which is a probability measure over the space of values of the mapping sin(·). A way to depict the limit is to identify it with the probability measure-valued map, say μ(·)(dξ) which assigns to each s ∈ [0, 1] the probability distribution μ0 (dξ) just deﬁned.

Univ. Trieste 31 (2000), supplemento 1, 1–69. [7] P. Billingsley, Convergence of Probability Measures. Wiley, New York, 1968. [8] N. N. Bogoliubov and Y. A. Mitropolsky, Asymptotic Methods in the Theory of Non-linear Oscillations. English Translation, Gordon and Breach, New York, 1961. [9] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Applied Mathematical Sciences 42. Springer-Verlag, New York, 1983. [10] P. Lochak and C. Meunier, Multiphase Averaging for Classical Systems.