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

By Vasile Staicu

This number of unique articles and surveys written by means of top specialists of their fields is devoted to Arrigo Cellina and James A. Yorke at the get together in their sixty fifth birthday. the amount brings the reader to the border of analysis in differential equations, a quick evolving department of arithmetic that, in addition to being a major topic for mathematicians, is likely one of the mathematical instruments such a lot used either via scientists and engineers.

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2 applied separately on the two subintervals. 6. We conclude by relating the previous considerations to singularly perturbed systems. 4) with x ∈ Rn and y ∈ Rm and when the desired time interval is t ∈ [0, ε−1 ]. 1) with the time dependency provided by the trajectory y(t), which in turn is coupled with x. This is, indeed, the general form used in the averaging problem, see Arnold [1], Lochak and Meunier [10], Verhulst [17]. 5) namely, a singularly perturbed equation; the desired time interval here is s ∈ [0, 1].

The main (and diﬃcult) theorem states that the set of solutions depends continuously upon the initial states in the upper semicompact sense. 3. 4. If F : X ❀ X is Marchaud, the solution map S is an upper semicompact evolutionary system. Viability Kernels and Capture Basins for Analyzing the Dynamic Behavior 33 The basic viability problem is reformulated in terms of the deﬁnition of viability kernels: Figure 1. Viability Kernels. Let K ⊂ X being regarded as a constrained environnement. The viability kernel Viab(K) of K is the set of initial states x0 ∈ K such that one evolution x(·) ∈ S(x0 ) starting at x0 is viable in K ViabS (K) := {x0 ∈ K | ∃x(·) ∈ S(x0 ) such that ∀t ≥ 0, x(t) ∈ K} When targets to be reached in ﬁnite time are involved, we introduce the concept of capture basins of targets: Figure 2.

When εj+1 ≤ ε ≤ εj the associated Young measure takes the value μi(j) (x) on part of the interval, say s ∈ [0, s ] and takes the value μi(j+1) (x) on the rest of the −1 −1 −1 . 3) is of order ε away from the solution of the autonomous equation on [0, ε−1 s ] and continues to be of order ε away from the solution of the corresponding autonomous equation on [ε−1 s , ε−1 ]. 2 applied separately on the two subintervals. 6. We conclude by relating the previous considerations to singularly perturbed systems.