Nontrivial periodic motions for resonant type asymptotically linear lattice dynamical systems

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:417

Article

Ma, Shiwang¹ 

[1] Nankai Univ, Sch Math Sci, Tianjin 300071, Peoples R China

关键词: An infinite lattice of particles;    Critical groups;    Morse theory;    Resonance;   

DOI  :  10.1016/j.jmaa.2014.03.046

来源: Elsevier

【 摘 要 】

In this paper, we consider the following one dimensional lattices consisting of infinitely many particles with nearest neighbor interaction (q) over bar (i)(t) = Phi(2-1)' (t, q(i-1)(t) - q(i)(t)) - Phi(i)' (t, q(i)(t) - q(i+1)(t)), i is an element of Z, where Phi(i)(t, x) = -(alpha(i)/2)vertical bar x vertical bar(2) + V-i(t, x) is T-periodic in t for T > 0 and satisfies Phi(i+N) = Phi(i) for some N is an element of N, q(i)(t) is the state of the i-th particle. Assume that alpha(i) = 0 for some i is an element of Z and V-i '(t, x) denoting the derivative of V-i respect to x is asymptotically linear with x both at origin and at infinity. We would like to point out that this system is resonant both at origin and at infinity and not studied up to now. Based on some new results concerning the precise computations of the critical groups, for a given m is an element of Z, we obtain the existence of nontrivial periodic solutions satisfying q(i+mN)(t + T) = q(i)(t) for all t is an element of R and i is an element of Z under some additional conditions. (C) 2014 Elsevier Inc. All rights reserved.

【 授权许可】

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