Multiple periodic solutions for lattice dynamical systems with superquadratic potentials. (English) Zbl 1318.37026

Sun, Jijiang; Ma, Shiwang

J. Differ. Equations 255, No. 8, 2534-2563 (2013).

Summary: In this paper, we consider one dimensional lattices consisting of infinitely many particles with nearest neighbor interaction. The autonomous dynamical system is described by the following infinite system of second order differential equations \[ \ddot {q}_i={\varPhi}_{i-1}'(q_{i-1}-q_i)-{\varPhi}_i'(q_i-q_{i+1}), \quad i\in \mathbb Z, \] where \({\varPhi}_i\) denotes the interaction potential between two neighboring particles and \(q_i(t)\) is the state of the \(i\)-th particle. Supposing \({\varPhi}_i\) is superquadratic at infinity, for all \(T>0\), we obtain a nonzero \(T\)-periodic solution of finite energy which may be nonconstant in some range of period. If in addition \({\varPhi}_i(x)\) is even in \( x\), we also obtain infinitely many geometrically distinct solutions for any period \(T>0\). In particular, a prescribed number of geometrically distinct nonconstant periodic solutions is obtained for some range of period. Since the functional associated to the above system is invariant under the actions of the non-compact group \(\mathbb Z\) and the continuous compact group \(S^1\) under our assumptions, in order to prove our results, we need to extend the abstract critical point theorem about strongly indefinite functional developed by T. Bartsch and Y. Ding [Math. Nachr. 279, No. 12, 1267–1288 (2006; Zbl 1117.58007)] to a more general class of symmetry.

MSC:

37K60	Lattice dynamics; integrable lattice equations
34C25	Periodic solutions to ordinary differential equations
70F45	The dynamics of infinite particle systems
34A33	Ordinary lattice differential equations

Keywords:

infinite particle system; nearest neighbor interaction; autonomous dynamical system

Citations:

Zbl 1117.58007

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