The damped Fermi-Pasta-Ulam oscillator

Hatvani, László: The damped Fermi-Pasta-Ulam oscillator. Electronic journal of qualitative theory of differential equations 61. pp. 1-11. (2019)

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Abstract

The system q¨k + γq˙k = V 0 (qk+1 − qk ) − V 0 (qk − qk−1 ) (k = 1, . . . , N − 2) is considered, where 0 < γ = const., 2 < N ∈ N, V : (A, B) → R (−∞ ≤ A < B ≤ ∞) is a strictly convex, two times continuously differentiable function. We connect to the system three kinds of boundary conditions: q0(t) = 0, qN−1(t) = L = const. > 0 (fixed endpoints – this is the original Fermi–Pasta–Ulam oscillator provided that the damping coefficient γ equals zero); q1(t) − q0(t) = L/(N − 1), qN−1(t) − qN−2(t) = L/(N − 1) (free endpoints); q0(t) = −(K − qN−2(t)), qN−1(t) = q1(t) + K, K = const. (cycle). We prove that the unique equilibrium state of the system with fixed endpoints is asymptotically stable. We also prove that the system with free endpoints and the cycle asymptotically stop at an equilibrium state along their arbitrary motion, i.e., for every motion there is q 1 ∈ R such that limt→∞ qk (t) = q 1 + (k − 1)r, limt→∞ q˙k (t) = 0 (k = 1, . . . , N − 2), where the constant r is defined by the equation V 0 (r) = 0.

Item Type: Article
Journal or Publication Title: Electronic journal of qualitative theory of differential equations
Date: 2019
Number: 61
Page Range: pp. 1-11
ISSN: 1417-3875
DOI: https://doi.org/10.14232/ejqtde.2019.1.61
Uncontrolled Keywords: Oszcillátorok
Additional Information: Bibliogr.: p. 10-11. ; összefoglalás angol nyelven
Date Deposited: 2019. Sep. 30. 10:12
Last Modified: 2019. Sep. 30. 10:12
URI: http://acta.bibl.u-szeged.hu/id/eprint/62285

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