Hatvani László:
*The damped Fermi-Pasta-Ulam oscillator.*
(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: | Journal |
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Publication full: | Electronic journal of qualitative theory of differential equations |

Date: | 2019 |

Number: | 61 |

ISSN: | 1417-3875 |

Page Range: | pp. 1-11 |

DOI: | 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: | 2021. Sep. 16. 10:42 |

URI: | http://acta.bibl.u-szeged.hu/id/eprint/62285 |

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