The damped Fermi-Pasta-Ulam oscillator
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 condition...
Elmentve itt :
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| Dokumentumtípus: | Folyóirat |
| Megjelent: |
2019
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| Sorozat: | Electronic journal of qualitative theory of differential equations
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| Kulcsszavak: | Oszcillátorok |
| doi: | 10.14232/ejqtde.2019.1.61 |
| Online Access: | http://acta.bibl.u-szeged.hu/62285 |
| Tartalmi kivonat: | 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. |
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| Terjedelem/Fizikai jellemzők: | 1-11 |
| ISSN: | 1417-3875 |