Energy transfer between modes in a nonlinear beam equation. (English. French summary) Zbl 1404.35277
Summary: We consider the nonlinear nonlocal beam evolution equation introduced by Woinowsky-Krieger [38]. We study the existence and behavior of periodic solutions: these are called nonlinear modes. Some solutions only have two active modes and we investigate whether there is an energy transfer between them. The answer depends on the geometry of the energy function which, in turn, depends on the amount of compression compared to the spatial frequencies of the involved modes. Our results are complemented with numerical experiments; overall, they give a complete picture of the instabilities that may occur in the beam. We expect these results to hold also in more complicated dynamical systems.
MSC:
| 35L35 | Initial-boundary value problems for higher-order hyperbolic equations |
| 34D20 | Stability of solutions to ordinary differential equations |
| 35A15 | Variational methods applied to PDEs |
| 74B20 | Nonlinear elasticity |
| 74K10 | Rods (beams, columns, shafts, arches, rings, etc.) |
| 35B10 | Periodic solutions to PDEs |
| 35R09 | Integro-partial differential equations |
| 35B35 | Stability in context of PDEs |
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