| Full text | |
| Author(s): |
Gameiro, Marcio
;
Lessard, Jean-Philippe
Total Authors: 2
|
| Document type: | Journal article |
| Source: | SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS; v. 16, n. 1, p. 687-728, 2017. |
| Web of Science Citations: | 7 |
| Abstract | |
In this paper, a method for computing periodic orbits of the Kuramoto-Sivashinsky PDE via rigorous numerics is presented. This is an application and an implementation of the theoretical method introduced in {[}J.-L. Figueras, M. Gameiro, J.-P. Lessard, and R. de la Llave, ``A framework for the numerical computation and a posteriori verification of invariant objects of evolution equations,{''} SIAM J. Appl. Dyn. Syst., to appear]. Using a Newton-Kantorovich-type argument (the radii polynomial approach), existence of solutions is obtained in a weighted l(infinity) Banach space of Fourier coefficients. Once a proof of a periodic orbit is done, an associated eigenvalue problem is solved and Floquet exponents are rigorously computed, yielding proofs that some periodic orbits are unstable. Finally, a predictor-corrector continuation method is introduced to rigorously compute global smooth branches of periodic orbits. An alternative approach and independent implementation of {[}J.-L. Figueras, M. Gameiro, J.-P. Lessard, and R. de la Llave, ``A framework for the numerical computation and a posteriori verification of invariant objects of evolution equations,{''} SIAM J. Appl. Dyn. Syst., to appear] appears in {[}J.-L. Figueras and R. de la Llave, ``Numerical computations and computer assisted proofs of periodic orbits of the Kuramoto-Sivashinsky equation,{''} SIAM J. Appl. Dyn. Syst., to appear]. (AU) | |
| FAPESP's process: | 10/00875-9 - Topological methods and rigorous numerics for bifurcations of dynamical systems |
| Grantee: | Marcio Fuzeto Gameiro |
| Support Opportunities: | Regular Research Grants |
| FAPESP's process: | 13/50382-7 - Rigorous computations for nonlinear partial differential equations |
| Grantee: | Marcio Fuzeto Gameiro |
| Support Opportunities: | Regular Research Grants |
| FAPESP's process: | 16/08704-5 - Rigorous computations in dynamics |
| Grantee: | Marcio Fuzeto Gameiro |
| Support Opportunities: | Regular Research Grants |
| FAPESP's process: | 13/07460-7 - Rigorous computations for PDEs |
| Grantee: | Marcio Fuzeto Gameiro |
| Support Opportunities: | Regular Research Grants |