A Posteriori Verification of Invariant Objects of Evolution Equations: Periodic Orbits in the Kuramoto-Sivashinsky PDE

作者：Gameiro Marcio^*; Lessard Jean Philippe

来源：SIAM Journal on Applied Dynamical Systems, 2017, 16(1): 687-728.

DOI：10.1137/16M1073789

• evolution equation
• periodic orbits
• contraction mapping
• Kuramoto-Sivashinsky
• rigorous computations
• interval analysis

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].

• 出版日期2017