Rigorous Numerics in Floquet Theory: Computing Stable and Unstable Bundles of Periodic Orbits

Abstract

In this paper, a new rigorous numerical method to compute fundamental matrix solutions of non-autonomous linear differential equations with periodic coefficients is introduced. Decomposing the fundamental matrix solutions Φ(t) by their Floquet normal forms, that is as product of real periodic and exponential matrices Φ(t) = Q(t)e Rt , one solves simultaneously for R and for the Fourier coefficients of Q via a fixed point argument in a suitable Banach space of rapidly decaying coefficients. As an application, the method is used to compute rigorously stable and unstable bundles of periodic orbits of vector fields. Examples are given in the context of the Lorenz equations and the ζ 3 -model.

Rigorous Computation of Non-uniform Patterns for the 2-Dimensional Gray-Scott Reaction-Diffusion Equation

“…Again, as a general statement, a larger growth for μ k allows a smaller choice of M. For two-dimensional PDE problems the coefficient α (3) k rapidly grows to large values, hence the parameter M needs to be selected quite large. Our situation is even worst due to the presence ofq = γ −1 that multiply the effect of α (3) k . To have a quantitative idea, consider the family of solutions labelled with (0, 3) in Fig.…”

“…The parameter M must be chosen large enough so that μ k remains reasonable larger than α (3) k for any k / ∈ F M . Again, as a general statement, a larger growth for μ k allows a smaller choice of M. For two-dimensional PDE problems the coefficient α (3) k rapidly grows to large values, hence the parameter M needs to be selected quite large.…”

Parameterization of Invariant Manifolds for Periodic Orbits I: Efficient Numerics via the Floquet Normal Form

“…These methods could be used in order to provide mathematically rigorous bounds on the initial data for the methods of the present work, Remark 1.2 (Validated Numerics). Combining the work of [80] with the techniques of the present work will lead to methods for obtaining mathematically rigorous bounds on the truncation error associated with our parameterization. See also the discussion in Section 5 of [47].…”

“…We make no attempt to review the relevant literature on spectral methods. The interested reader can consult for example [64,48,59,71,72,80].…”

“…In fact [80] develops a mathematically rigorous computer assisted method for bounding the error associated with spectral Fourier approximations of the Floquet normal form. These methods could be used in order to provide mathematically rigorous bounds on the initial data for the methods of the present work, Remark 1.2 (Validated Numerics).…”