Rigorous continuation of periodic solutions of differential equations (Olivier Hénot, École Polytechnique)

The goal of the talk is to present a new computer-assisted method to perform (global) continuation of periodic orbits. The strategy is fully spectral in that it consists in verifying the contraction of an appropriate fixed-point operator in the vicinity of a high-order Chebyshev interpolant of the branch. First, we review rigorous numerics and computer-assisted proof techniques based on the Newton-Kantorovich Theorem. Then, we formulate a general framework to rigorously compute periodic orbits, as well as a continuum of them.

As a first application, we demonstrate Marchal's conjecture in celestial mechanics: the P_{12} family emanating from the Lagrange equilateral triangle ends at the Figure 8 choreography discovered by Moore in 1993.

As a second application, we follow a family of steady-states of a nonlinear PDE modelling the density of several species in a bounded environment, known as the Shigesada-Kawasaki-Teramoto (SKT) system.

We will further discuss how this strategy generalizes seamlessly to multi-dimensional parameter continuation; if time permits, we investigate a 2-parameter family of steady-states of the Cahn-Hilliard equation.