Jean-Philippe Lessard

Validated Continuation for Equilibria of PDEs

One of the most efficient methods for determining the equilibria of a continuous parameterized family of differential equations is to use predictor-corrector continuation techniques. In the case of partial differential equations this procedure must be applied to some finite-dimensional approximation, which of course raises the question of the validity of the output. We introduce a new technique that combines the information obtained from the predictor-corrector steps with ideas from rigorous computations and verifies that the numerically produced equilibrium for the finite-dimensional system can be used to explicitly define a set which contains a unique equilibrium for the infinite-dimensional partial differential equation. Using the Cahn-Hilliard and Swift-Hohenberg equations as models we demonstrate that the cost of this new validated continuation is less than twice the cost of the standard continuation method alone.

Rigorous numerics for symmetric connecting orbits: even homoclinics of the Gray-Scott equation

In this paper we propose a rigorous numerical technique for the computation of symmetric connecting orbits for ordinary differential equations. The idea is to solve a projected boundary value problem (BVP) in a function space via a fixed point argument. The formulation of the projected BVP involves a high order parameterization of the invariant manifolds at the steady states. Using this parameterization, one can obtain explicit exponential asymptotic bounds for the coefficients of the expansion of the manifolds. Combining these bounds with piecewise linear approximations, one can construct a contraction in a function space whose unique fixed point corresponds to the wanted connecting orbit. We have implemented the method to demonstrate its effectiveness, and we have used it to prove the existence of a family of even homoclinic orbits for the Gray–Scott equation.

Global smooth solution curves using rigorous branch following

In this paper, we present a new method for rigorously computing smooth branches of zeros of nonlinear operators f: ℝ l 1 × B 1 → ℝ l 2 ×B 2 , where B 1 and B 2 are Banach spaces. The method is first introduced for parameter continuation and then generalized to pseudo-arclength continuation. Examples in the context of ordinary, partial and delay differential equations are given.

Chaotic braided solutions via rigorous numerics: chaos in the Swift-Hohenberg equation

We prove that the stationary Swift–Hohenberg equation has chaotic dynamics on a critical energy level for a large (continuous) range of parameter values. The first step of the method relies on a computer assisted, rigorous, continuation method to prove the existence of a periodic orbit with certain geometric properties. The second step is topological: we use this periodic solution as a skeleton, through which we braid other solutions, thus forcing the existence of infinitely many braided periodic orbits. A semiconjugacy to a subshift of finite type shows that the dynamics is chaotic.

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

We present an efficient numerical method for computing Fourier--Taylor expansions of (un)stable manifolds associated with hyperbolic periodic orbits. Three features of the method are that (1) we obtain accurate representation of the invariant manifold as well as the dynamics on the manifold, (2) it admits natural a posteriori error analysis, and (3) it does not require numerically integrating the vector field. Our approach is based on the parameterization method for invariant manifolds, and studies a certain partial differential equation which characterizes a chart map of the manifold. The method requires only that some mild nonresonance conditions hold. The novelty of the present work is that we exploit the Floquet normal form in order to efficiently compute the Fourier--Taylor expansion. A number of example computations are given including manifolds in phase space dimension as high as ten and manifolds which are two and three dimensional. We also discuss computations of cycle-to-cycle connecting orbits ...

Rigorous Numerics for Nonlinear Differential Equations Using Chebyshev Series

A computational method based on Chebyshev series to rigorously compute solutions of initial and boundary value problems of analytic nonlinear vector fields is proposed. The idea is to recast solutions as fixed points of an operator defined on a Banach space of rapidly decaying Chebyshev coefficients and to use the so-called radii polynomials to show the existence of a unique fixed point near an approximate solution. As applications, solutions of initial value problems in the Lorenz equations and symmetric connecting orbits in the Gray--Scott equation are rigorously computed. The symmetric connecting orbits are obtained by solving a boundary value problem with one of the boundary values in the stable manifold.

Rigorous numerics for analytic solutions of differential equations: the radii polynomial approach

Judicious use of interval arithmetic, combined with careful pen and paper estimates, leads to effective strategies for computer assisted analysis of nonlinear operator equations. The method of radii polynomials is an efficient tool for bounding the smallest and largest neighborhoods on which a Newton-like operator associated with a nonlinear equation is a contraction mapping. The method has been used to study solutions of ordinary, partial, and delay differential equations such as equilibria, periodic orbits, solutions of initial value problems, heteroclinic and homoclinic connecting orbits in the C category of functions. In the present work we adapt the method of radii polynomials to the analytic category. For ease of exposition we focus on studying periodic solutions in Cartesian products of infinite sequence spaces. We derive the radii polynomials for some specific application problems, and give a number of computer assisted proofs in the analytic framework.

Stationary Coexistence of Hexagons and Rolls via Rigorous Computations

In this work we introduce a rigorous computational method for finding heteroclinic solutions of a system of two second order differential equations. These solutions correspond to standing waves between rolls and hexagonal patterns of a two-dimensional pattern formation PDE model. After reformulating the problem as a projected boundary value problem (BVP) with boundaries in the stable/unstable manifolds, we compute the local manifolds using the parameterization method and solve the BVP using Chebyshev series and the radii polynomial approach. Our results settle a conjecture by Doelman et al. [European J. Appl. Math., 14 (2003), pp. 85--110] about the coexistence of hexagons and rolls.

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

In this paper, a rigorous method to compute Floquet normal forms of fundamental matrix solutions of nonautonomous linear differential equations with periodic coefficients is introduced. The Floquet normal form of a fundamental matrix solution

Validated continuation over large parameter ranges for equilibria of PDEs

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