arXiv subjects

Jean-Philippe Lessard

Publications and source records attributed to Jean-Philippe Lessard.

At least 37 records · Page 2Linked to original sources

arXiv2022

Periodic orbits in Hořava-Lifshitz cosmologies

We consider spatially homogeneous Hořava-Lifshitz (HL) models that perturb General Relativity (GR) by a parameter $v\in (0,1)$ such that GR occurs at $v=1/2$. We describe the dynamics for the extremal case $v=0$, which possess the usual Bianchi hierarchy: type $\mathrm{I}$ (Kasner circle of equilibria), type $\mathrm{II}$ (heteroclinics that induce the Kasner map) and type $\mathrm{VI_0},\mathrm{VII_0}$ (further heteroclinics). For type $\mathrm{VIII}$ and $\mathrm{IX}$, we use a computer-assisted approach to prove the existence of periodic orbits which are far from the Mixmaster attractor and thereby we obtain a new behaviour which is not described by the BKL picture of bouncing Kasner-like states.

gr-qc↗

arXiv2022

Saddle-Type Blow-Up Solutions with Computer-Assisted Proofs: Validation and Extraction of Global Nature

In this paper, blow-up solutions of autonomous ordinary differential equations (ODEs) which are unstable under perturbations of initial points, referred to as saddle-type blow-up solutions, are studied. Combining dynamical systems machinery (e.g., compactifications, time-scale desingularizations of vector fields) with tools from computer-assisted proofs (e.g., rigorous integrators, the parameterization method for invariant manifolds), these blow-up solutions are obtained as trajectories on local stable manifolds of hyperbolic saddle equilibria at infinity. With the help of computer-assisted proofs, global trajectories on stable manifolds, inducing blow-up solutions, provide a global picture organized by global-in-time solutions and blow-up solutions simultaneously. Using the proposed methodology, intrinsic features of saddle-type blow-ups are observed: locally smooth dependence of blow-up times on initial points, level set distribution of blow-up times, and decomposition of the phase space playing a role as separatrixes among solutions, where the magnitude of initial points near those blow-ups does not matter for asymptotic behavior. Finally, singular behavior of blow-up times on initial points belonging to different family of blow-up solutions is addressed.

math.DS↗

arXiv2022

Validated forward integration scheme for parabolic PDEs via Chebyshev series

In this paper we introduce a new approach to compute rigorously solutions of Cauchy problems for a class of semi-linear parabolic partial differential equations. Expanding solutions with Chebyshev series in time and Fourier series in space, we introduce a zero finding problem $F(a)=0$ on a Banach algebra $X$ of Fourier-Chebyshev sequences, whose solution solves the Cauchy problem. The challenge lies in the fact that the linear part $\mathcal{L} = DF(0)$ has an infinite block diagonal structure with blocks becoming less and less diagonal dominant at infinity. We introduce analytic estimates to show that $\mathcal{L}$ is an invertible linear operator on $X$, and we obtain explicit, rigorous and computable bounds for the operator norm $\| \mathcal{L}^{-1}\|_{B(X)}$. These bounds are then used to verify the hypotheses of a Newton-Kantorovich type argument which shows that the (Newton-like) operator $\mathcal{T}(a)=a - \mathcal{L}^{-1} F(a)$ is a contraction on a small ball centered at a numerical approximation of the Cauchy problem. The contraction mapping theorem yields a fixed point which corresponds to a classical (strong) solution of the Cauchy problem. The approach is simple to implement, numerically stable and is applicable to a class of PDE models, which include for instance Fisher's equation and the Swift-Hohenberg equation. We apply our approach to each of these models.

math.NA↗

arXiv2021

Persistence of Periodic Orbits under State-dependent Delayed Perturbations: Computer-assisted Proofs

A computer-assisted argument is given, which provides existence proofs for periodic orbits in state-dependent delayed perturbations of ordinary differential equations (ODEs). Assuming that the unperturbed ODE has an isolated periodic orbit, we introduce a set of polynomial inequalities whose successful verification leads to the existence of periodic orbits in the perturbed delay equation. We present a general algorithm, which describes a way of computing the coefficients of the polynomials and optimizing their variables so that the polynomial inequalities are satisfied. The algorithm uses the tools of validated numerics together with Chebyshev series expansion to obtain the periodic orbit of the ODE as well as the solution of the variational equations, which are both used to compute rigorously the coefficients of the polynomials. We apply our algorithm to prove the existence of periodic orbits in a state-dependent delayed perturbation of the van der Pol equation.

math.DS↗

arXiv2021

Singularities and heteroclinic connections in complex-valued evolutionary equations with a quadratic nonlinearity

In this paper, we consider the dynamics of solutions to complex-valued evolutionary partial differential equations (PDEs) and show existence of heteroclinic orbits from nontrivial equilibria to zero via computer-assisted proofs. We also show that the existence of unbounded solutions along unstable manifolds at the equilibrium follows from the existence of heteroclinic orbits. Our computer-assisted proof consists of three separate techniques of rigorous numerics: an enclosure of a local unstable manifold at the equilibria, a rigorous integration of PDEs, and a constructive validation of a trapping region around the zero equilibrium.

math.DS↗

arXiv2021

Spatial relative equilibria and periodic solutions of the Coulomb $(n+1)$-body problem

We study a classical model for the atom that considers the movement of $n$ charged particles of charge $-1$ (electrons) interacting with a fixed nucleus of charge $μ>0$. We show that two global branches of spatial relative equilibria bifurcate from the $n$-polygonal relative equilibrium for each critical values $μ=s_{k}$ for $k\in \lbrack 2,...,n/2]$. In these solutions, the $n$ charges form $n/h$-groups of regular $h$-polygons in space, where $h$ is the greatest common divisor of $k$ and $n$. Furthermore, each spatial relative equilibrium has a global branch of relative periodic solutions for each normal frequency satisfying some nonresonant condition. We obtain computer-assisted proofs of the existence of several spatial relative equilibria on global branches away from the $n$-polygonal relative equilibrium. Moreover, the nonresonant condition of the normal frequencies for some spatial relative equilibria is verified rigorously using computer-assisted proofs.

math.DS↗

arXiv2021

Microscopic Patterns in the 2D Phase-Field-Crystal Model

Using the recently developed theory of rigorously validated numerics, we address the Phase-Field-Crystal (PFC) model at the microscopic (atomistic) level. We show the existence of critical points and local minimizers associated with "classical" candidates, grain boundaries, and localized patterns. We further address the dynamical relationships between the observed patterns for fixed parameters and across parameter space, then formulate several conjectures on the dynamical connections (or orbits) between steady states.

math.NA↗

arXiv2020

From the Lagrange polygon to the figure eight I: Numerical evidence extending a conjecture of Marchal

The present work studies the continuation class of the regular $n$-gon solution of the $n$-body problem. For odd numbers of bodies between $n = 3$ and $n = 15$ we apply one parameter numerical continuation algorithms to the energy/frequency variable, and find that the figure eight choreography can be reached starting from the regular $n$-gon. The continuation leaves the plane of the $n$-gon, and passes through families of spatial choreographies with the topology of torus knots. Numerical continuation out of the $n$-gon solution is complicated by the fact that the kernel of the linearization there is high dimensional. Our work exploits a symmetrized version of the problem which admits dense sets of choreography solutions, and which can be written as a delay differential equation in terms of one of the bodies. This symmetrized setup simplifies the problem in several ways. On one hand, the direction of the kernel is determined automatically by the symmetry. On the other hand, the set of possible bifurcations is reduced and the $n$-gon continues to the eight after a single symmetry breaking bifurcation. Based on the calculations presented here we conjecture that the $n$-gon and the eight are in the same continuation class for all odd numbers of bodies.

math.DS↗

arXiv2020

Global dynamics in nonconservative nonlinear Schrödinger equations

In this paper, we study the global dynamics of a class of nonlinear Schrödinger equations using perturbative and non-perturbative methods. We prove the semi-global existence of solutions for initial conditions close to constant. That is, solutions will exist for all positive time or all negative time. The existence of an open set of initial data which limits to zero in both forward and backward time is also demonstrated. This result in turn forces the non-existence of any real-analytic conserved quantities. For the quadratic case, we prove the existence of two (infinite) families of nontrivial unstable equilibria and prove the existence of heteroclinic orbits limiting to the nontrivial equilibria in backward time and to zero in forward time. By a time reversal argument, we also obtain heteroclinic orbits limiting to the nontrivial equilibria in forward time and to zero in backward time. The proofs for the quadratic equation are computer-assisted and rely on three separate ingredients: an enclosure of a local unstable manifold at the equilibria, a rigorous integration of the flow (starting from the unstable manifold) and a proof that the solution enters a validated stable set (hence showing convergence to zero).

math.AP↗

arXiv2020

Free vibrations in a wave equation modeling MEMS

We study a nonlinear wave equation appearing as a model for a membrane (without viscous effects) under the presence of an electrostatic potential with strength $λ$. The membrane has a unique stable branch of steady states $u_λ$ for $λ\in\lbrack0,λ_{\ast}]$. We prove that the branch $u_λ$ has an infinite number of branches of periodic solutions (free vibrations) bifurcating when the parameter $λ$ is varied. Furthermore, using a functional setting, we compute numerically the branch $u_λ$ and their branches of periodic solutions. This approach is useful to validate rigorously the steady states $u_λ$ at the critical value $λ_{\ast}$.

math.AP↗

arXiv2020

Torus knot choreographies in the $n$-body problem

We develop a systematic approach for proving the existence of choreographic solutions in the gravitational $n$ body problem. Our main focus is on spatial torus knots: that is, periodic motions where the positions of all $n$ bodies follow a single closed which winds around a $2$-torus in $\mathbb{R}^3$. After changing to rotating coordinates and exploiting symmetries, the equation of a choreographic configuration is reduced to a delay differential equation (DDE) describing the position and velocity of a single body. We study periodic solutions of this DDE in a Banach space of rapidly decaying Fourier coefficients. Imposing appropriate constraint equations lets us isolate choreographies having prescribed symmetries and topological properties. Our argument is constructive and makes extensive use of the digital computer. We provide all the necessary analytic estimates as well as a working implementation for any number of bodies. We illustrate the utility of the approach by proving the existence of some spatial choreographies for $n=4,5,7$, and $9$ bodies.

math.DS↗

arXiv2020

Rigorous verification of Hopf bifurcations via desingularization and continuation

In this paper we present a general approach to rigorously validate Hopf bifurcations as well as saddle-node bifurcations of periodic orbits in systems of ODEs. By a combination of analytic estimates and computer-assisted calculations, we follow solution curves of cycles through folds, checking along the way that a single nondegenerate saddle-node bifurcation occurs. Similarly, we rigorously continue solution curves of cycles starting from their onset at a Hopf bifurcation. We use a blowup analysis to regularize the continuation problem near the Hopf bifurcation point. This extends the applicability of validated continuation methods to the mathematically rigorous computational study of bifurcation problems.

math.DS↗

arXiv2019

Rigorous numerics for nonlinear heat equations in the complex plane of time

In this paper, we introduce a method for computing rigorous local inclusions of solutions of Cauchy problems for nonlinear heat equations for complex time values. Using a solution map operator, we construct a simplified Newton operator and show that it has a unique fixed point. The fixed point together with its rigorous bounds provides the local inclusion of the solution of the Cauchy problem. The local inclusion technique is then applied iteratively to compute solutions over long time intervals. This technique is used to prove the existence of a branching singularity in the nonlinear heat equation. Finally, we introduce an approach based on the Lyapunov-Perron method to calculate part of a center-stable manifold and prove that an open set of solutions of the Cauchy problem converge to zero, hence yielding the global existence of the solutions in the complex plane of time.

math.DS↗

arXiv2019

Traveling wave oscillatory patterns in a signed Kuramoto-Sivashinsky equation with absorption

In this paper, a partial proof of a conjecture raised by Galaktionov and Svirshchevskii concerning existence and global uniqueness of an asymptotically stable periodic orbit in a fourth-order piecewise linear ordinary differential equation is presented. The fourth-order equation comes from the study of traveling wave patterns in a signed Kuramoto-Sivashinsky equation with absorption. The proof is twofold. First, the problem of solving for the periodic orbit is transformed into a zero finding problem in four dimensional space, which is solved with a computer-assisted proof based on Newton's method and the contraction mapping theorem. Second, the rigorous bounds about the periodic orbit in phase space are combined with the theory of discontinuous dynamical systems to prove that the orbit is asymptotically stable.

math.DS↗

arXiv2019

Spontaneous periodic orbits in the Navier-Stokes flow

In this paper, a general method to obtain constructive proofs of existence of periodic orbits in the forced autonomous Navier-Stokes equations on the three-torus is proposed. After introducing a zero finding problem posed on a Banach space of geometrically decaying Fourier coefficients, a Newton-Kantorovich theorem is applied to obtain the (computer-assisted) proofs of existence. The required analytic estimates to verify the contractibility of the operator are presented in full generality and symmetries from the model are used to reduce the size of the problem to be solved. As applications, we present proofs of existence of spontaneous periodic orbits in the Navier-Stokes equations with Taylor-Green forcing.

math.AP↗

arXiv2017

Stability and Uniqueness of Slowly Oscillating Periodic Solutions to Wright's Equation

In this paper, we prove that Wright's equation $y'(t) = - αy(t-1) \{1 + y(t)\}$ has a unique slowly oscillating periodic solution (SOPS) for all parameter values $α\in [ 1.9,6.0]$, up to time translation. Our proof is based on a same strategy employed earlier by Xie [27]; show that every SOPS is asymptotically stable. We first introduce a branch and bound algorithm to control all SOPS using bounding functions at all parameter values $α\in [ 1.9,6.0]$. Once the bounding functions are constructed, we then control the Floquet multipliers of all possible SOPS by solving rigorously an eigenvalue problem, again using a formulation introduced by Xie. Using these two main steps, we prove that all SOPS of Wright's equation are asymptotically stable for $α\in [ 1.9,6.0]$, and the proof follows. This result is a step toward the proof of the Jones' Conjecture formulated in 1962.

math.DS↗

arXiv2017

Polynomial interpolation and a priori bootstrap for computer-assisted proofs in nonlinear ODEs

In this work, we introduce a method based on piecewise polynomial interpolation to enclose rigorously solutions of nonlinear ODEs. Using a technique which we call a priori bootstrap, we transform the problem of solving the ODE into one of looking for a fixed point of a high order smoothing Picard-like operator. We then develop a rigorous computational method based on a Newton-Kantorovich type argument (the radii polynomial approach) to prove existence of a fixed point of the Picard-like operator. We present all necessary estimates in full generality and for any nonlinearities. With our approach, we study two systems of nonlinear equations: the Lorenz system and the ABC flow. For the Lorenz system, we solve Cauchy problems and prove existence of periodic and connecting orbits at the classical parameters, and for ABC flows, we prove existence of ballistic spiral orbits.

math.DS↗

arXiv2017

Continuation of homoclinic orbits in the suspension bridge equation: a computer-assisted proof

In this paper, we prove existence of symmetric homoclinic orbits for the suspension bridge equation $u""+βu" + e^u-1=0$ for all parameter values $β\in [0.5,1.9]$. For each $β$, a parameterization of the stable manifold is computed and the symmetric homoclinic orbits are obtained by solving a projected boundary value problem using Chebyshev series. The proof is computer-assisted and combines the uniform contraction theorem and the radii polynomial approach, which provides an efficient means of determining a set, centered at a numerical approximation of a solution, on which a Newton-like operator is a contraction.

math.DS↗