# Jason Mireles JamesFlorida Atlantic University | FAU · Department of Mathematical Sciences

Jason Mireles James

PhD

## About

91

Publications

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1,272

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Introduction

I am Associate Professor in the Department of Mathematical Sciences at Florida Atlantic University. My research focuses on nonlinear analysis and dynamical systems theory, and I'm particularly interested in numerical methods for studying invariant manifolds and connecting dynamics. I'm also quite interested in methods of computer assisted proof in analysis.

Additional affiliations

August 2014 - present

August 2014 - July 2019

January 2010 - July 2014

Education

August 2003 - December 2009

## Publications

Publications (91)

We develop computer assisted arguments for proving the existence of transverse homoclinic connecting orbits, and apply these arguments for a number of non-perturbative parameter and energy values in the spatial equilateral circular restricted four body problem. The idea is to formulate the desired connecting orbits as solutions of certain two point...

This work provides a systematic recipe for computing accurate high order Fourier expansions of quasiperiodic invariant circles (and systems of such circles) in area preserving maps. The recipe requires only a finite data set sampled from the quasiperiodic circle. Our approach, being based on the parameterization method of [HdlL07, HdlL06a, HdlL06b]...

The goal of this paper it to prove existence theorems for one parameter families (branches) of ejection-collision orbits in the planar circular restricted three body problem (CRTBP), and to study some of their bifurcations. The orbits considered are ejected from one primary body and collide with the other (as opposed to more local ejections-collisi...

This work develops a functional analytic framework for making computer assisted arguments involving transverse heteroclinic connecting orbits between hyperbolic periodic solutions of ordinary differential equations. We exploit a Fourier-Taylor approximation of the local stable/unstable manifold of the periodic orbit, combined with a numerical metho...

For the three body problem with equal masses, we prove that the most symmetric continuation class of Lagrange's equilateral triangle solution, also referred to as the $P_{12}$ family of Marchal, contains the remarkable figure eight choreography discovered by Moore in 1993, and proven to exist by Chenciner and Montgomery in 2000. This settles a conj...

This work develops a computational framework for proving existence, uniqueness, isolation , and stability results for real analytic fixed points of m-th order Feigenbaum-Cvitanović renormalization operators. Our approach builds on the earlier work of Lanford, Eckman, Wittwer, Koch, Burbanks, Osbaldestin, and Thurlby [37, 26, 23, 8, 9], however the...

This work provides a systematic recipe for computing accurate high order Fourier expansions of quasiperiodic invariant circles in area preserving maps. The recipe requires only a finite data set sampled from the quasiperiodic circle. Our approach, being based on the parameterization method, uses a Newton scheme to iteratively solve a conjugacy equa...

We present a computational method for studying transverse homoclinic orbits for periodic solutions of delay differential equations, a phenomenon that we refer to as the {\em Poincar\'{e} scenario}. The strategy is geometric in nature, and consists of viewing the connection as the zero of a nonlinear map, such that the invertibility of its Fr\'{e}ch...

This work concerns the dynamics of a certain class of delay differential equations (DDEs) which we refer to as state dependent delay maps. These maps are generated by delay differential equations where the derivative of the current state depends only on delayed variables, and not on the un-delayed state. However, we allow that the delay is itself a...

A new method for approximating unstable manifolds for parabolic PDEs is introduced, which combines the parameterization method for invariant manifolds with finite element analysis and formal Taylor series expansions, and is applicable to problems posed on irregular spatial domains. The param-eterization method centers on an infinitesimal invariance...

We develop a multiple shooting parameterization method for studying stable/unstable manifolds attached to periodic orbits of systems whose dynamics is determined by an implicit rule. We represent the local invariant manifold using high order polynomials and show that the method leads to efficient numerical calculations. We implement the method for...

This paper studies the local unstable manifold attached to an equilibrium solution of a system of delay differential equations (DDEs). Two main results are developed. The first is a general method for computing the formal Taylor series coefficients of a function parameterizing the unstable manifold. We derive linear systems of equations whose solut...

This paper considers two point boundary value problems for conservative systems defined in multiple coordinate systems, and develops a flexible a-posteriori framework for computer assisted existence proofs. Our framework is applied to the study collision and near collision orbits in the circular restricted three body problem. In this case the coord...

This paper develops validated computational methods for studying infinite dimensional stable manifolds at equilibrium solutions of parabolic PDEs, synthesizing disparate errors resulting from numerical approximation. To construct our approximation, we decompose the stable manifold into three components: a finite dimensional slow component, a fast-b...

We combine the parameterization method for invariant manifolds with the finite element method for elliptic PDEs,to obtain a new computational framework for high order approximation of invariant manifolds attached to unstable equilibrium solutions of nonlinear parabolic PDEs. The parameterization method provides an infinitesimal invariance equation...

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 periodi...

Normally hyperbolic invariant manifolds theory provides an efficient tool for proving diffusion in dynamical systems. In this paper we develop a methodology for computer assisted proofs of diffusion in a-priori chaotic systems based on this approach. We devise a method, which allows us to validate the needed conditions in a finite number of steps,...

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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt}...

Normally hyperbolic invariant manifolds theory provides an efficient tool for proving diffusion in dynamical systems. In this paper we develop a methodology for computer assisted proofs of diffusion in a-priori chaotic systems based on this approach. We devise a method, which allows us to validate the needed conditions in a finite number of steps,...

The present work studies the robustness of certain basic homoclinic motions in an equilateral restricted four-body problem. The problem can be viewed as a two-parameter family of conservative autonomous vector fields. The main tools are numerical continuation techniques for homoclinic and periodic orbits, as well as formal series methods for comput...

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 two-torus in R3 . After changing to rotating coordinates and exploiting symmetr...

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 c...

The set of transverse homoclinic intersections for a saddle-focus equilibrium in the planar equi-lateral restricted four body problem admit certain simple homoclinic oribts which form the skeleton of the complete homoclinic intersection-or homoclinic web. In the present work the planar restricted four body problem is viewed as an invariant subsyste...

The present work studies the robustness of certain basic homoclinic motions in an equilateral restricted four body problem. The problem can be viewed as a two parameter family of conservative autonomous vector fields. The main tools are numerical continuation techniques for homoclinic and periodic orbits, as well as formal series methods for comput...

This paper develops validated computational methods for studying infinite dimensional stable manifolds at equilibrium solutions of parabolic PDEs. Our approach is constructive and combines the parameterization method with Lyapunov-Perron operators. More precisely, we decompose the stable manifold into three components: a finite dimensional slow com...

This paper develops validated computational methods for studying infinite dimensional stable manifolds at equilibrium solutions of parabolic PDEs. Our approach is constructive and combines the parameterization method with Lyapunov-Perron operators. More precisely, we decompose the stable manifold into three components: a finite dimensional slow com...

We make a detailed numerical study of a three dimensional dissipative vector field derived from the normal form for a cusp-Hopf bifurcation. The vector field exhibits a Neimark–Sacker bifurcation giving rise to an attracting invariant torus. Our main goals are to (A) follow the torus via parameter continuation from its appearance to its disappearan...

The set of transverse homoclinic intersections for a saddle-focus equilibrium in the planar equilateral restricted four-body problem admit certain simple homoclinic orbits which form the skeleton of the complete homoclinic intersection -- or homoclinic web. In the present work, the planar restricted four-body problem is viewed as an invariant subsy...

We present a new approach to validated numerical integration of systems delay differential equations. We focus on the case of a single constant delay though the method generalizes to systems with multiple lags. The method provides mathematically rigorous existence results as well as error bounds for both the solution and the Fréchet derivative of t...

\In this work, we develop a high order polynomial approximation scheme for the lo- cal unstable manifold attached to a linearly unstable traveling wave. The wave profile is itself an equilibrium solution of a nonlinear parabolic partial differential equation (PDE) formulated on the real line, and its unstable manifold describes the dynamics of smal...

This work develops validated numerical methods for linear stability analysis at an equilibrium solution of a system of delay differential equations (DDEs). The case of a single constant delay is considered. The method downplays the role of the scalar transcendental characteristic equation in favor of a functional analytic approach exploiting the st...

We make a detailed numerical study of the dynamics of a three dimensional dissipative vector field exhibiting a Neimark-Sacker bifurcation. Our main goals are to follow the attracting invariant torus born out of this bifurcation to its destruction in subsequent appearance of a chaotic attractor, and also to study the stable/unstable manifolds of th...

We make a detailed numerical study of the dynamics of a three dimensional dissipative vector field exhibiting a Neimark-Sacker bifurcation. Our main goals are to follow the attracting invariant torus born out of this bifurcation to its destruction in subsequent appearance of a chaotic attractor, and also to study the stable/unstable manifolds of th...

This work studies existence and regularity questions for attracting invariant tori in three dimensional dissipative systems of ordinary differential equations. Our main result is a constructive method of computer assisted proof which applies to explicit problems in non-perturbative regimes. We obtain verifiable bounds on the regularity of the attra...

The present work develops validated numerical methods for analyzing continuous branches of connecting orbits – as well as their bifurcations – in one parameter families of discrete time dynamical systems. We use the method of projected boundaries to reduce the connecting orbit problem to a finite dimensional zero finding problem for a high dimensio...

We describe a method for computing an atlas for the stable or unstable manifold attached to an equilibrium point, and implement the method for the saddle-focus libration points of the planar equilateral restricted four body problem. We employ the method at the maximally symmetric case of equal masses, where we compute atlases for both the stable an...

We develop a systematic approach for proving the existence of spatial choreogra-phies in the gravitational n body problem. 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 p...

We develop a systematic approach for proving the existence of spatial choreographies in the gravitational $n$ body problem. 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...

We use validated numerical methods to prove the existence of spatial pe- riodic orbits in the equilateral restricted four body problem. We study each of the vertical Lyapunov families (up to symmetry) in the triple Copenhagen problem, as well as some halo and axial families bifurcating from planar Lyapunov families. We consider the system with both...

In this paper we develop mathematically rigorous computer assisted techniques for studying high order Fourier-Taylor parameterizations of local stable/unstable mani-folds for hyperbolic periodic orbits of analytic vector fields. We exploit the numerical methods developed in [1] in order to obtain a high order Fourier-Taylor series expansion of the...

We describe a method for computing an atlas for the stable or unstable manifold attached to an equilibrium point, and implement the method for the saddle-focus libration points of the planar equilateral restricted four body problem. We employ the method at the maximally symmetric case of equal masses, where we compute atlases for both the stable an...

This work is a translation from French of the memoir "Connaissance actuelle des orbites dans le problème des trois corps" written by Elis Strömgren in 1933 about his research at the Copenhagen observatory. This work is often referred to in contemporary works however it appears to be only available in French. This is a modest attempt by the authors...

These are the MatLab codes associated with the paper. See the readme file.

This paper develops a Chebyshev–Taylor spectral method for studying stable/unstable manifolds attached to periodic solutions of differential equations. The work exploits the parameterization method — a general functional analytic framework for studying invariant manifolds. Useful features of the parameterization method include the fact that it can...

We prove the existence of chaotic motions in a planar restricted four body problem, establishing that the system is not integrable. The idea of the proof is to verify the hypotheses of a topological forcing theorem. The forcing theorem applies to two freedom Hamiltonian systems where the stable and unstable manifolds of a saddle-focus equilibrium i...

We develop a validated numerical procedure for continuation of local stable/unstable manifold patches attached to equilibrium solutions of ordinary differential equations. The procedure has two steps. First we compute an accurate high order Taylor expansion of the local invariant manifold. This expansion is valid in some neighborhood of the equilib...

The goal of these notes is to illustrate the use of validated numerics as a tool for studying the dynamics near and between equilibrium solutions of ordinary differential equations. We examine Taylor methods for computing local stable/unstable manifolds and also for expanding the flow in a neighborhood of a given initial condition. The Taylor metho...

This work concerns efficient numerical methods for computing high order Taylor and Fourier-Taylor approximations of unstable manifolds attached to equilibrium and periodic solutions of delay differential equations. In our approach we first reformulate the delay differential equation as an ordinary differential equation on an appropriate Banach spac...

We consider the problem of computing stable/unstable manifolds attached to periodic orbits of maps, and develop quasi-numerical methods for polynomial approximation of the manifolds to any desired order. The methods avoid function compositions by exploiting an idea inspired by multiple shooting schemes for periodic orbits. We consider a system of c...

We develop and implement a semi-numerical method for computing high order Tay-lor approximations of the unstable manifold at a hyperbolic fixed point of a compact infinite dimensional analytic map. Even though the method involves several layers of truncation our goal is to obtain a representation of the invariant manifold which is accurate in a lar...

We develop techniques for computing the (un)stable manifold at a hyperbolic equilibrium of an analytic vector field. Our approach is based on the so-called parametrization method for invariant manifolds. A feature of this approach is that it leads to a-posteriori analysis of truncation errors which, when combined with careful management of round of...

In this work we study, from a numerical point of view, the (un)stable manifolds of a certain class of dynamical systems called hybrid maps. The dynamics of these systems are generated by a two stage procedure: the first stage is continuous time advection under a given vector field, the second stage is discrete time advection under a given diffeomor...

In this work we implement a rigorous computer-assisted technique
for proving existence of periodic solutions of nonlinear differential
equations with non-polynomial nonlinearities. We exploit ideas from the
theory of automatic differentiation in order to formulate an augmented
nonlinear system which has only polynomial nonlinearities.
We valid...

In this work we develop a method for computing mathematically rigorous enclosures of some one dimensional manifolds of heteroclinic orbits for nonlinear maps. Our method exploits a rigorous curve following argument build on high order Taylor approximation of the local stable/unstable manifolds. The curve following argument is a uniform interval New...

We develop and implement computer assisted arguments which establish the existence of heteroclinic/homoclinic connecting orbits between fixed points of compact infinite dimensional maps. The argument is based on a-posteriori analysis of a certain “finite time boundary value problem”. A key ingredient in the analysis is the representation of local s...

This work treats a functional analytic framework for computer assisted Fourier analysis which can be used to obtain mathematically rigorous error bounds on numerical approximations of solutions of differential equations. An abstract a-posteriori theorem is employed in order to obtain existence and regularity results for C^k problems with 0 < k ≤ ∞...

We develop a method for computing polynomial approximations of unstable
manifolds at equilibrium solutions of parabolic PDEs. These polynomials have a
finite number of variables, even though they map into an infinite dimensional
state space. We implement this method numerically, and develop explicit
a-posteriori error bounds. By combining the a-pos...

This work describes a method to rigorously compute the real Floquet normal form decomposition of the fundamental matrix solution of a system of linear ODEs having periodic coefficients. The Floquet normal form is validated in the space of analytic functions. The technique combines analytical estimates and rigorous numerical computations and no rigo...

In this work we develop some automatic procedures for computing high order polynomial expansions of local (un)stable manifolds for equilibria of differential equations. Our method incorporates validated truncation error bounds, and maximizes the size of the image of the polynomial approximation relative to some specified constraints. More precisely...