Computer Assisted Proof in Analysis
Part of my research deals with nonlinear analysis of operator equations coming from applied mathematics and dynamical systems theory. The computer is a powerful tool for studying these equations, and I am especially interested in methods for obtaining mathematically rigorous results in collaboration with the computer. Many of these methods have an `a-posteriori' flavor, i.e. first we computer a good numerical approximation and then we try to get explicit bounds between the approximate and the true solution. In practice this requires a delicate balance between deliberate numerics and pen and paper mathematical analysis.
[1] A rigorous implicit C1 Chebyshev integrator for delay equations. With J.P. Lessard (Submitted)
[2] A functional analytic approach to validated numerics for eigenvalues of delay equations. With J.P. Lessard (to appear in the Journal of Computational Dynamics)
[3] Computer assisted proofs of contracting invariant tori for ODEs. With Maciej Capinski and Emmanuel Fleurantin (to appear in Discrete and Continuous Dynamical Systems)
[4] Torus Knot Choreographies in the n-body problem. With Renato Calleja, Carlos Garcia-Azpeitia, and J.P. Lessard (Submitted).
[5] Spatial periodic oribts in the equilateral circular restricted four body problem: computer assisted proofs of existence. With Jaime Burgos-Garcia and J.P. Lessard. Celestial Mechanics and Dynamical Astronomy. 131:2 (2019).
[6] Chaotic motions in the restricted four body problem via Devaney's saddle-focus homoclinic tangle theorem. With Shane Kepley. Journal of Differential Equations. Volume 266, Issue 4, February 2019. Pages 1709-1755.
[7] Validated numerics for continuation and bifurcation of connecting orbits for maps. With Ronald Adams (to appear in Qualitative Theory of Dynamical Systems).
[8] Analytic continuation of local (un)stable manifolds with rigorous computer assisted error bounds. With Shane Kepley and Bill Kalies. SIAM Journal on Applied Dynamical Systems, Vol 17, No. 1, pp. 157--202 (2018)
[9] Validated numerics for equilibria of analytic vector fields: invariant manifolds and connecting orbits. (To appear in the AMS Proceedings of Symposia in Applied Mathematics).
[10] Parameterization of invariant manifolds for periodic orbits (II): a-posteriori analysis and computer assisted error bounds. With Roberto Castelli and J.P. Lessard (to appear in the Journal of Dynamics and Differential Equations).
[11] Fourier-Taylor Parameterization of Unstable Manifolds for Parabolic Partial Differential Equations: Formalism, Implementation, and Rigorous Validation with Christian Reinhardt. Indagationes Mathematicae. Volume 30, Issue 1, January 2019. Pages 39-80.
[12] Computer assisted Fourier analysis in sequence spaces of varying regularity with J.P. Lessard. SIAM Journal on Mathematical Analysis, Vol 49, Issue 1, pp. 530-561. (2017).
[13] Connecting orbits for compact infinite dimensional maps: computer assisted proofs of existence with Rafael de la Llave. SIAM Journal on Applied Dynamical Systems, Vol. 15, No. 2, pp. 1268-1323 (2016).
[14] Fourier-Taylor Approximation of Unstable Manifolds for Compact Maps: Numerical Implementation and Computer Assisted Error Bounds.Foundations of Computational Mathematics, Vol 17, Issue 6, pp. 1467-1523 (2017).
[15] Analytic Enclosure of the Fundamental Matrix Solution. With Roberto Castelli and J.P. Lessard. Applications of Mathematics, vol 60 (2015), issue 6, pp. 617--636.
[16] Stationary coexistence of hexagons and rolls via rigorous computatoins. With J.B. van den Berg, A. Deschenes, and J.P. Lessard SIAM Journal on Applied Dynamical Systems, Vol 14, No. 2, pp. 942-979. (2015)
Click here for the associated codes.
[17] Computer Assisted Proof of Transverse Saddle-to-Saddle Connecting Orbits for First Order Vector Fields. With J.-P. Lessard and C. Reinhardt. Journal of Dynamics and Differential Equations. Volume 26, Issue 2, pp 267--313. (2014)
[18] Rigorous Numerics for Analytic Solutions of Differential Equations: the Radii Polynomial Approach. With Allan Hungria and J.P. Lessard. Mathematics of Computation, Volume 85, Number 299, May 2016, Pages 1427-1459.
[19] Automatic differentiation for Fourier series and the radii polynomial approach. With Julian Ransford and J.P. Lessard. Physica D: Nonlinear Phenomena, Vol 334, Number 1, pp. 174-186, Novermber (2016)
[20] Rigorous A-Posteriori Computation of (Un)Stable Manifolds and Connecting Orbits for Analytic Maps. With K. Mischaikow. SIAM Journal on Applied Dynamical Systems, Volume 12, Number 2, pp. 957-1006. (2013)
[21] Computer Assisted Error Bounds for Linear Approximation of (Un)Stable Manifolds and Rigorous Validation of Higher Dimensional Transverse Connecting Orbits. Communications in Nonlinear Science and Numerical Simulation, Vol 22, pp. 1102-1133. (2015)
[22] Rigorous Numerics for Symmetric Connecting Orbits: Even Homoclinics of the Gray-Scott Equation. With J. B. van den Berg, J.-P. Lessard, and K. Mischaikow. SIAM Journal on Mathematical Analysis, Volume 43, Issue 4, pp. 1557-1594. (2011)
[23] Computational Proofs in Dynamics. With K. Mischaikow. Appears in the Springer Encyclopedia of Applied and Computational Mathematics. Bjorn Engquist Editor.
Invariant Manifolds
Invariant sets are the fundamental building blocks for understanding dynamical systems. The stable and unstable manifolds associated with hyperbolic equilibria and periodic orbits of differential equations are canonical examples. I am interested in ``high order'' methods such as the Parameterization Method for computing these invariant manifolds. I am also interested in methods for numerically studying the intersections of these manifolds, as intersections of stable and unstable manifolds give rise to connecting orbits between invariant sets. These computational methods can be used to study (for example) chaotic dynamics, transport phenomena, and Morse homology.
[24] Homoclinic dynamics in a spatial restricted four body problem: blue skies into Smale horseshoes for vertical Lyapunov families. With Maxime Murray (Submitted).
[25] Parameterization method for unstable manifolds of standing waves on the line. With Blake Barker and Jalen Morgan (Submitted).
[26] Finite element approximation of invariant manifolds by the parameterization method. With Jorge Gonzalez and Necibe Tuncer (Submitted).
[27] Resonant tori, transport barriers, and chaos in a vector field with a Neimark-Sacker bifurcation. With Emmanuel Fleurantin (to appear in Communications in Nonlinear Science and Numerical Simulation).
[28] Homoclinic dynamics in a restricted four body problem: transverse connections for the saddle-focus equilibrium set. With Shane Kepley. Celestial Mechanics and Dynamical Astronomy, (2019) 131: 13. https://doi.org/10.1007/s10569-019-9890-8
[29] Chebyshev-Taylor parameterization of stable/unstable manifolds for periodic orbits: implementation and applications. With Maxime Murray. International Journal of Bifurcation and Chaos, Vol 27, No. 14 (2017).
[30] Parameterization method for unstable manifolds of delay differential equations. With Chris Groothedde, Journal of Computational Dynamics, Vol 4, Issue 1 (2017).
[31] Validted computation of heteroclinic sets. With Maciej Capinski SIAM Journal on Applied Dynamical Systems, Volume 16, Issue 1, pp. 375--409 (2017).
[32] High-order parameterization of stable/unstable manifolds for long periodic orbits of maps. With Jorge Gonzalez SIAM Journal on Applied Dynamical Systems, Vol. 16, No. 3, pp. 1748 -- 1795 (2017).
[33] High-order parameterization of (un)stable manifolds for hybrid maps: implementation and application with Vincent Naudot and Qiuying Lu. Communications in Nonlinear Science and Numerical Simulation, Volume 53, pp. 184--201, December (2017).
[34] Computation of maximal local (un)stable manifold patches by the Parameterization Method. With Maxime Breden, and J.P. Lessard, Indagationes Mathematicae, Vol 27, Issue 1, January 2016, pages 340-367.
[35] Computing (un)stable manifolds with validated error bounds: non-resonant and resonant spectra. With Christian Reinhardt, and J.B. van den Berg. Journal of Nonlinear Science, Vol. 26 (2016), pp. 1055--1095.
(click to view manuscript)
[36] Parameterization of slow-stable manifolds and their invariant vector bundles: theory and numerical implementation. with J.B. van den Berg Discrete and Continuous Dynamical Systems, Vol. 36, No. 9, pp. 4637--4664 (2016).
[37] Parameterization of invariant manifolds for periodic orbits (I): efficient numerics via the Floquet normal form. with Roberto Castelli and J.P. Lessard. SIAM Journal on Applied Dynamical Systems, Vol. 14, No. 1, pp. 132-167. (2015)
[38] Polynomial Approximation of One Parameter Families of (Un)Stable Manifolds with Rigorous Computer Assisted Error Bounds. Indagationes Mathematicae, Vol 26, Issue 1, pp. 225-265. (2015)
[39] Computation of Heteroclinic Arcs with Application to the Volume Preserving Henon Family. With Hector Lomeli. SIAM Journal on Applied Dynamical Systems, Volume 9, Issue 3, pp. 919--953.(2010)
[40] Quadratic Volume-Preserving Maps: (Un)Stable Manifolds, Hyperbolic Dynamics, and Vortex-Bubble Bifurcations. Journal of Nonlinear Science, Volume 23, Number 4, 2013, pp. 585--615 (2013).
KAM Theory
Many analytic methods for studying smooth invariant manifolds are based on the observation that parameterizations of the manifolds satisfy certain functional equations. In the case of invariant manifolds associated with a mix of stable and unstable eigendirections the functional equations admit the "small divisors" of KAM theory. Studying these equations requires a delicate mix of geometry, number theory, and hard analysis. We prove some `a-posteriori' existence theorems for symplectic and volume preserving diffeomorphisms.
[41] Parameterization of Invariant Manifolds by Reducibility for Volume Preserving and Symplectic Maps. With R. de la Llave. Discrete and Continuous Dynamical Systems. Volume 32, Number 12, December 2012. Pages 4321-4360.
Set Oriented Numerics
Another approach to numerical discretization of dynamical systems is to represent phase space as a collection of `cells' and then to represent the dynamics as a directed graph over these cells/nodes. This is the so called set oriented approach. In this setting combinatorial properties of the directed graph provide insight into the underlying dynamics. I have done some work on the application of set oriented methods to conservative systems, where the Morse Set/Gradient Set dichotomy breaks down.
[42]
Adaptive Set-Oriented Computation of Topological Horseshoe Factors in Area and Volume Preserving Maps.
SIAM Journal on Applied Dynamical Systems, Vol 9. Issue 4. 2010 pp. 1164-1200.
