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] With Jiaqi Yang, Joan Gimeno, and J.P. Lessard. (Submitted).
[2] With Jonathan Jaquette, and J.B. van den Berg. \\ Journal of Dynamics and Differential Equations (Accepted Feb 2022) \\ https://doi.org/10.1007/s10884-022-10146-1 (appeared online March 2022).
[3] with Olivier Henot and J.P. Lessard \\ Journal of Dynamics and Differential Equations (Accepted April 2021) \\ https://doi.org/10.1007/s10884-021-10002-8 (appeared online May 2021).
[4] With Maciej Capinski, Jorge Gonzalez, and Jean-Pierre Marco. Communications in Nonlinear Science and Numerical Simulations. Volume 106 (2022), Paper No. 105970, 21pp. \\ https://doi.org/10.1016/j.cnsns.2021.105970
[5] With J.P. Lessard. Journal of Dynamics and Differential Equations (Accepted July 2020) \\ https://doi.org/10.1007/s10884-020-09880-1 (appeared online August 2020)
[6] With J.P. Lessard. Journal of Computational Dynamics. Vol 7, Num 1, pp. 123-158. June 2020.
[7] With Maciej Capinski and Emmanuel Fleurantin. Discrete and Continuous Dynamical Systems. December 2020, 40(12):6681 - 6707.
[8] With Renato Calleja, Carlos Garcia-Azpeitia, and J.P. Lessard. Nonlinearity. Vol 34, Num 1, January 2021.
[9] With Jaime Burgos-Garcia and J.P. Lessard. Celestial Mechanics and Dynamical Astronomy. 131:2 (2019).
[10] With Shane Kepley. Journal of Differential Equations. Volume 266, Issue 4, February 2019. Pages 1709-1755.
[11] With Ronald Adams. Qualitative Theory of Dynamical Systems 18(2), July 2017.
[12] With Shane Kepley and Bill Kalies. SIAM Journal on Applied Dynamical Systems, Vol 17, No. 1, pp. 157--202 (2018)
[13] Chapter 2 of the AMS Proceedings of Symposia in Applied Mathematics, Volume 74.
[14] With Roberto Castelli and J.P. Lessard. The Journal of Dynamics and Differential Equations. 30, 1525-1581 (2018).
[15] with Christian Reinhardt. Indagationes Mathematicae. Volume 30, Issue 1, January 2019. Pages 39-80.
[16] with J.P. Lessard. SIAM Journal on Mathematical Analysis, Vol 49, Issue 1, pp. 530-561. (2017).
[17] with Rafael de la Llave. SIAM Journal on Applied Dynamical Systems, Vol. 15, No. 2, pp. 1268-1323 (2016).
[18] Foundations of Computational Mathematics, Vol 17, Issue 6, pp. 1467-1523 (2017).
[19] With Roberto Castelli and J.P. Lessard. Applications of Mathematics, vol 60 (2015), issue 6, pp. 617--636.
[20] 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)
for the associated codes.
[21] With J.-P. Lessard and C. Reinhardt. Journal of Dynamics and Differential Equations. Volume 26, Issue 2, pp 267--313. (2014)
[22] With Allan Hungria and J.P. Lessard. Mathematics of Computation, Volume 85, Number 299, May 2016, Pages 1427-1459.
[23] With Julian Ransford and J.P. Lessard. Physica D: Nonlinear Phenomena, Vol 334, Number 1, pp. 174-186, November (2016)
[24] With K. Mischaikow. SIAM Journal on Applied Dynamical Systems, Volume 12, Number 2, pp. 957-1006. (2013)
[25] Communications in Nonlinear Science and Numerical Simulation, Vol 22, pp. 1102-1133. (2015)
[26] 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)
[27] 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.
[28] with Archana Neupane Timsina (Submitted).
[29] With Jorge Gonzalez and Necibe Tuncer (Submitted). \\ https://arxiv.org/abs/2203.03115
[30] With Renato Calleja, Carlos Garcia-Azpeitia, and J.P. Lessard. Celestial Mechanics and Dynamical Astronomy. 133, Num 10 (2021).
[31] With Wouter Hetebrij. Celestial Mechanics and Dynamical Astronomy. 133, Num: 4 (2021).
[32] With Maxime Murray Celestial Mechanics and Dynamical Astronomy. 132, Num: 38 (2020).
[33] With Blake Barker and Jalen Morgan. SIAM Journal on Applied Dynamical Systems. Vol. 19, No. 3, 1758-1797 (2020).
[34] With Emmanuel Fleurantin Communications in Nonlinear Science and Numerical Simulation. 85 (2020), 105226, 29 pp.
[35] With Shane Kepley. Celestial Mechanics and Dynamical Astronomy, 131:13, March (2019)
[36] With Maxime Murray. International Journal of Bifurcation and Chaos, Vol 27, No. 14 (2017).
[37] With Chris Groothedde, Journal of Computational Dynamics, Vol 4, Issue 1 (2017).
[38] With Maciej Capinski SIAM Journal on Applied Dynamical Systems, Volume 16, Issue 1, pp. 375--409 (2017).
[39] With Jorge Gonzalez SIAM Journal on Applied Dynamical Systems, Vol. 16, No. 3, pp. 1748 -- 1795 (2017).
[40] with Vincent Naudot and Qiuying Lu. Communications in Nonlinear Science and Numerical Simulation, Volume 53, pp. 184--201, December (2017).
[41] With Maxime Breden, and J.P. Lessard, Indagationes Mathematicae, Vol 27, Issue 1, January 2016, pages 340-367.
[42] With Christian Reinhardt, and J.B. van den Berg. Journal of Nonlinear Science, Vol. 26 (2016), pp. 1055--1095.
[43] with J.B. van den Berg Discrete and Continuous Dynamical Systems, Vol. 36, No. 9, pp. 4637--4664 (2016).
[44] with Roberto Castelli and J.P. Lessard. SIAM Journal on Applied Dynamical Systems, Vol. 14, No. 1, pp. 132-167. (2015)
[45] Indagationes Mathematicae, Vol 26, Issue 1, pp. 225-265. (2015)
[46] With Hector Lomeli. SIAM Journal on Applied Dynamical Systems, Volume 9, Issue 3, pp. 919--953.(2010)
[47] 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.
[48] 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.
[49] SIAM Journal on Applied Dynamical Systems, Vol 9. Issue 4. 2010 pp. 1164-1200.
