Jonathan Jaquette, PhD
Postdoctoral Associate
Boston University College of Arts and Sciences
Dept of Mathematics and Statistics


Pronouns: he/him/his



My research focuses on developing a global understanding of how complex systems change over time, and bridging the gap between what can be proven mathematically and what can be computed numerically.

Nonlinear differential equations are rarely explicitly solvable by hand. Instead of searching for arbitrary solutions, the dynamical systems viewpoint is to focus one's analysis on the qualitative behavior of invariant sets. For example, while a periodic orbit's geometry may be sensitive to perturbations, its topology (e.g. being homeomorphic to a circle) is much more robust. With abstract theorems one can describe in great detail the dynamics on and around generic invariant sets. However for a specific differential equation, verifying the hypotheses of such a theorem often requires hard quantitative analysis.

I am particularly interested in infinite dimensional dynamical systems and understanding their dynamics through a holistic study of a system's invariant sets. Computationally, this draws on a variety of numerical techniques from dynamical systems, partial differential equations, nonlinear optimization and algebraic topology. Analytically, this often involves proving theorems with explicitly verifiable hypotheses (e.g. rather than assuming "there exists some e>0", concretely quantifying how small e must be). The impetus for this is not bookkeeping for bookkeeping's sake. But rather to build a complete picture of a complex system from disparate pieces.

For example, standard numerical methods can solve an initial value problem for an ODE and provide local error bounds at each step. However a global error bound on the final solution requires the cumulative error be quantified. This quickly becomes a nontrivial problem in chaotic systems, where arbitrarily close initial conditions will inevitably diverge, and the difficulties compound in partial differential equations where the phase space is infinite dimensional.

To that end, validated numerics have been developed to keep track of all the sources of error inherent to numerical calculations. To bridge the gap between numerics and a computer assisted proof, a problem must be translated into a list of the conditions that the computer can check. Most famously used to solve the four color theorem, computer assisted proofs have been employed to great effect in dynamics, proving results such as the universality of the Feigenbaum constants and Smale's 14th problem on the nature of the Lorenz attractor.

Publications listed below are automatically derived from MEDLINE/PubMed and other sources, which might result in incorrect or missing publications. Faculty can login to make corrections and additions.

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  1. Jaquette J. Quasiperiodicity and blowup in integrable subsystems of nonconservative nonlinear Schrodinger equations. Journal of Dynamics and Differential Equations. 2022; 1-25. View Publication
  2. Jaquette J, Beck M. Validated spectral stability via conjugate points. SIAM Journal on Applied Dynamical Systems. 2022; 21:366-404. View Publication
  3. Jaquette J, Lessard JP, Takayasu A. Global dynamics in nonconservative nonlinear Schrodinger equations. Advances in Mathematics. 2022; 398. View Publication
  4. Jaquette J, Takayasu A, Lessard JP, Okamoto H. Rigorous numerics for nonlinear heat equations in the complex plane of time. Numerische Mathematik. 2022; 1-58. View Publication
  5. Jaquette J, van den Berg JB, Mireles James JD. Validated Numerical Approximation of Stable Manifolds for Parabolic Partial Differential Equations. Journal of Dynamics and Differential Equations,. 2022; 1-61. View Publication
  6. Jaquette J, Lessard JP, Takayasu A. Singularities and heteroclinic connections in complex-valued evolutionary equations with a quadratic nonlinearity. Commun Nonlinear Sci Numer Simulat. 2022; 107. View Publication
  7. Jaquette J, Schweinhart B. FRACTAL DIMENSION ESTIMATION WITH PERSISTENT HOMOLOGY: A COMPARATIVE STUDY. Commun Nonlinear Sci Numer Simul. 2020 May; 84. PMID: 32256012; PMCID: PMC7117095; DOI: 10.1016/j.cnsns.2019.105163;
     
  8. Jaquette J. A proof of Jones’ conjecture. Journal of Differential Equations. 2019; 266(6):3818-3859. View Publication
  9. Jaquette J, van den Berg JB . A proof of Wright’s conjecture. Journal of Differential Equations. 2018; 264(12):7412-7462. View Publication
  10. Jaquette J, Lessard JP, Mischaikow K. Stability and uniqueness of slowly oscillating periodic solutions to Wright’s equation. Journal of Differential Equations. 2017; 263(11):7263-7286. View Publication
Showing 10 of 11 results. Show More

This graph shows the total number of publications by year, by first, middle/unknown, or last author.

Bar chart showing 11 publications over 4 distinct years, with a maximum of 6 publications in 2022

YearPublications
20172
20181
20192
20226


2022 SIAM: Early Career Travel Award
2021 SIAM: Early Career Travel Award
2020 European Commission: Seal of Excellence
2019 Brandeis University: Susan Lindquist Award
2019 SIAM: Early Career Travel Award
2019 Rutgers University Graduate School: TA/GA Professional Development Fund Award
2017 Rutgers University Graduate School: TA/GA Professional Development Fund Award
2017 SIAM: Student Travel Award
2016 Rutgers University Graduate School: Special Study Award
2012 Swarthmore College: Lockwood Fellow
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