Continuation of slowly oscillating periodic solutions (SOPS). In 1962, Jones numerically observed that for a large class of initial conditions, the solutions of Wright’s delay equation y’(t)=-αy(t-1)[1+y(t)] all seemed to converge to a single SOPS. That lead to Jones Conjecture, which states that Wright’s equation has a unique SOPS for every α>π/2. In [1], we reformulate this conjecture and we use a method called validated continuation to rigorously compute a global continuous branch of SOPS of Wright's equation. Using this method, we show that a part of this branch does not possess any fold point, partially answering the new reformulated conjecture.

Jones Conjecture. In [2], we prove that Wright’s equation has a unique SOPS for all α ∈ I:=[1.9,6], up to time translation. Our proof is based on a same strategy employed earlier in [Xie, J. Differential Equations 103 (2) (1993) 350-374]: show that every SOPS is asymptotically stable. We first introduce a branch and bound algorithm to control all SOPS using bounding functions at all α ∈ I. Once the bounding functions are constructed, we 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 ∈ I, and the proof follows. This result is a step toward the proof of Jones Conjecture.

Multiple delays. Studying DDEs with multiple time lags presents severe complications. While the difficulties in understanding their dynamics mostly come from the infinite dimensionality of the state space and the presence of nonlinearities, it is well accepted that many techniques available for studying the dynamics of DDEs with one delay do not work for DDEs with more delays or distributed delays. In [3], we develop rigorous numerical methods to prove existence of periodic solutions in DDEs with multiple time lags. We present a proof of coexistence of three periodic solutions for a modified Wright’s equation with two time lags, and present rigorous computations of several nontrivial periodic solutions for a modification of Wright’s equation with three time lags.

Periodic solutions in a delayed van der Pol equation. In [4] we introduce a method to prove existence of several rapidly and slowly oscillating periodic oscillations of a delayed van der Pol oscillator. The proof is a combination of careful pen and paper analytic estimates, the contraction mapping theorem and a computer program using interval arithmetic. Using this approach we extend some existence results obtained by Nussbaum in [Ann. Mat. Pura Appl., 4(101), 263-306, 1974].

[1] J.-P. Lessard. Recent advances about the uniqueness of the slowly oscillating periodic solutions of Wright’s equation. Journal of Differential Equations, 248 (5): 992–1016, 2010. (pdf)

[2] J. Jaquette, J.-P. Lessard and K. Mischaikow. Stability and uniqueness of slowly oscillating periodic solutions to Wright’s equation, Journal of Differential Equations, 263(11): 7263-7286, 2017. (pdf)

[3] G. Kiss and J.-P. Lessard. Computational fixed point theory for differential delay equations with multiple time lags. Journal of Differential Equations, 252 (4): 3093–3115, 2012. (pdf)

[4] G. Kiss and J.-P. Lessard. Rapidly and slowly oscillating periodic oscillations of a delayed van der Pol oscillator. Journal of Dynamics and Differential Equations, 29(4): 1233-1257, 2017. (pdf)

[5] J.B. van den Berg, C. Groothedde and J.-P. Lessard. A general method for computer-assisted proofs of periodic solutions in delay differential equations. Preprint, 2018.

Contact

Department of Math. and Stat.

McGill University

Burnside Hall, Room 1119

805 Sherbrooke West

Montreal, QC, H3A 0B9, CANADA

jp.lessard@mcgill.ca

Phone: (514) 398-3804

Positions Available

@ Ph.D. level:

1. • I recommend that you read some of my papers before contacting me.
   

2. • I will not reply to generic emails.
   

@ Postdoc level:

1. • Openings are available through the
   CRM-ISM Postdoctoral program.

Jean-Philippe Lessard

Associate Professor

McGill University

Department of Mathematics and Statistics

Research Projects