Forcing Theorems via Morse-Conley-Floer Homology

From a mathematical view point, the advantage of rigorous numerics over simulations is that the outcomes can be used as components in the building of mathematics. This is often expressed in the form of forcing theorems: if one finds a certain type of solution, then this implies (by analytic theory) other properties of the dynamical system. The most famous result of this type is the theorem period-3 implies chaos for interval maps. Mathematics in general is riddled with such statements, where the assumptions in the theorems are in practice very hard if not impossible to check for any specific system, at least not by hand. It is in overcoming this obstacle that part of my research is dedicated to combine rigorous numerics with Morse-Conley-Floer theory to obtain new forcing theorems in differential equations.

Chaotic braided solutions via Morse-Conley theory and rigorous numerics. We prove in [1] that the stationary Swift-Hohenberg equation has chaotic dynamics on a critical energy level for a large (continuous) range of parameter values. The first step of the method relies on a computer-assisted, rigorous, continuation method to prove the existence of a periodic orbit with certain geometric properties. The second step is topological: we use this periodic solution as a skeleton, through which we braid other solutions, thus forcing the existence of infinitely many braided periodic orbits. A semi-conjugacy to a subshift of finite type shows that the dynamics is chaotic.

[1] J.B. van den Berg and J.-P. Lessard. Chaotic braided solutions via rigorous numerics: chaos in the Swift-Hohenberg equation. SIAM Journal on Applied Dynamical Systems, 7 (3): 988–1031, 2008. (pdf)


[2] J.B. van den Berg, M. Gameiro, J.-P. Lessard and R. Vandervorst. Computing relative indices of critical points in strongly indefinite problems, 2017. In preparation.

Computational Morse-Floer theory. In the upcoming paper [2], we discuss how to compute relative indices of critical points in strongly indefinite variational problems using computer-assisted proof techniques. For all its popularity and success, Morse-Floer homology is renowned for being difficult to compute explicitly. Recent developments in rigorous numerics brings the opportunity to compute at least some of the ingredients of the Morse-Floer homology construction explicitly, namely critical points and their relative (or Morse) indices. This requires understanding the spectral flow along a homotopy between two linear operators on some infinite dimensional space. Using relatively simple model equations, we show that equilibria and their relative indices are computable.

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:

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@ Postdoc level:

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Jean-Philippe Lessard

Associate Professor

McGill University

Department of Mathematics and Statistics

Research Projects