Numerical Analysis Seminars: Numerical computation of slow manifolds and canard trajectories by the continuation of boundary-value problems
7 March 2014 14:00 in CM105
In this talk, I will present a numerical strategy to compute attracting and repelling slow manifolds associated with slow-fast dynamical systems in dimension 3, with 2 slow variables. The method is based on the numerical continuation of one-parameter families of two-point boundary-value problems. Attracting and repelling slow manifolds are perturbations the attracting and repelling sheets of the fast nullsurface of the vector field (a.k.a critical manifold). In regions of phase space where the critical manifold is not normally hyperbolic, slow manifolds may intersect transversally along special solutions called "canards", and give rise to complicated dynamics. I will review the classical canard phenomenon in planar slow-fast systems of Van der Pol type, as well as recent results related to canards in "2 slow / 1 fast" systems. I will then emphasise the use of numerical continuation to complement the study of canards in such systems, their dynamics and underlying bifurcation structure. A few examples will be given, in the context of neuronal models displaying complicated oscillatory solutions (mix of sub-threshold oscillations and spikes).
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