# Slow-Fast Systems with an Equilibrium Near the Folded Slow Manifold

*2024, Volume 29, Number 2, pp. 376-403*

Author(s):

**Gelfreikh N. G., Ivanov A. V.**

We study a slow-fast system with two slow and one fast variables. We assume that
the slow manifold of the system possesses a fold and there is an equilibrium of the system in
a small neighborhood of the fold. We derive a normal form for the system in a neighborhood
of the pair “equilibrium-fold” and study the dynamics of the normal form. In particular, as
the ratio of two time scales tends to zero we obtain an asymptotic formula for the Poincaré
map and calculate the parameter values for the first period-doubling bifurcation. The theory is
applied to a generalization of the FitzHugh – Nagumo system.

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