The entry-exit function and geometric singular perturbation theory

By Peter De Maesschalck, Stephen Schecter. 2015

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For small $\epsilon>0$, the system $\dot x = \epsilon$, $\dot z = h(x,z,\epsilon)z$, with $h(x,0,0) 0$ for $x>0$, admits solutions that approach the $x$-axis while $x 0$. The limiting attraction and repulsion points are given by the well-known entry-exit function. For $h(x,z,\epsilon)z$ replaced by $h(x,z,\epsilon)z^2$, we explain this phenomenon using geometric singular perturbation theory. We also show that the linear case can be reduced to the quadratic case, and we discuss the smoothness of the return map to the line $z=z_0$, $z_0>0$, in the limit $\epsilon\to0$.

Subfields: Computer Networks and Communications, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/eddb75773ff47d50242d5300fbc0ba4edf9fdab0