Quantitative jump theorem

P. I. Kaleda

Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 72 (2011) no. 2, pp. 223-247

Résumé

The jump theorem proved by Mishchenko and Pontryagin more than fifty years ago is one of the fundamental results in the theory of relaxation oscillations. Its statement is asymptotic in character. In this paper we present a quantitative analogue of it. This means the following. The jump theorem describes the map along trajectories (the Poincaré map) from a transversal ‘before the jump’ to a transversal ‘after the jump’. This map is exponentially contracting, and its deviation from the jump point with respect to the slow coordinate is of order $\varepsilon^{2/3}$, where $\varepsilon$ is the small parameter in the fast-slow system. These estimates are asymptotic. Normalizing the system by choosing the scale, we prove that for all $\varepsilon$ no greater than $e^{-12}$, the Poincaré map is defined, its deviation lies in the interval $\varepsilon^{2/3}[e^{-6},e^3]$, and the map itself is a contraction with a coefficient that does not exceed $e^{-k(\varepsilon)}$, where $k(\varepsilon)\ge1/(6\varepsilon)-10^3$. The main tool used in the investigation is the method of blowup with different weights, in the form described by Krupa and Szmolyan.