Two Contrasting Examples of Multidimensional Differential Systems with Lyapunov Extreme Instability

A. A. Bondarev

Matematičeskie zametki, Tome 115 (2024) no. 1, pp. 24-42

Résumé

Using specific examples, we constructively show that, in dimensions greater than $1$, the Lyapunov extreme instability of a differential system, i.e., the property that the phase curves of all nonzero solutions starting sufficiently close to zero leave any prescribed compact set, does not imply that these solutions go arbitrarily far away from zero in the sense of Perron or in the upper limit sense as $t\to\infty$. Namely, we construct two Lyapunov extremely unstable systems such that all solutions of the first system tend to zero, while the solutions of the second system are divided into two types: all nonzero solutions starting in the closed unit ball tend to infinity in the norm, and all the other solutions tend to zero. Further, both systems constructed in the paper have zero first approximation along the zero solution.

Détail

Keywords: differential system, Lyapunov stability, Perron stability, upper limit stability, nonlinear system, asymptotic properties of solutions.