Moscow Mathematical Journal

Volume 18, Issue 4, October–December 2018 pp. 681–692.

Instability, Asymptotic Trajectories and Dimension of the Phase Space

Authors: V.V. Kozlov (1) and D.V. Treschev (2)
Author institution:(1) Steklov Mathematics Institute, 8 Gubkina street, 11991, Moscow, Russia
(2) Steklov Mathematics Institute, 8 Gubkina street, 11991, Moscow, Russia and Lomonosov Moscow State University

Summary:

Suppose the origin x=0 is a Lyapunov unstable equilibrium position for a flow in ℝⁿ. Is it true that there always exists a solution t↦x(t), x(t)≠0 asymptotic to the equilibrium: x(t)→0 as t→−∞? The answer to this and similar questions depends on some details including the parity of n and the class of smoothness of the system. We give partial answers to such questions and present some conjectures.

2010 Math. Subj. Class. 37B25, 58F10, 70H14.

Keywords: Laypunov stability, asymtotic trajectories, quasihomogeneous systems.

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