# 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 ℝ^{n}. 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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