V.V. Kozlov and D.V. Treschev, "Instability, asymptotic trajectories and dimension of the phase space", Moscow Mathematical Journal, 18:4, 681--692 (2018).
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.