Instability, asymptotic trajectories and dimension of the phase space.(English)Zbl 1416.37030

Summary: Suppose the origin $$x=0$$ is a Lyapunov unstable equilibrium position for a flow in $$\mathbb{R}^n$$. Is it true that there always exists a solution $$t\mapsto x(t), x(t)\neq 0$$ asymptotic to the equilibrium: $$x(t)\rightarrow 0$$ as $$t\rightarrow-\infty$$? 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.

MSC:

 37C75 Stability theory for smooth dynamical systems 37C20 Generic properties, structural stability of dynamical systems
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