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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.

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