Can Phase Trajectories Intersect with Equilibria in Dynamical Systems?

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Phase trajectories in dynamical systems generally do not intersect in any dimension of phase space. Near equilibrium points, trajectories form closed loops, indicating stability, while the trajectory at equilibrium is simply a point. The discussion raises questions about the nature of equilibrium, particularly regarding spiral points that suggest more complex behavior. There is uncertainty about whether equilibrium points should be considered isolated. Overall, the relationship between phase trajectories and equilibria remains a complex topic in dynamical systems.
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Generally speaking, phase trajectories can not intersect in any dimension phase space.
But how about rest points. can trajectories truly entry equilibria.

THX!
 
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The phase trajectories close to equilibrium points are little circles around them. They don't intersect each other. The phase trajectory of a system in equilibrium is just a point.
 
but it sounds strange that no trajectory intersect with equilibrium.
there are some other kinds of equilibrium points, like spiral point, which seems not just circles around point.
Is that finally attach equilibrium, since i cannt find any good reason that why equilibrium should be isolate?
 

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