Can Phase Trajectories Intersect with Equilibria in Dynamical Systems?

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SUMMARY

Phase trajectories in dynamical systems do not intersect in any dimension of phase space. Equilibrium points, or rest points, are represented as singular points where trajectories converge but do not cross. Near these equilibrium points, trajectories form closed loops, such as circles around stable equilibria. The discussion raises questions about the nature of different types of equilibrium points, including spiral points, and challenges the notion that equilibria must remain isolated.

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  • Knowledge of equilibrium points and their classifications
  • Basic grasp of stability analysis in systems
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Students and researchers in mathematics, physics, and engineering focusing on dynamical systems, as well as anyone interested in the behavior of trajectories in relation to equilibrium points.

enricfemi
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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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