I have an ordinary differential system of dimension larger than 2 that contains a locally-asymptotically-stable unique fixed point. Additionally, the system is bounded.

Now, suppose that I can show the eigenvalues of the linearized system about the fixed point are strictly real and negative. This would mean near the equilibrium, the system is a sink. This seems to rule out the possibility of a limit cycle. Because if there is a limit cycle, it must be surrounding the fixed point and this would make the trajectory toward the fixed point spiral-like.

As a comment pointed out: if a limit cycle exists in 3 and higher dimensions, it does not have to be a surface-like completely surrounding the fixed point. However, it will surround the fixed point in some direction. This would lead to the trajectories in the respective direction being spiral-like, which is excluded by the strictly real eigenvalues.

If my reasoning is correct, can I then conclude that the fixed point is globally-asymptotically-stable?

  • 3
    $\begingroup$ What is a limit cycle in an autonomous 3D (for example) system of ODEs? How can it surround the equilibrium? $\endgroup$
    – user539887
    Apr 27, 2019 at 6:40
  • $\begingroup$ @user539887 I think I know what you mean. Although, if the eigenvalues are strictly positive, from any direction, the trajectory toward the fixed point would be non-spiral. The limit cycle does not have to "completely surround" the fixed point. If the limit cycle surrounds some part of the fixed point, then from the respective direction, the trajectory would be spiral-like. $\endgroup$
    – Paichu
    Apr 28, 2019 at 18:22
  • $\begingroup$ If all the eigenvalues have positive real parts then there is no trajectory tending toward the equilibrium (in positive time), whether in a spiraling way or not. Again, I repeat: what is the limit cycle, and how can it surround (in particular, incompletely) the equilibrium? Perhaps you mean there is a two-dimensional invariant manifold on which a limit cycle surrounds the equilibrium? $\endgroup$
    – user539887
    Apr 29, 2019 at 6:14
  • $\begingroup$ @user539887 I apologize. I mean the eigenvalues are strictly real and negative. And for the limit cycle, please correct me if I am wrong, I am thinking as long as "the 2D projection" of the limit cycle surrounds the fixed point, then in some directions, the trajectory would be spiral-like. $\endgroup$
    – Paichu
    Apr 29, 2019 at 14:56
  • $\begingroup$ While being a node indeed prevents trajectories from spiraling around equilibrium, it rules out limit cycles only in some vicinity of an equilibrium. Nothing prevents spiraling of trajectories near the limit cycle that is far from equilibrium and going to a node after that. $\endgroup$
    – Evgeny
    Apr 29, 2019 at 19:39


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