I have an ordinary differential system of dimension 3 that contains a locally-asymptotically-stable unique fixed point. Additionally, the system is strictly-positively invariant and 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 nodal sink. This seems to rule out the possibility of a limit cycle.

Because I think if a limit cycle exists in 3 and higher dimensions; while it does not have to be a surface-like completely surrounding the fixed point, it will surround the fixed point in some directions. This would lead to the trajectories in the respective direction being spiral-like, which is excluded by the strictly real eigenvalues.

If my understanding and reasoning are correct, can I then conclude that the fixed point is globally-asymptotically-stable? If so, please give me some reference on this, perhaps a theorem. If not, could you please suggest some alternatives to address the global stability of such a situation?

limit cyclein 3D, for example? Some surface? Or (as I proposed in my last comment) a limit cycle on the two-dimensional invariant manifold? $\endgroup$ – user539887 Apr 29 at 8:57