# Global stability question for system with a unique locally-asymptotically-stable steady state

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?

• "Additionally, the system is strictly-positively invariant and bounded" can you remind what this means? Apr 28, 2019 at 19:49
• @fedja Yes. For context, I am studying a biological system. So by "strictly-positively invariant", I mean all of the variables are greater than 0 and never becomes negative for all $t > 0$. For "bounded", I mean all variables are bounded by positive constants (<$\infty$) for all $t>0$. If you are interested, the form of the system is presented here: mathoverflow.net/questions/330210/… Apr 28, 2019 at 19:52
• In $3$ dimensions, besides fixed points and limit cycles there can be strange attractors. Apr 28, 2019 at 21:32
• I repeat for the second time my comment I made (and once repeated) below your question Global stability question for system with a unique locally-asymptotically-stable steady state on MSE: What do you understand by a limit cycle in 3D, for example? Some surface? Or (as I proposed in my last comment) a limit cycle on the two-dimensional invariant manifold? Apr 29, 2019 at 8:57
• @user539887 Please correct me if I am wrong, I think as long as "the 2D projection" of the limit cycle surrounds the fixed point, then in some directions, the trajectory would be spiral-like. Apr 29, 2019 at 14:57