thank you for your suggestions on expanding upon our current figure/simulations as well as adding more intuition and

We sincerely thank all the reviewers for their meticulous work and their helpful and detailed comments. We will gladly incorporate them! Thank you for this suggestion. For the Stable manifold theorem in line 30, we refer to Chapter 5 Theorem III.7 in "Global Stability of Dynamical Systems" by M. Shub and in line 130, we refer to Section 2.7 in "Differential Equations and Dynamical Systems" by L. "Might be interesting to add a note... Is this correct?" "Does the paper... stuck in a saddle point" The last parts of the statements of Theorem 4.1 and Corollary 4.2 assert "This is of course... that area of study."