binzhang

Please accept my deepest respect and sincere gratitude for taking the time to enlighten me. Your explanation has been truly eye-opening. I now realize that my previous understanding was confined to low-dimensional systems. To summarize, in higher-dimensional systems, the eigenvalues from other dimensions can influence the stability of equilibrium points, resulting in scenarios where local equilibria undergo fold bifurcations without any actual change in their stability. Once again, thank you very...

Dear experts, In my study of a 3D system, while plotting the one-parameter bifurcation diagram, I found that the equilibrium points on both sides of the fold bifurcation (FB) are unstable (and the fold bifurcation point is non-degenerate), which contradicts the definition of a fold bifurcation. However, MatCont's numerical window explicitly shows that the eigenvalues of the equilibrium points on both sides have at least one positive real part (i.e., both are unstable). I also verified this using...

Perhaps you could examine the one-parameter bifurcation diagrams on other planes or visualize the three-dimensional structure of the bifurcation diagram. Since your system is three-dimensional, the disappearance of the limit cycle might be caused by a homoclinic bifurcation occurring when it collides with the equilibrium curve on another plane.

Perhaps you could examine the one-parameter bifurcation diagrams on other planes or visualize the three-dimensional structure of the bifurcation diagram. Since your system is three-dimensional, the disappearance of the limit cycle might be caused by a homoclinic bifurcation occurring when it collides with the equilibrium curve on another plane.