Linear stability analysis fails to determine the local stability property of a non-hyperbolic equilibrium point as there is a emergence of a centre subspace (other than stable and unstable subspace) of the linearized system corresponding to the eigenvalue whose real part is zero. Centre manifold of the corresponding nonlinear system may not be unique. So, what is the exact procedure to analyze such kind of situation?
The short answer is that there is no "exact" procedure for analyzing non-hyperbolic equilibrium points. However, there are some work using blowing up/down methods. You may look at this methods in a differential equation literature. You can check the way we used in one of our paper. "Bifurcations and global dynamics in a predator–prey model with a strong Allee effect on the prey, and a ratio-dependent functional response." P Aguirre, JD Flores, E González-Olivares - Nonlinear Analysis: Real World Applications, 2014. In this paper we analyzed the stability of the origin using blow up/down techniques.
You are welcome! We have used that technique basically to de-singularize the origin in some of our models.
Okay sir. I will read your article carefully and try to understand the technique. Thanks for your assistance.
I follow this procedure in this article no hyperbolic equilibra in the three charged body problem , blow up is other good possibility.
Here is a couple of references where there is a comprehensive analysis of the non-hyperbolic singularity.
- F. Berezovskaya, G. Karev, R. Arditi, Parametric analysis of the ratio-dependent predator–prey model, J. Math. Biol. 43 (2001) 221–246.
- Claudio Arancibia-Ibarra, José D. Flores & Peter van Heijster. "Stability analysis of a modified Leslie-Gower predation model with weak Allee effect in the prey." Frontiers in Applied Mathematics and Statistics: Dynamical Systems. January 2022.
- Claudio Arancibia-Ibarra, Pablo Aguirre, José Flores & Peter van Heijster. "Bifurcation Analysis of a predator-prey model with intra-specific interactions and ratio-dependent functional response." Applied Mathematics and Computation. Vol. 402. March 2021.
Use a Lyapunov functin or study the Hopf bifurcation. Also, you can apply the center manifold theory to study it.
the non-hyperbolic singular points may be nilpotent or linearly singular points where
The Nilpotent singular point is when all eigenvalues are equal to zero and the Jacobian matrix is not equal to 0. - The Linearly zero singular point is when all eigenvalues are equal to zero and Jacobian matrix is equal to 0.
To analyze the stabilty of these equilibrium points in the plane, you use blow ups techniques for linearly zero singular points, whereas for the nilpotent singular points you use theorem 3.5 of Freddy Dumortier, Jaume Llibre, Joan C. Artés "Qualitative Theory of Planar Differential Systems".
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