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Can stable heteroclinic cycles (SHCs) exist in 2-dimensional space for the dynamical system given by $\dot{\mathbf{x}} = f(\mathbf{x})$ where $\mathbf{x} \in \mathbb{R}^2$? How can we prove it?

All the examples of SHCs that I've come across are in 3-dimensions or higher (e.g. Guckenheimer-Holmes cycle,Generalized Lotka-Volterra models). According to Rabinovich et al (2010), the condition for the existence of SHCs in $d$ dimensions is:

The eigen values of the Jacobian are ordered according to $\lambda_1 > 0 > Re\{\lambda_2\} > \ldots Re\{\lambda_d\}$ at each fixed point. Further, the ratio $\frac{-Re\{\lambda_2\}}{\lambda_1} > 1$ at all fixed points.

However, I am unable to prove whether or not this condition can be satisfied when $d=2$. Also, I've tried in vain to find any examples of stable heteroclinic cycles in 2-dimensional dynamical systems.

I'm new to dynamical systems and facing difficulty in identifying the relevant literature. I'd be glad if anyone can guide me to the relevant references and/or examples.

Thanks!

srujana311
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  • If you want a particular example of attracting heteroclinic cycle, you can use the method from this answer and apply it to an equation of math pendulum $\ddot{x} = - \sin{x}$. Any other conservative system with a level set containing two saddle equilibria will do okay too. A possible reason why 2D heteroclinic cycles are not often discussed in this context is that they are not structurally stable, while Guckenheimer-Holmes cycle is. – Evgeny Jul 23 '21 at 16:17
  • It worked. Thanks a lot! – srujana311 Jul 31 '21 at 17:37
  • @Evgeny Could you please mention the relevant literature for the concepts you discussed in your paper? Also, is it possible to perturb non-hamiltonian systems to obtain attracting heteroclinic cycles? For example, I'd like to perturb the dynamical system with $\dot{x} = y - y^3, \quad \dot{y} = -x - y^2$. – srujana311 Jul 31 '21 at 17:46
  • You can apply the same idea to a non-Hamiltonian/non-conservative case if you can write the equations for a set that contains separatrices. It's pretty easy to do in Hamiltonian/conservative case hence why I've used it. I think I've picked this trick from Meiss' book "Differential dynamical systems". By the way, for what concepts do you want literature references? – Evgeny Jul 31 '21 at 19:57
  • @Evgeny Thanks! I was referring to perturbing the dynamical system to obtain a globally attracting heteroclinic/homoclinic loop. Could you please elaborate on how to perturb the equations for the non-Hamiltonian case? – srujana311 Aug 03 '21 at 04:40
  • The idea is basically the same. When you have a system $\dot{x} = P(x, y), ; \dot{y} = Q(x, y)$ you can get a rotated system $\dot{x} = P(x, y) - \alpha (x, y) Q(x, y), ; \dot{y} = Q(x, y) + \alpha (x, y) P(x, y)$. Where $\alpha(x, y) = 0$ the vector field is unchanged, and if $\alpha(x, y) \equiv 0$ on some flow invariant set of original system, then the original system and the rotated system share this invariant set. If you know the equations of heteroclinic cycle or homoclinic loop in form $F(x, y) = 0$ you can probably choose $\alpha (x, y) = \pm F(x, y)$. – Evgeny Aug 03 '21 at 16:40

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