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I would like to build a four-dimensional dynamical system that has the following behavior:

enter image description here

Here, $x_1, x_2, x_3$ and $x_4$ are the four dimensions, and each axis has a fixed point that should be a saddle. Thus, trajectories spiral towards either than $1-2-3$ plane or the $2-3-4$ plane, and cycle near each of the fixed points. Trajectories in the $1-2-4$ plane should go to fixed point 2. I'm venturing a guess that such a system can be described by:

$$dx_i = x_i\left(b_i - \sum_{j=1}^{n}a_{ij}x_j^2 \right) dt$$

or in matrix form: $$dx = x(b-x^TAx)dt$$. Here, $b_i > 0$, and $a_{ij}>0$ I've linearized around each fixed point and have the following 4 eigenvalues for each fixed point (denote $\lambda_{ij} = b_j-\frac{a_{ji}}{a_{ii}}b_i)$ :

  1. $-2b_1, \lambda_{12}, \lambda_{13}, \lambda_{14}$
  2. $-2b_2, \lambda_{21}, \lambda_{23}, \lambda_{24}$
  3. $-2b_3, \lambda_{31}, \lambda_{32}, \lambda_{34}$
  4. $-2b_4, \lambda_{41}, \lambda_{42}, \lambda_{43}$

From the drawing of what I want (after normalizing everything so $b_i =1$, I believe I should have:

  1. $\lambda_{41} = \lambda_{14} = 0$
  2. $\lambda_{12}, \lambda_{23}, \lambda_{31}, \lambda_{34}, \lambda_{42} > 0$
  3. $\lambda_{21}, \lambda_{32}, \lambda_{13}, \lambda_{43}, \lambda_{24} < 0$

Let's define $\alpha > 1$ and $0 < \beta < 1$. First I tried using these parameters:

$$ A \equiv \left(\begin{array}{rr} \,1 & \alpha & \beta & 1 \\ \, \beta & 1 & \alpha & \beta \\ \, \alpha & \beta & 1 & \alpha \\ \, 1 & \alpha & \beta & 1 \end{array}\right) $$

The parameters satisfy points $1-3$, but due to the symmetry of $x_1$ and $x_4$, I don't get this cycling in the two planes that I want. So then I tried this:

$$ A \equiv \left(\begin{array}{rr} \,1 & \alpha & \beta & 1 \\ \, \beta & 1 & \alpha & \beta \\ \, \alpha & \beta & 1 & \alpha \\ \, 1 & \alpha+0.1 & \beta & 1 \end{array}\right) $$

Doing this, I found that all the trajectories converge onto the $1-2-3$ plane and nothing converges onto the $2-3-4$. If I do this:

$$ A \equiv \left(\begin{array}{rr} \,1 & \alpha & \beta & 1 \\ \, \beta & 1 & \alpha & \beta \\ \, \alpha & \beta & 1 & \alpha \\ \, 1 & \alpha-0.1 & \beta & 1 \end{array}\right) $$

In all these examples, I took $\alpha = 1.2$ and $\beta = 0.9$. Now everything converges onto the $2-3-4$ plane and nothing converges onto the $1-2-3$ plane. Thus, I'm having a hard time finding parameters so that trajectories, depending on their initial condition, will converge to the either the $1-2-3$ cycle or the $2-3-4$ cycle, but not where every initial condition leads to only one of the cycles. I'm also not sure why that extra $+0.1$ or the $-0.1$ is causing one of the cycles to be favored over the other -- based on the eigenvalues, that extra $+0.1$ or $-0.1$ shouldn't change the sign of the eigenvalue $(\lambda_{42})$ so I would think the behavior should stay the same, not switch which cycle trajectories go to. Can anyone help come up with parameters that leads to the behavior above (if possible), and explain why the extra $0.1$ is making such a difference?

Brenton
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  • Just for the remark: if you want both heteroclinic cycles to be asymptotically stable in their respective subspaces ($x_4 = 0$ and $x_1 = 0$), you should check saddle values of saddles and the product of these values for saddles in cycle. I haven't seen the treatment of this condition here and I don't remember whether Kirk and Silber asked this cycles to be asymptotically stable in this subspaces (I think they did, but I have to take a look at article again). – Evgeny Aug 02 '15 at 06:32
  • @Evgeny Ah yes I believe I know what you're referring to. I believe that has to do with whether or not the perimeter of the cycle is attracting or repelling. Hmmm, maybe I need to find parameters so both cycles are attracting? – Brenton Aug 02 '15 at 21:41
  • Regarding my remark. Quote from Kirk&Silber (page 4): "Each of the cycles in the network exists in a three-dimensional subspace of $\mathbb{R}^4$; in the subspace the cycle is a generalization of one proposed by dos Reis (Structural stability of equivariant vector fields on two-manifolds) (see also Guckenheimer and Holmes, Structurally stable heteroclinic cycles)". Since they're referring to GH-cycle, I think they ask their subcycles to be asymptotically stable in respective subspaces. – Evgeny Aug 03 '15 at 08:04
  • @Evgeny Something seems off. I don't see anyways to have a cycle in each plane. In the picture above, there's no way for the $1-2-4$ plane to have a cycle at all. If I reverse the $2-4$ direction, then the $2-3-4$ cycle disappears. So I don't see how they got a cycle in each subspace – Brenton Aug 03 '15 at 17:56
  • Sorry, I misleadead you a little bit :( I thought that you're still realizing Kirk-Silber network as it was written in their article. Now I see that it's compeletely not the same. Sorry again... – Evgeny Aug 03 '15 at 18:14
  • @Evgeny I think I am still trying to realize the Kirk-Silber network. In their figure 1 schematic, it looks like they have 2 cycles: $1-2-3$ and $1-2-4$. I just don't see a dynamical system that gives this – Brenton Aug 03 '15 at 18:18
  • @Evgeny This is actually confusing to me in the paper: "Because the cycles have a common heteroclinic connection, it is not possible for both cycles to attract nearly all nearby trajectories, but we find that any of the other combinations of (i). (ii) and (iii) can occur for the pair." Does this mean one cycle always dominates over the other at attracting trajectories? – Brenton Aug 03 '15 at 18:46
  • Yep, it seems to be so. I'm afraid that comments on MSE (and even posts) won't be enough to handle this discussion soon :) – Evgeny Aug 03 '15 at 19:48

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