I would like to build a four-dimensional dynamical system that has the following behavior:
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 $\lambda_{ij} = b_j-\frac{a_{ji}}{a_{ii}}b_i)$ :
- $-2b_1, \lambda_{12}, \lambda_{13}, \lambda_{14}$
- $-2b_2, \lambda_{21}, \lambda_{23}, \lambda_{24}$
- $-2b_3, \lambda_{31}, \lambda_{32}, \lambda_{34}$
- $-2b_4, \lambda_{41}, \lambda_{42}, \lambda_{43}$
From the drawing of what I want (after normalizing everything so $b_i =1$, I believe I should have:
- $\lambda_{41} = \lambda_{14} = 0$
- $\lambda_{12}, \lambda_{23}, \lambda_{31}, \lambda_{34}, \lambda_{42} > 0$
- $\lambda_{21}, \lambda_{32}, \lambda_{13}, \lambda_{43}, \lambda_{24} < 0$
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?