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In my previous post (i.e., Limit of a Discrete Dynamical System) the following system was considered:

$$\left[\begin{array}{c}x_{t+1}\\y_{t+1}\end{array}\right]=\left[\begin{array}{c}y_{t}/b\\x_{t}-1+\left(\frac{a}{b^2}\right)y_t^2\end{array}\right]\;\;\;\text{with}\;\;\;\left[\begin{array}{c}x_{0}\\y_{0}\end{array}\right]=\left[\begin{array}{c}0\\0\end{array}\right]$$

and where $a=-0.5$ and $b=-1.00030$.

By this system the kind people out there showed me how the fixed point is derived, how the Jacobian fits into the analysis, and how eigenvalues and eigenvectors are used to evaluate the dynamic of the fixed point. In this follow on post the same sort of system is considered except that the value of the parameter $b$ is changed to:

$$b=-1.00050$$

In this case, the following graphic suggests that the system settles, in an asymptotic sense, to a periodic orbit of five distinct points. My question is: Is there an analytic way to derive these five points?

Dynamic system with asymptotic attracting periodic orbit

So, if I mimic the analysis that was done in the previous post, by pretending that there are fixed points for this system (which, in this case, is where $a=-0.50$ and $b=-1.00050$), I get what I shall refer to as fake fixed points (ffp). It turns out that the fake fixed points are very similar (i.e. close to) the fixed points of the system that was considered in the first post. I calculated just the two $y$ values:

$$y1=-0.585872209\;\;\text{and}\;\;y2=-3.417128291$$

Now let $\left(x_{ffp},y_{ffp}\right)$ be the fake fixed point that corresponds to the case where $y=y1=-0.585872209$. This choice was made because the graphic suggests a centering of the system's dynamic on that fake fixed point. The Jacobian of this system is:

$$J\!\left(x,y\right)=\left[\begin{array}{cc}0&\frac{1}{b}\\1&2\left(\frac{a}{b^2}\right)y\end{array}\right]\;\;\;\text{where}$$

$$\text{det}\;J\!\left(x,y\right)=-\frac{1}{b}=0.99950025\;\;\forall{\left(x,y\right)}\in\text{R}\!\!\times\!\!\text{R}$$

Since the determinent is non-zero, an expression for eigenvalues are found:

$$\text{det}\left(J\!\left(x,y\right)-\lambda I\right)=0\;\;\text{implies that}$$

$$\lambda_{1,2}=\frac{1}{b}\left[\left(\frac{a}{b}\right)y\pm\sqrt{\left(\frac{a}{b}\right)^2y^2+b}\;\;\right]$$

where it is noted that the eigenvalues depend on $y$. Since the limit points are cyclic, we are interested in the eigenvalues where:

$$\left(\frac{a}{b}\right)^2y^2+b<0\;\;\text{which implies that}\;\;y<\bigg|\!\!\frac{b}{a}\!\!\bigg|\sqrt{|b|}.\;\;\text{Therefore,}$$

$$y<2.001500187$$

The graphic suggests that the system does not, in the limit, converge to a single fixed point -- and, neither does it diverge away from the fake fixed point $\left(x_{ffp},y_{ffp}\right)$. Therefore, if we choose $y=0$, then 1.) the real part of $\lambda_{1,2}$ will be zero, 2.) $y$ will be less than $2.001500187$, and 3.) the resulting values of the eigenvalues $\lambda_{1,2}$ will result in a system that forever orbits around $\left(x_{ffp},y_{ffp}\right)$ within a bounded region of $\text{R}\!\!\times\!\!\text{R}$. So, when $y=0$ then:

$$\lambda_{1,2}=\pm\left(\frac{1}{\sqrt{|b|}}\right)i=\pm(0.999750094)i$$

So what does this mean? In my opinion, nothing much. I could see the asymptotic limit cycle nature of the system by just looking at the graphic of the orbit. I am no closer to answering the question of analytically deriving the five cyclic limit points of this system than when I started out at the begining of this post. (However, it was fun going through the calculations.) Does anyone out there have any ideas on how to derive these five cyclic limit points analytically? Thanks.

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  • $\begingroup$ Very nice question, +1! $\endgroup$ – Robert Lewis Sep 10 '14 at 5:33
  • $\begingroup$ Maybe looking at $(x_{t+5}, y_{t+5})$ in terms of $(x_t,y_t)$ might help? :-) $\endgroup$ – copper.hat Sep 10 '14 at 6:19

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