# Bifurcation in 3 dimensions (simple)

I am Doing a project i have a toy system that describes a bifurcation in 3 dimensions i am posting this in part because i can no longer understand what i have written down ( its been awhile) i have drawn a horrible picture attached below of what the stage's looks like.

Now onto the interesting bit.

Some translation of variables that may help in no particular order as well as some useful information about the system.

(r,0,0) on $\sum_{1}$ corresponds to the homo-clinic orbit. The surface $\sum_{1}$ is defined in such a way that each ode will only intersect this surface once and that it must intersect this surface at least once.

$\nu_{n+1}/\nu_{n} = -e^{-np/w}$

An Aproximation of the flow between the 2 surfaces $\sum_{1} \to \sum_{0}$ is governed by the following differential equations.

$x^{'}=-px+wy$

$y^{'}=-py-wx$

$z^{'}=\lambda z$

$\theta^{'}=w$

$r^{'}=-p$

The basic idea was to fiddle with some parameters $\sum_{0} \to \sum_{1}$ an look at bifurcations.

$\begin{pmatrix} x \\ y\\ \end{pmatrix}^{'}= A \begin{pmatrix} x \\ y\\ \end{pmatrix} + \begin{pmatrix} r \\ 0\\ \end{pmatrix} + \begin{pmatrix} a\nu \\ b\nu\\ \end{pmatrix}$

Where $\nu$ is our bifurcation parameter.

First we wanted to build a mapping matrix A.

Staring by setting out intial conditions we have $\theta_{0}=0$ and $r_{0}=x_{0}$ and $z_{0}=z_{0}$

$z(t)=z_{0} e^{\lambda t}$ $r(t)=x_{0} e^{-p t}$ $\theta(t)=wt$

$z(T)=h$

$T= 1/(\lambda) \log (h/z_{0})$

$x=x_{0}e^{-p \log(h/z_{0})/ \lambda} \cos(w \log(h/z_{0})/ \lambda)$

$y=x_{0}e^{-p \log(h/z_{0})/ \lambda} \sin(w \log(h/z_{0})/ \lambda)$

$x=x_{0}(z_{0}/h)^{p/ \lambda} \cos(w T)$ $y=x_{0}(z_{0}/h)^{p/ \lambda} \sin(w T)$

$z_{new}=F(z_{old})$

$\begin{pmatrix} x \\ z\\ \end{pmatrix}_{new} = C_{1} \begin{pmatrix} a&b \\ c&d\\ \end{pmatrix} \begin{pmatrix} \cos(w T) \\ \sin(w T)\\ \end{pmatrix} + \begin{pmatrix} r \\ 0\\ \end{pmatrix} + \begin{pmatrix} a\nu \\ b\nu\\ \end{pmatrix}$ Where $C_{1}=x_{0}(z_{0}/h)^{\delta}$ and $\delta=(p/\lambda)$

This step im a bit lost on i believe that the K is just some correctional value for the c,d and coupled with the phase shift identity to put this as all one equation which is fine but i need to know exactly what it is to run iterations.

$z_{new} = K z_{0}^{\delta} \cos(wT- \alpha) +b \nu$ i dont think i expanded this right but...

$z_{new}=x_{0}(z_{0}/h)^{p/\lambda} (c(\cos((wlog(h/z_{0})/\lambda)) +d(\sin((wlog(h/z_{0})/\lambda) +b\nu$ but i dont know what c and d are so i moved back to $z_{new} = x_{0}(z_{0}/h)^{p/\lambda} z_{0}^{p/\lambda} \cos(wT- \alpha) +b \nu$ i think i am looking at (p/\lamda) >1 which if we assume from here it was very easy to understand the periodic values when this equaled 0

I have never seen anything in 3d and only one example of a bifurcation in 2-Dimensions but my intuition tells me that the important value of this bifurcation in the way its setup are going to be when $b\nu=0$ i believe without proof of course that this will be where all the fun happens im not sure where i defined it but in case i hadn't but ill look at $b>0$ and where $nu \in R$ Intuition tells me that when $\nu>0$ we will get many fp in fact as this appear to be like a 1 dimensional analog i think we will get 2 bifurcation points generated on certain values of $n\pi/2$ and that each of these will be saddle and node bifurcations. again i also believe that when $\nu<0$ we will have only 1 fp in the system ( anything starting at z=0 will end up at (0,0,0) ) this will be the most boring case when the unstable manifold on the z axis has too much strength and will merely tangent all solutions off to +/- infinity depending on whether u have + or - z values of course. As shown in the picture below.

i believe the homo clinic orbit corresponds to $z_{new}=0$ clearly if we pick $x_{0}=0$ ( $y_{0}=0$ already ) any flow of the solution $z_{new}=x_{0}(z_{0}/h)^{p/\lambda} (c(\cos((wlog(h/z_{0})/\lambda)) +d(\sin((wlog(h/z_{0})/\lambda) +b\nu$ when yields $z_{new}= 0 +b\nu$ and this is true for all initial values $z_{0}$ remove $\lambda=0$ i believe that when $\nu=0$ as well we will have our homo clinic orbit.

i) My question is Can someone help verify that this is whats going on?

ii) help me translate where im at to the point that i can put this mapping into a computer and run some iterations.

I would like to find a stable and unstable fp in the $b>0$ and $\nu>0$ case and verify the homo-clinical orbit and lastly if possible find a closed orbit inside of this region.

I have very little understanding of computers so not only do i need a means of solving for the ode but i would also like to know what software i should put it into that's as simple as possible to use. much of this could be completely wrong my prof explained it to me in about 30 minutes. Any comments or confusion on the notation is welcome but please be as specific and obvious as possible i am physics undergrad and am liable to not understand anything you say at any time.

EDIT:

For those that are looking at this with 2 heads its kind of something like $x^{'}= y$ $y^{'}=-x -(y/3)$ $z^{'}=z - (x^{2}+y^{2})$

This Matrix A would be the linearization of our non linear system and would corresponding to a spiral sink in the x-y plane and an unstable manifold on the z axis you then add some parameter on z so that as z gets large all of a sudden z gets overwhelmed by the spiral sink.

$A=\begin{pmatrix} 0&1&0 \\ -1&-(1/3)&0\\ 0&0&1\\ \end{pmatrix}$

this may attain an orbit like the one described below the one used and explained is toy system with several assumptions to make it easier to work with.

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