# Stability region of a polynomial map

I come from the field of accelerator physics, where we study the dynamics of electrons traversing a circular ring guided by magnetic fields. Suppose one knows the one turn map which takes an electron from one phase space ($x_0$,$p_{x0}$ in one-D) point back to another ($x_1$,$p_{x1}$) and that one can express this map as a power series which terminates at some order $N$:

$x_1 = a_{10} x_0 + a_{01}p_0 + a_{20} x_0^2 + a_{11} x_0 p_0 +\dots + a_{nm}x_0^n p_0^m+\dots$ $p_1 = b_{10} x_0 + b_{01}p_0 + b_{20} x_0^2 + b_{11} x_0 p_0 +\dots + b_{nm}x_0^n p_0^m+\dots$

and $a_{nm}=b_{nm}=0$ for $n+m>N$. A typical case would be

$a_{10}=b_{01}=\cos\theta$, $a_{01}=-b_{10}=\sin\theta$,

for some real angle $\theta$. One may also impose additional requirements (in the case of non-radiating particles) that the map be symplectic, but the case with some damping is important as well. And this problem is the 1-D (2-D phase space) case, and the real problem is 3-D. The question is about the long term stability of this system. Can one formulate the stability region within the $x-p$ plane in terms of this map without actually computing the iterated map for each point? My sense is that a solution to this problem is still unknown to accelerator physicists (e.g. those working on the LHC and those working on 3rd generation light sources), but I wonder if it has been solved by mathematicians. Certainly for special cases, such as the Henon map and related maps much is known, but I don't know if a general solution exists. The method of normal form is the standard approach within these fields, but does not supply a solution. There are also methods for judging chaoticity of an orbit, but these methods are also inadequate from my understadning. For the case with damping (for light sources) the medium term stability (say 5000 terms) is of interest as well. Apologies if the problem is not well enough defined, but I wonder if someone has some knowledge of recent (or old!) solutions, or approaches to analysis.

Perhaps another useful thing to say is that it is assumed that the point $(0,0)$ is a fixed point of the map. So one may phrase this as finding the stability region expressed as a function of the polynomial map, in the vicinity of a fixed point.

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You have used the words "...long term...". What is length in this context? Time? And what is the iteration here? Is it $(x_i,p_i)$? – user13838 Nov 8 '11 at 22:58
The system is iterated, so that if we write $\vec z_i = (x_i,p_i)$ and the map is written $\vec z_{i+1} = M(\vec z_{i})$, then $\vec z_{i+2} = M(M(\vec z_i))$. Time is the number of iterations. So the question is what happens over many iterations- either large, or the limit as the number of iterations goes to infinity. – Boaz Nov 8 '11 at 23:02
I don't have any answer but nonlinear stability tools from control theory might help. I was thinking about the map$$\pmatrix{x[k+1]}=\pmatrix{1\\x[k]\\x^2[k]\\ \vdots}\pmatrix{0 &a_{01} &\ldots &a_{0m}\\a_{10} &a_{11} & &\\\vdots & & &}\pmatrix{1\\p[k]\\p^2[k]\\\vdots}$$. We can define a similar equation for $p[k+1]$. Then, maybe(!), one might come up with a region of attraction argument with Sum-of-Squares techniques. But 5000 seems to much to handle via those methods. I will try to ome up with something. – user13838 Nov 9 '11 at 1:23
Thanks Percusse. I have wondered whether non-linear control theory had solved this problem. In thinking about this problem (which is called the "dynamic aperture" in accelerator physics) I worked with a colleague on a method to linearize it and then use linear algebra to analyze. We made interesting progress, I think, but didn't solve the stability problem. See here for what we tried: osti.gov/bridge/product.biblio.jsp?osti_id=958709 I was curious if this had been done before. – Boaz Nov 9 '11 at 5:42
I couldn't finish reading the paper but looks very familiar, to be honest, to what is being done occasionally in nonlinear systems theory. I was thinking more along the lines of this type arguments. – user13838 Nov 9 '11 at 10:50