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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 $(x_i,p_i)$?2011-11-08
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    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.2011-11-08
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    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.2011-11-09
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    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: http://www.osti.gov/bridge/product.biblio.jsp?osti_id=958709 I was curious if this had been done before.2011-11-09
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    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](http://www.cds.caltech.edu/~utopcu/images//e/e0/TP-ITAC-2009.pdf).2011-11-09
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    Thanks for the comment. I will have a look at your reference.2011-11-09
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    Its helpful to learn the language of another field, so I can ask the right questions. e.g. Lyapunov function, sum of squares method, region of attraction, etc. One difference that I see is that the problem is phrased in terms of differential equations, instead of iterated maps, but I realize that often similar methods may apply to both ways of formulating the dynamical system. For storage rings, we mainly have a map (there are local differential equations, with periodic coefficients, but this may be summarized by a one turn map.)2011-11-09
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    Ah of course, I forgot to mention that the difference equations are handled under the phrase *discrete time systems* . Most of the notions follow mutatis mutandis for discrete time systems.2011-11-09
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    without a specific form for your maps, not much can be said. For instance, there are polynomial maps with no region of bounded stability (on an open dense set of full measure, every orbit escapes to infinity, and otherwise the orbits are chaotic).2017-05-19
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    Interesting, thanks @Glougloubarbaki. I do assume a stable fixed point, i.e. the linearized map about the fixed point is stable. Does your example apply in this case? My vague understanding of the KAM theorem is that it says that there should still be some region of stability, even when the higher order terms are added.2017-05-22
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    I did not see that. If you have a stable fixed point (stable in the sense that $|P'(x)|<1$ and $P(x)=x$), then yes, you have an open basin of attraction for that fixed point. however there is almost always no simple description of it (its boundary will typically be a fractal known as the Julia set of $P$.2017-05-22
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    It is possible however to describe the behaviour of nearby points near $x$ under iteration depending on the value of $P(x)$2017-05-22
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    Thanks @Glougloubarbaki . So, "dynamic aperture" in accelerator physics is the same as "Julia set". It may not be simple to describe, but perhaps there may be fast ways to calculate it or control its size?2017-05-23
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    Here's a reference where the Henon map is studied by an accelerator physicist. http://inspirehep.net/record/336261?ln=zh_CN M. Giovannozi, “Analysis of the stability domain for the Henon Map” CERN SL /92-23 It would be interest to know there if there is a substantial body of literature on this from the dynamical systems or chaotic dynamics perspective.2017-05-23

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