# Uses of power series maps

I'm from the field of accelerator physics, specifically related to circular storage rings for synchrotron light sources. High energy electrons circulate around the ring, guided by magnetic fields. The electrons circulate billions of times and one wants to predict the stability. You can describe the motion of the electrons at one point in the ring in terms of phase space (position, momentum space). Each turn around the ring, the particle returns to a new position and momentum, and this defines a map in phase space called the "one-turn map". We may assume that there is a fixed point at the origin, and so it can be expanded in a power series. Thus, one wants to know about the stability of iterated power series maps. There are many hard questions about this, and the topic has an old history. Numerous libraries have been implemented- to implement so called Truncated Power Series Algebra. (See e.g. this paper about zlib by Y. Yan. More background on the physics and one approach to analysis is the normal form approach, e.g. Bazzani et. al. here.) The question is how to use such a library, and how to solve the stability problem. The main approach used in beam dynamics has been normal form analysis, which I don't believe has been successful. I wonder if some kind of spectral methods have been developed in other fields (perhaps along the lines of something like this?). Can someone think of another domain where the long term stability of iterated power series maps with a fixed point at the origin are analyzed, so we might share knowledge or get some fresh ideas? One example I know of is the work of Fishman and "Accelerator Modes" in atomic physics. Are there others? What other systems can be modeled as a kicked rotor, or a Henon map?

• I think it might be helpful to elaborate a little on your terminology. For example, I'm familiar with all of the mathematical concepts you mentioned, but I can't quite visualize what you mean in this context by "a map of phase space." I'm sure in your particular field this requires no explanation, but people from other specialties may realize they actually know how to help you if you do a little bit more explaining what you mean. – Colin K Nov 30 '11 at 1:46
• That's a good point actually: since presumably this site will be bringing together people from many different scientific disciplines, it will be especially important to define field-specific terms (or at least link to explanations). – David Z Nov 30 '11 at 2:00
• Agreed, Collin and David. Thanks for the comments. Phase space is position-momentum space. Think of one position in the ring, and the electron has a transverse position and momentum (velocity). After it goes around the ring one time, it will have a new position and velocity. So its called a one-turn map. If it was linear, it would be like a harmonic oscillator, which traces out an ellipse in phase space. For the case where its circular, the map would have the form x_1 = cos(theta) x_0 + sin(theta) p_0 and p_1 = -sin(theta) x_0 + cos(theta) p_0. Does that clarify? – Boaz Nov 30 '11 at 11:02
• I added a few references to literature in beam physics and computation, and added a short definition of phase space. – Boaz Nov 30 '11 at 12:05
• Incidentally, I asked a similar question on Stack Exchange, Mathematics, here. There I was asking about solutions to the stability question from a mathematical point of view. Here, I was wondering whether the same problem exists in other scientific subjects, since it seems somewhat general, but hasn't been connected to much outside of beam dynamics. One area I know about is accelerator modes in atomic physics. Are there others? – Boaz Nov 30 '11 at 16:56

You probably know this already, but it sounds like something from the world of chaos theory and fractals? (hence it is computationally "difficult")

To your question, have you looked at the world of planetary mechanics and N-body problems? These are also forced to use iterative solutions, and the fundamental underlying physics is N^2, although the force sources are typically allowed to move around as well - just to complicate things further.

It is a long time since I've looked at them, but your mention of phase maps of stability sound very much like Henon Maps. I'm sure these must have wider applications, but they are usually described in terms of planetary stabilities (eg. the stability of a second moon in a planet-moon system).

• Yes, the Henon map is exactly the kind of thing we have in accelerator beam dynamics. The problem with the analogy to the N-body problem is that the space is much larger there. The "phase-space" is 6xN dimensional, whereas for the single particle in a storage ring it is just 6-dimensional in the general case. I'm curious about what other domains end up with something like a Henon-map to model the dynamics. Along the chaos theory route, I thought of looking into population dynamics theory as well. Thanks for the answer. – Boaz Nov 30 '11 at 11:13

You could look into asymptotic behavior of discrete Dynamical Systems. There is both a rich theoretical literature on this topic in mathematics and more applied literature in physics and computer science.

• Thanks Mrocklin. I have looked a bit at the general literature, and not found a solution, or maybe it was too mathematical, and I didn't find the same problem posed in a way I could understand it. – Boaz Nov 30 '11 at 11:08
• Here are some questions from this field: (1) Do you form orbits - i.e. after several iterations do you come back to the same spot? (2) Is your system sensitive to small perturbations - i.e. if we start a state a little bit off of your start state will it end up in a wildly different place? (3) Do some sorts of perturbations act wildly while others are tame? Providing answers to these sorts of questions may provide some insight on the properties of your physical system. – MRocklin Nov 30 '11 at 14:15
• (1) Near the origin, the dynamics are stable and form closed orbits. Going out further, one sometimes finds other islands of stability. And then even further away, the dynamics are unstable, i.e. unbounded. (2) Some aspects are sensitive and some are not. The stable orbits are not so sensitive to any kind of perturbation. (3) The perturbations typically act periodically with some frequency. Some frequencies cause resonances which can dramatically change the dynamics even for small perturbations. But knowing ahead of time which such frequencies are dangerous is not well understood. – Boaz Nov 30 '11 at 16:38

It might be useful to look into Taylor model methods; this seems to be a nice overview article. Try if COSY infinity can do what you want.

• Thanks Erik. Yes, I'm somewhat familiar with COSY infinity. The article you link to looks useful for an overview of the methods of using power series to compute different functions and to find bounds on errors, etc. My question however is about what systems (besides circular storage rings) can be modeled by power series and how one solves for the stability region. I don't think normal form methods can do it, for example. Its been an influential theme in beam dynamics, but I don't see that it has solved the problem. – Boaz Nov 30 '11 at 19:35