A tutorial on the use of StdMap is my article Visual Explorations of Dynamics. A basic reference for the mathematics behind the standard map is my review article (Meiss, 1992). You can also consult a number of textbooks, such as (Meyer and Hall, 1992), (Lichtenberg and Lieberman, 1992), (Arrowsmith and Place, 1990), and (MacKay and Meiss, 1987).
The program is able to find periodic orbits when given “rotation numbers.” To find such an orbit choose the Find -> Periodic Orbit menu. You must then choose a “frequency” (p, q), where p and q are integers in the dialog that pops up. This means that the orbit rotates “p” times around the cylinder in “q” iterations. “Around” for the standard map means that x has increased by one, so a (p,q) orbit satisfies
For the Hénon map, we measure “around” by counting how many times the point rotates around the (elliptic) fixed point.
You can approximate orbits with irrational frequency ω by looking at periodic orbits that limit on that frequency, p/q → ω.
A cantorus is the remnant of an invariant circle. For an integrable, or slightly perturbed integrable mapping (e.g. k small for the standard map), the quasiperiodic orbits with “sufficiently” irrational frequencies cover a circle, forming invariant circles (according to the KAM theorem). When you increase k the invariant circles are destroyed, but there are still quasiperiodic orbits: they just cover a Cantor set instead of a circle.
To find such a cantorus, set k = 1.0 for the standard map, and select the “Farey Path” menu. What then happens is that you are asked to select a pair of orbits. The default is (0,1) and (1,1). These orbits form the head of a Farey tree. Then you are asked to select a Farey path, the default being “LRLRLRLRLR...”. This path leads to the frequency, ω= 1/ γ 2, where γ = (1 + √5)/2, the golden mean. This happens to be the “most irrational” frequency for the standard map, that is, its invariant circle lasts the longest, up to k = 0.97163540631. It is no coincidence that this parameter value is the default one for the standard map.
Any Farey path that never terminates, and doesn’t eventually become all L’s or all R’s, corresponds to an irrational, and thus either an invariant circle or cantorus. Farey paths that are eventually “LLLLL...” or “RRRRR....” correspond to homoclinic orbits to periodic points.
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Each of the maps in the program is reversible. This means that there is an involution S that inverts the map: STS = T-1, where S is an transformation (S2 = I) that reverses orientation. A good example is simply S(x,y) = (-x,y). The implications of this for our purposes are that symmetric periodic orbits can be found by a one-dimensional, instead of a two-dimensional search. In general S will have a fixed set which is a curve (we call it a symmetry line, though it might not be straight). Periodic orbits that have points on the fixed set are called “symmetric.” The program only finds symmetric periodic orbits. Threre is a family of symmetries associated with any symmetry, S = S1. The second member is S2= TS. This is also reversor for T. Additionally, since maps on the cylinder have a translation symmetry R(x,y) = (x+1,y), (so that RT = TR) there are other symmetries S3 = SR, and S4 = TSR. The fixed lines of these four symmetries allow us to find periodic orbits of different kinds. The command Find-›Symmetry Lines... will plot these for lines and their iterates. Check “Inverse Iteration” to plot their inverse images. |
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The curve dialog allows you to sketch curves and iterate them with your map |
Actually, associated with any symmetry S = S1, there is a family of symmetries The first member is S2= TS. This is also reversor for T. Additionally, since maps on the cylinder have a translation symmetry R(x,y) = (x-1,y), there are other symmetries S3 = SR, and S4 = TSR. The fixed lines of these four symmetries allow us to find periodic orbits of different kinds. The command “Symmetry Lines...” will plot these for lines and their iterates. Check “Inverse Iteration” to plot their inverse images.
There are two primary kinds of periodic orbits, stable (elliptic) or minimax orbits and unstable (hyperbolic) minimizing orbits. For more info on this see (Meiss, 1992). These orbits are found by searching for an orbit that starts on different symmetry sets. The program can automatically choose which symmetry line is appropriate for minimizing and minimax orbits, though this doesn’t work for some parameter values: indeed some of the maps do not always have a “dominant” symmetry line since their twist reverses, see (Dullin, Meiss and Sterling 2004).
For the standard map, we choose
The fixed set of S is Fix(S) = {x = 0}. The four fundamental symmetry lines are given by
It turns out that when k > 0 the minimax orbits (they are elliptic for small k) have points on Fix(S). This line is called the “dominant” line. The minimizing orbits have points on Fix(TS) for q even and Fix(SR) for q odd. The dominant line becomes Fix(SR) when k < 0.
Another simplification is that symmetric orbits have points on two of the symmetry lines, and so one can find the orbits by iterating for half the period (q). This is what we do.
For more information on symmetry see Devaney, 1976; Kook and Meiss, 1989; Quispel and Roberts, 1988; Sevryuk, 1986).
Arrowsmith, D. K. and C. M. Place (1990), An Introduction to Dynamical Systems, (Cambridge University Press, Cambridge).
Devaney, R. (1976), “Reversible Diffeomorphisms and Flows,” Trans. Am. Math. Soc. 218: 89-113.
Dullin, H. R., J. D. Meiss, et al. (1991-2006). “Symbolic Codes for Rotational Orbits.” SIAM J. Appl. Dyn. Sys. 4: 515-562.
Kook, H. T. and J. D. Meiss (1989), “Periodic Orbits for Reversible, Symplectic Mappings”, Physica D 35: 65-86.
Lichtenberg, A. J. and M. A. Lieberman (1992). Regular and Chaotic Dynamics. New York, Springer-Verlag.
MacKay, R. S. and J. D. Meiss (1987), Hamiltonian Dynamical Systems: a reprint selection, (Adam-Hilgar Press,London).
Meiss, J. D. (2008), “Visualizing Explorations of Dynamics: The Standard Map”, arXiv Preprint
Meiss, J. D. (1992), “Symplectic Maps, Variational Principles, and Transport,” Rev. Mod. Phys. 64: 795-848.
Meyer, K. R. and G. R. Hall (1992). Introduction to the Theory of Hamiltonian Systems. New York, Springer-Verlag.
Quispel, G. R. W. and J. A. G. Roberts (1988), “Reversible Mappings of the Plane,” Phys. Lett. 132: 161-163.
Sevryuk, M. B. (1986), Reversible Systems, Lecture Notes in Mathematics, (Springer-Verlag, New York).