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The Standard STANDARD MAP The standard map also describes the relativistic The standard or Taylor–Chirikov map is a family of cyclotron, and is the equilibrium condition for a area-preserving maps, z = f(z)where z = (x, y) is the chain of masses connected by harmonic springs in original position and z = (x,y) the new position after a periodic potential—the Frenkel–Kontorova model application of the map, which is defined by introduced in 1938 (Meiss, 1992). Similar maps include Chirikov’s separatrix map (valid near the separatrix of k a resonance), the Kepler map (describing the motion x = x + y − ( πx), sin 2 of comets under the influence of Jupiter as well as 2π (1) k a classical hydrogen atom in a microwave field), y = y − sin(2πx). 2π and the Fermi map (for a ball bouncing between oscillating walls) (Lichtenberg & Lieberman, 1992). Here x is a periodic configuration variable (usually The higher-dimensional version is the Froeshlé map computed modulo 1), y ∈ R is the momentum variable, (See Symplectic maps). and the parameter k represents the strength of a nonlinear kick. This map was first proposed in 1968 by Symmetries Bryan Taylor and then independently obtained by Boris Chirikov to describe the dynamics of magnetic field The standard map f has a number of symmetries that lines. The standard map and Hénon’s area-preserving lead to special dynamical behavior. To see these, it is quadratic map are extensively studied paradigms for convenient to lift the map from the cylinder to the plane chaotic Hamiltonian dynamics. by extending the angle variable x to R. The standard map is an “exact symplectic” map of Let Tm,n(x, y) = (x + m, y + n) be the translation the cylinder. Because x(x, y) is a monotone func- by an integer vector (m, n).Asf is periodic, its lift has f ◦ T = T ◦ f tion of y for each x, it is also an example of a discrete translation symmetry m,0 m,0 . a monotone twist map (See Aubry–Mather the- More unusually, the standard map also has a discrete f ◦T = T ◦f(x,y) ory). Every twist map has a Lagrangian generat- vertical translation symmetry 0,n n,n . ing function, and the standard map is generated Identifying orbits equivalent under these symmetries by F(x,x) = 1 (x − x)2 + (k/4π 2) cos(2πx), so that implies that standard map can be thought of as acting 2 T ={− 1 ≤ x,y < 1 } y =−∂F/∂x and y = ∂F/∂x . The map can also be on the torus 2 2 . obtained from a discrete Lagrangian variational princi- The standard map also commutes with the reflection ple as follows. Define the discrete action for any config- S(x,y)= ( − x, − y). This can be used to identify uration sequence ...,xt−1,xt ,xt+1,... as the formal the lower half plane with the upper one, and sum to restrict the map to the space S ={(x, y) : − 1 ≤ x<1 , ≤ y ≤ 1 } ( − 1 ,y)≡ ( 1 ,y) 2 2 0 2 identifying 2 2 and each half of the upper and lower boundaries: A[...,xt− ,xt ,xt+ ,...]= F(xt ,xt+ ). (2) 1 1 1 (x, ) ≡ ( − x, ) (x, 1 ) ≡ ( − x, 1 ) t 0 0 , 2 2 . The map on the S ( ± 1 , ) two sphere is singular at the corners 2 0 and Then an orbit is a sequence that is a critical point of A; ( ± 1 , 1 ) 2 2 . this gives the discrete Euler–Lagrange equation The standard map is also reversible: it is conjugate to its inverse Rf R − 1 = f − 1 (Lamb & Roberts, 1998). k One reversor is R1(x, y) = ( − x,y − (k/2π)sin(2πx)); xt+1 − 2xt + xt−1 =− sin(2πxt ). (3) n 2π this generates a family of reversors R = f ◦ R1. These reversors are involutions, R2 = id, thus f can be written This second difference equation is equivalent to (1) as the composition of two involutions f = (f ◦ R)◦ R. upon defining yt = xt − xt−1. Finally, the composition of a symmetry and a reversor The standard map is an exact or approximate de- is also a reversor, so that, for example R2 = SR is also scription of many physical systems, including the a reversor. “kicked rotor.” Consider a rigid body with mo- Symmetric orbits are invariant under a symmetry ment of inertia I that is free to rotate in a hor- or a reversor. This is particularly interesting since izontal plane about its center of mass. Suppose symmetric orbits must have points on the fixed sets that an impulsive torque (θ)=−A sin(θ) is ap- of the reversor, Fix(R) ={z : z = R(z)} or on Fix(f R). plied to the rotor at times nT , n ∈ Z. Let (θj ,Lj ) Because these fixed sets are curves, symmetric orbits be angular position and angular momentum at time are particularly easy to find. Rimmer showed that jT − ε for ε → 0 + . At time T later these become the bifurcations of symmetric orbits are special; for (θj+1,Lj+1) = (θj + (T /I)Lj+1,Lj + (θj )). Scal- example, they undergo pitchfork bifurcations (See ing variables appropriately gives (1). Bifurcations). 1 STANDARD MAP Dynamics manifolds. For each minimizing m/n orbit, these When k = 0, the dynamics of the standard map intersect and enclose the minimax orbit, forming an are integrable: the momentum y is an invariant. island chain or resonance. The intersection of the On each invariant circle C0 ={(x, y) : y = ω}, the manifolds is transverse, though the angle between them ω is exponentially small in k (Gelfreich & Lazutkin, angle after t iterates is given by xt = x0 + ωt mod 1, thus the dynamics is that of the constant rotation, 2001). A number of island chains are easily visible Rω(θ) = θ + ω, on the circle with rotation number ω. 0 in computer simulations. In the color figure (See When ω is rational, every orbit on Cω is periodic; otherwise, they are quasi-periodic and densely cover Standard map: Figure 1 in color plate section), we show a number of orbits of the standard map for k = 0.6 the circle. T When |k|1, Moser’s version of the KAM theorem on the torus . In the figure, each of the blue curves implies that most of these invariant circles persist; is formed from many iterates on a rotational invariant circle like those predicted by the KAM theorem. The that is, there is a rotational invariant circle Cω on green orbits are secondary and tertiary circles arising which the dynamics is conjugate to the rotation Rω (See Hamiltonian dynamics). KAM theory applies from resonances. to circles with Diophantine rotation number, that is, When stable and unstable manifolds intersect ω ∈{ :|n − m| >c/nτ ;∀m, n ∈ Z,n =/, 0} for transversely, some iterate of the map has a Smale some τ ≥ 1 and c>0. This excludes, of course, all of the horseshoe. This implies that there is, at least, a cantor set rational rotation numbers as well as intervals about each of chaotic orbits. Umberger & Farmer (1985) showed rational, but still leaves a positive measure set. While it numerically that there is a fat fractal set on which is difficult to obtain reasonable estimates for the interval the dynamics has a positive Lyapunov exponent. The of k for which all Diophantine circles (with given c and proof of this statement is still illusive. The regions τ occupied by chaotic orbits appear to grow in measure as ) persist, in 1985 Herman showed analytically that k there is at least one invariant circle when |k|≤0.029, increases. Numerically, it appears that a single initial and de la Llave & Rana (1990) used a computer-assisted condition densely covers each “zone of instability” proof to extend this result up to 0.91. a chaotic zone bounded by invariant circles. Chaotic Some of the periodic orbits on the rational circles trajectories that were only slightly visible in the C0 also persist for nonzero k. Indeed, the Poincaré– previous color figure (gold orbits near the stable and m/n unstable manifolds of the resonances) dominate the Birkhoff theorem implies that there are at least two dynamics when k = 2.0(See Standard map: Figure 2 period n orbits (with positive and negative Poincaré in the color plate section). At this value of k there are no indices, respectively). Aubry–Mather theory implies rotational invariant circles. In the figure, the gold region that orbits with rotation number m/n can be found is filled by a single trajectory with 1.5(10)6 iterates. It variationally; one is a global minimum of the action appears to densely cover most of phase space, though (2), and the other is a minimax point (a saddle of there are still a number of secondary and tertiary islands A with one downward direction). For example, when visible. k>0, ( 1 , 0) is a minimizing fixed point, and (0, 0) is a 2 There are also many elliptic periodic orbits that minimax fixed point. The reversibility of the standard are created for nonzero k. For example, the (0, 0) map implies that there must be symmetric periodic fixed point undergoes a period-doubling bifurcation orbits for each ω = m/n as well. Indeed, it is observed at k = 4, creating a period two orbit. More generally, that the minimax periodic orbits always have a point on when the eigenvalues of any elliptic period-n orbit are the line Fix(R) ={y = 0}, the “dominant” symmetry ± π ω λ ± = e 2 i then new orbits are born that encircle line.
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