Area-preserving mapping

A mapping is said to be symplectic or canonical if it preserves the differential form dp A dq which defines the symplectic structure in the phase space. Differential forms provide a geometric interpretation of symplectic-ness in terms of conservation of areas which follows from Liouville s theorem [14]. In one-degree-of-freedom example symplecticness is the preservation of oriented area. An example is the harmonic oscillator where the t-flow is just a rigid rotation and the area is preserved. The area-preserving character of the solution operator holds only for Hamiltonian systems. In more then one-degree-of-freedom examples the preservation of area is symplecticness rather than preservation of volume [5]. 

Two-dimensional Area-Preserving Maps Consider a Hamiltonian of the form... 

Since the absolute value of the Jacobian J = a qn+i,Pn+i)/d qn,Pn) = 1, we see that this discrete-time map is indeed area-preserving. 

This is strictly true only for two-dimensional area-preserving maps in dimensions iV > 4, chaotic orbits may leak through KAM surfaces by a process called Arnold diffusion (see [licht83j). 

The key to effective mixing lies in producing stretching and folding, an operation that is referred to in the mathematics literature as a horseshoe map. Horseshoe maps, in turn, imply chaos. The 2D case is the simplest. The equations of motion for a two-dimensional area preserving flow can be written as... 

If the dynamic mapping is also area-preserving (i.e., dxdp = dx dp ), (8.44) is satisfied if we choose... 

We have shown elsewhere that the different bifurcation scenarios can be conveniently discussed in terms of area-preserving mappings generated by the action function [10]... 

What guarantees the area-preserving nature of your classical map... 

The classical dynamics of a system can also be analyzed on the so-caUed Poincare surface of section (PSS). Hamiltonian flow in the entire phase space then reduces to a Poincare map on a surface of section. One important property of the Poincare map is that it is area-preserving for time-independent systems with two DOFs. In such systems Poincare showed that all dynamical information can be inferred from the properties of trajectories when they cross a PSS. For example, if a classical trajectory is restricted to a simple two-dimensional toms, then the associated Poincare map will generate closed KAM curves, an evident result considering the intersection between the toms and the surface of section. If a Poincare map generates highly erratic points on a surface of section, the trajectory under study should be chaotic. The Poincare map has been a powerful tool for understanding chemical reaction dynamics in few-dimensional systems. 

This choice of G X) is designed to mimic some aspects of a molecular process. The classical phase space is two-dimensional. Let (X ,P ) be the position and momentum of the particle just before the rath kick. Then the kicking field induces an area-preserving map... 

To conclude this section we discuss the baker s map (Farmer et al. (1983)) as an example for an area preserving mapping in two dimensions. Area preservation is of utmost importance for Hamiltonian systems, since Liouville s theorem (Landau and Lifechitz (1970), Goldstein (1976)) guarantees the preservation of phase-space volume in the course of the time evolution of a Hamiltonian system. The baker s map is a transformation of the unit square onto itself. It is constructed in the following four steps illustrated in Fig. 2.5. 

Because of the impulsive nature of the <5-kick drive, the time evolution of a point 9, y) over one cycle of the external perturbation can be written in the form of an area-preserving mapping... 

Figure 5.71. Geographical distribution of Januctry average space heating requirements in the year 2050, based on satellite measurements of temperature (on a gross scale of an approximately 50-km grid) in combination with the scenario s assumed building standards. The variations thus reflect both dimate differences across Denmark and differences in heated space per unit cell. The map uses the Mollweide area-preserving projection, in contrast to the straight longitude-latitude co-ordinate system of, e.g.. Fig. 5.70 (Sorensen ei al., 2001).

In contrast, when a = 7 the baker s map is area-preserving. area(/ ( )) = area(7 ). Now the square 5 is mapped onto itself, with no gaps be-... 

This distinction between a < and a = exemplifies a broader theme in nonlinear dynamics. In general, if a map or flow contracts volumes in phase space, it is called dissipative. Dissipative systems commonly arise as models of physical situations involving friction, viscosity, or some other process that dissipates energy. In contrast, area-preserving maps are associated with conservative systems, particularly with the Hamiltonian systems of classical mechanics. 

The distinction is crucial because area-preserving maps cannot have attractors (strange or otherwise). As defined in Section 9.3, an attractor should attract all orbits starting in a sufficiently small open set containing it that requirement is incompatible with area-preservation. 

Area-preserving baker s map) Consider the dynamics of the baker s map in the area-preserving case 0 = 7. 

This mixture of regularity and chaos is typical for area-preserving maps (and for Hamiltonian systems, their continuous-time counterpart). 

Computer project) Explore the area-preserving Henon map (6 = 1). 

Henon, M. (1969) Numerical study of quadratic area-preserving mappings. Quart. 

Moser, J. (1962). On invariant curves of area-preserving mappings of an annulus. Nachr. Akad. Wiss. Gott,. II Math. Phys. Kl 1962, 1-20. 

In order to calculate the ratio 5 between the semi-minor and the semi-major axes of the ellipses surrounding the fixed points of a periodic orbit we consider a two dimensional area preserving mapping and a point [x, y) of a periodic orbit of frequency P/Q. Let us write the Jacobian of the Qth iteration of the map as follows ... 

MacKay, R. S. (1993). Renormalisation in Area Preserving Maps. World Scientific. 

Figure 6.5 Sketch of the area preserving baker s map. The square is squeezed in the horizontal and stretched in the vertical direction, then it is cut in half to cover the original unit square.

Figure 14 Area preservation on the Poincare map. The region bounded by 7, is mapped onto the region bounded by 72 by the area-preserving mapping U such that AI = A2. 2 denotes the energetic periphery.

Another, slightly more technical ( ) way to understand area preservation is to recall that the coordinates of all the points on the Poincare map are specified by coordinates that are canonically conjugate. Because both the initial and final coordinates of a family of trajectories propagated for one mapping are so specified, there must exist a generating function that transforms the coordinates of the initial points into those of the final points. Such a generating function is necessarily a canonical transformation. All canonical transformations preserve the norm of the vectors they transform it can be shown that this property is equivalent to area preservation on the Poincare map. ... 

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