Area preserving map

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

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). 

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]... 

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... 

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. 

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

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 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.

Universal Behavior in Families of Area-Preserving Maps. 

This is an area-preserving map. The multipliers of the fixed point 0(x = y = 0) satisfy the relations... 

---