The stability of an area-preserving mapping

Abstract

In many cases, the evolution of a Hamiltonian system can be represented by an area-preserving mapping of the plane onto itself. The stability or instability of the dynamical system is reflected in the derived mapping. The mappingT(x, y)=(x', y'):

$$\begin{gathered} x\prime = x + a(y - y^3 ) \hfill \\ y\prime = y - a(x\prime - x\prime ^3 ) \hfill \\ \end{gathered} $$

was studied in order to determine methods of readily compartmentalizing the plane into regions of stable and unstable behavior under many applications ofT, without resorting to costly and frequently inaccurate methods requiring computation of thousands of maps.

The concept of separatrices, which exist for perfectly integrable systems, is replaced by the more general idea of extended eigenvectors from hyperbolic fixed points of the mappings, where the eigenvectors are those of the matrix representing the mapping linearized in the neighborhood of the fixed point. It was demonstrated by Bartlett that these extended eigenvectors, or eigencurves, from neighboring hyperbolic fixed points, may intersect each other to form intricate networks of intersecting loops. This area will be stable if the ratio of loop area to cell area is very small, of the order of 10⁻⁵ for the above mapping. Generally, if the oscillation cannot be seen, one should act as if the entire area of the cell is stable.