ABSTRACT
This chapter initiates the study of dynamical systems in more than one dimension. In the one-dimensional case it is usually convenient to restrict attention to maps of an interval; for typical two-dimensional systems. For one-dimensional maps the Lyapunov exponent is defined by tracking the image of an interval of negligible length. The geometry of that situation is simple; using linear approximation a small interval maps to another. For two-dimensional maps one could know that expansion and contraction is non-uniform; for this reason it is necessary to track a small ellipse. The chapter shows that, to linear approximation, the image of a small ellipse is another, although the lengths of the axes and their orientation are changed. Two methods are available to demonstrate this fact; elementary calculation and linear algebra. An important special class of two-dimensional dynamical systems is obtained by treating the vector components as real and imaginary parts of a complex variable.
