In the first three chapters we presented nonlinear theory by way of examples.
We now present the theory in a more
general setting. Many of the fundamental ideas have already been illustrated
in the one-dimensional setting. For example, in one-dimension we found that
the stability of a fixed point is determined by the derivative at the
fixed point. The same result holds in higher dimensions. However, the actual
computational machinery needed is far more intricate because the derivative
of an n-dimensional map is an 
matrix.
Another key idea we have already introduced is hyperbolicity, hyperbolic sets, and symbolic analysis (see section 2.11). In this chapter we present a detailed study of the Smale horseshoe, which is the canonical example of a ``hyperbolic chaotic invariant set.'' A thorough understanding of this example is essential to the analysis of a chaotic repeller or attractor. We conclude this chapter with a discussion of sensitive dependence on initial conditions. This brings us back to the Lyapunov exponent, which we define for a general n-dimensional dynamical system.
