ABSTRACT
In this chapter we consider solutions of systems of nonlinear differential equations and the iteration of nonlinear functions of more than one variable. We also consider the phase portraits for both types of nonlinear systems. For nonlinear systems, even the analysis near a fixed point is more complicated. Rather than just using inequalities from the Mean Value Theorem, we must use the more complicated linear theory from the last chapter and nonlinear theorems from differential calculus like the Inverse Function Theorem, Implicit Function Theorem, and the Contraction Mapping Theorem. We are able to prove that if the linearization is hyperbolic, then the nonlinear map or differential equation is topologically conjugate to the linearization. This theorem is simple in one dimension, but requires a proof using the Contraction Mapping Theorem in higher dimensions. We also introduce the nonlinear invariant manifold which is tangent to contracting directions of the linearization, called the stable manifold, and the corresponding invariant manifold which is tangent to the expanding directions, called the unstable manifold. The proof that these manifolds exist is again a nontrivial fact which needs an involved proof.
