The Kolmogorov-Arnold-Moser (KAM) theorem presents limits to the notion of chaos. This is the deepest result to date on this subject. This chapter starts with a review of Newton's method for solving non-linear equations, itself the iteration of a function. It introduces a notion of distance from the fixed point of Newton's iteration, which is used to prove convergence for good initial conditions. This is then applied to a model problem, that of bringing a matrix to normal form. KAM theory for maps of the circle to itself is developed next. Amazingly, maps close to rational rotations are chaotic while those close to irrational rotations are not. A digression into number theory explores the connection to Diophantine equations. Finally, the chapter solves the Hamilton-Jacobi equation by Newton's iteration, establishing that there are small perturbations of integrable systems that are not chaotic.
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