  The traditional Newton method for solving nonlinear operator equations in
Banach spaces is discussed within the context of the continuous Newton method.
This setting makes it possible to interpret the Newton method as a discrete
dynamical system and thereby to cast it in the framework of an adaptive step
size control procedure. In so doing, our goal is to reduce the chaotic behavior
of the original method without losing its quadratic convergence property close
to the roots. The performance of the modified scheme is illustrated with
various examples from algebraic and differential equations.
