Applications of Scaling Ideas to Dynamics. II. From Periodic Motion to Unbounded Chaos: Investigations of the Simple Pendulum

The simplest example of the onset of chaos in a Hamiltonian system is provided by the standard or Chirikov-Taylor model. As a nonlinearity parameter, k, is increased the long term behavior of the momentum, p, is examined. At k = 0, p is conserved. For k < kc, for all starting points, p is of bounded variation. For some starting points its behavior is periodic, for others quasi-periodic, for others chaotic. At some critical value of k, unbounded chaotic variation first appears. A scaling analysis to describe this onset is described.