From Order to Chaos

In Chapter 6, we studied integrable systems. The description of their motion is very simple. With a correct choice of the coordinates in phase space (angle-action variables), each trajectory is a helix which rolls, at constant angular velocity, around a torus with constant actions. In Chapter 7, we emphasized the catastrophe of small divisors: the perturbation theory is singular if the ratio of certain frequencies is close to a rational number. In this chapter, we investigate this situation and show that, depending upon the initial conditions, the motion can be regular and predictable or, in complete contrast, chaotic and unpredictable.