The Breakdown of KAM Trajectories

Hamiltonian systems with two degrees of freedom, for example, two coupled oscillators, are the simplest class of conservative systems that may exhibit a nontrivial dynamics. In this introduction, we briefly review the phenomenology associated with that class of systems. If the coupling between the two oscillators is linear, the system is integrable and the motion decouples into normal modcs. These are best described in terms of action-angle variables:' I&)- I S I,(r)- I $ 0'0)- dG % + 0:; e 2 w- % ( f i t + 05 (1) because the energy is conserved, E- E ( I f, 13, the motion in phase-space is confined to a torus. If one considers a cross section of the torus (Poincarb section, see Raum I), then the dynamics defines an area-preserving mapping of that section into itself: en+,- 4 + W n + J 2~s In+, - I, (2) where the winding number Q(In+,) is the ratio of the two normal modes frequencies, Q- wI/*. Two types of motion are possible: (1) Cyclic motion: If Q is rational (Q- p / q) , there exist an infinite number of different cyclic orbits. For any initial condition (8, lo), the sequence {Om) repeats itself with period q: On+,- 0, [mod 2 4. (2) KAM curve: If Q is irrational, the sequence {On} never repeats, but the points 0, fill out a curve r. If the coupling between the two oscillators is nonlinear, the system is in general nonintegrable. According to the Poincar&Birkhoff theorem? only two cyclic orbits remain for every rational winding number (Q- p / q) : an elliptic orbit that nearby points tend to cycle around and a hyperbolic one that nearby points are repelled from. Some KAM curves (irrational Q) may still exist depending on the strength of the nonlinear coupling and their proximity to a rational orbit (a destabilizing factor). However, the most important qualitative and quantitative difference between th