ARNOLD DIFFUSION IN HAMILTONIAN SYSTEMS WITH THREE DEGREES OF FREEDOM

We review, by means of a simple example, the mechanism for a very general form of self-generated stochastic motion-the Arnold diffusion-which occurs in nearintegrable Hamiltonian systems with three or more degrees of freedom.’ Without loss of generality, we consider autonomous systems for which the Hamiltonian is explicitly independent of time. Nonautonomous systems in N degrees of freedom can be made autonomous in N + I degrees of freedom by introducing an extended phase space.’ “Near-integrable” Hamiltonians have the form H = Ho + tH, with H , integrable, N not integrable, and the perturbation strength t << 1. The generic behavior of near-integrable systems with two degrees of freedom is now reasonably well known.*-‘ A finite fraction of the trajectories of such systems are the integrable trajectories of KAM theory, with the remaining fraction appearing to be stochastic. The integrable trajectories depend discontinuously on initial conditions. Stochastic and integrable trajectories are intimately comingled, with some stochastic trajectory lying arbitrarily close to every point both i n the four-dimensional phase space and in the two-dimensional surface of section. The stochastic trajectories form in the neighborhood of resonances of the motion between the two degrees of freedom. They appear as thin layers of stochasticity surrounding the separatrices of the motion associated with these resonances. The thickness of the layers increases with increasing perturbation strength. For weak perturbation, stochastic layers associated with different resonances are isolated from each other by K A M surfaces. The motion is stable. lying either in a K A M surface or within a thin stochastic layer bounded by nearby K A M surfaces. As the perturbation increases, the thickness of the layers expands, leading to resonance overlap. the destruction of the last KAM surface separating the layers. This signals the sudden appearance of strong sfochasririry i n the motion, in which the previously separated layers merge and the trajectory freely moves across the layers. The nature of the motion in systems with three or more degrees of freedom is similar to the above in most respects. Stochastic and integrable (KAM) trajectories are intimately comingled in the 2N-dimensional phase space. Stochastic layers form near the separatrices associated with resonances of the motion among the degrees of freedom. For strong perturbation, resonance overlap leads to motion across the layers and the presence of strong stochasticity. I n the limit of weak perturbation, however, resonance overlap does not occur. A new physical bchavior of the motion then makes