RCAAM Activities Report for 2012 (in Greek and English)
The Speed of Arnold Diffusion
The phenomenon of Arnold diffusion is important in a number of applications in solar system dynamics and plasma physics. The above figure, published in Efthymiopoulos and Harsoula, Physica D, 2013 (in press), shows a visualization of Arnold diffusion in a set of variables arising after a so-called optimal normal form construction using a special computer-algebraic program. According to theory, at a crossing domain of multiple resonances, the normal form construction allows to express the equations of motion as a Hamiltonian system of two degrees of freedom. This is perturbed, however, by a third degree of freedom appearing in the so-called `remainder' of the normal form. After a very slow drift in this third dimension (corresponding, in the figure, to the long edge of the framed parallelepiped), the orbits by-pass the barriers (invariant tori) of the 2D dynamics, and thus transit from one resonance to another. The timescale for this phenomenon is millions, or billions of characteristic periods of the orbits (see Efthymiopoulos and Harsoula 2013 for details).