The fractal structure of quasi-periodic solutions of hamiltonian systems

After a simple example giving an outlook of the whole complexity of solutions, the classical Kolmogorov-Arnold-Moser (KAM) demonstration is outlined with some useful extensions. The usual Arnold tori have a dimensionality equal to the number of degrees of freedom and a generalization of the KAM demonstration shows the very general existence of quasi-periodic motions on invariant tori with a smaller dimensionality. These latter invariant tori corresponds to resonances and lead to a well-ordered picture of solutions. The solutions are classified into three main types: (a) the periodic and quasi-periodic orbits that are ‘first-order stable’; (b) the chaotic orbits; (c) the open orbits coming from infinity and going back to infinity. The solutions of the first type separate the two other types, but they fill a nowhere dense set when the Hamiltonian problem of interest is non-integrable. This set has a structure looking like a fractal but without an exact scale property. It is conjectured (‘Arnold diffusion conjecture’) that all non-integrable analytic autonomous Hamiltonian systems lead to these same types of picture and classification.