Large-Scale Stochasticity in Hamiltonian Systems

Large scale stochasticity (LSS) in Hamiltonian systems is defined on the paradigm Hamiltonian H(v, x, t) = v²/2 − M cos x − P cos k(x − t) which describes the motion of one particle in two electrostatic waves. A renormalization transformation T_r is described which acts as a microscope that focusses on a given KAM (Kolmogorov–Arnold–Moser) torus in phase space. Though approximate, T_r yields the threshold of LSS in H with an error of 5–10%. The universal behaviour of KAM tori is predicted: for instance the scale invariance of KAM tori and the critical exponent of the Lyapunov exponent of Cantori. The Fourier expansion of KAM tori is computed and several conjectures by L Kadanoff and S Shenker are proved. Chirikov's standard mapping for stochastic layers is derived in a simpler way and the width of the layers is computed. A simpler renormalization scheme for these layers is defined. A Mathieu equation for describing the stability of a discrete family of cycles is derived. When combined with T_r, it allows to prove the link between KAM tori and nearby cycles, conjectured by J Greene and, in particular, to compute the mean residue of a torus. The fractal diagrams defined by G Schmidt are computed. A sketch of a methodology for computing the LSS threshold in any two-degree-of-freedom Hamiltonian system is given.