Stochasticity in classical Hamiltonian systems: Universal aspects

Abstract For beginners: This review presents universal aspects of stochasticity of simple (1.5- or 2-degree-of-freedom) Hamiltonian systems. Stochasticity is the seemingly erratic wandering of orbits of non-integrable Hamiltonian systems over some part of phase space, accompanied by exponential divergence of nearby orbits. It is a large-scale phenomenon that spreads over larger and larger regions of phase space by the successive breakups of barriers called Kolmogorov-Arnold-Moser (KAM) tori when some perturbation to an integrable Hamiltonian is increased. The main emphasis of this review is on the breakup of KAM tori which is described by a renormalization group for Hamiltonians of the KAM type. This paper also reports recent progress in describing chaotic transport which is the large scale manifestation of stochasticity, but this is not the last word to chaos. The central model of this paper is the Hamiltonian of one particle in two longitudinal waves, H p (v,x,t) = v 2 2−M cos x − P cos k(x − t) , which is a paradigm for simple Hamiltonian systems. Simple approximate renormalization schemes for KAM tori of Hp are derived, and the way to exactly renormalize a general Hamiltonian of the KAM type is explained as well. For experts: The breakup of KAM tori and chaotic transport are the main topics of this review which aims at updating the picture of Hamiltonian stochasticity as described, for instance, in Lichtenberg and Lieberman's textbook. The exact way to renormalize a Hamiltonian of the KAM type for a given torus is explained, and simple computable approximate schemes are described as well. The connection with schemes for area-preserving maps is made. Orbits are characterized by a zoning number whose value is determined by both the spatial periodicity of primary resonances and the rotation number of a KAM torus. Universal classes of KAM tori are related to a periodic continued fraction expansion of the zoning number. When the expansion ends with ones onle, the torus is termed noble. The robustness of noble tori is shown to imply a hierarchy in the denumerable set of fixed points of the renormalization group. A simple renormalization scheme for the standard map describes high-order fixed points of the renormalization groups for KAM tori and for period n-tupling. When the perturbation to a given integrable Hamiltonian is increased enough, a KAM torus develops a dense set of gaps and is called a Cantorus. The fact that time is exactly renormalized in schemes for Hamiltonian systems, makes it possible to compute critical exponents for Cantori by simple scaling arguments. Accurate calculations of thresholds of global stochasticity are obtained by the combination of the exact renormalization scheme and of an approximate expression for the noble stable manifold. A perturbation theory reduces a general Hamiltonian to a many-wave Hamiltonian. The width of stochastic layers is computed by a method different from Melnikov-Chirikov's one, but which agrees with it; it is defined in terms of rotation numbers. Three regimes of chaotic transport are described and the corresponding scaling laws are given: the Cantorus-with-small-hole regime, the quasi-linear regime, and the trapping regime. For the reader to whom this abstract is too short and the whole article is too long, we recommend sections 1 and 2.