Chaotic trajectories in the standard map. The concept of anti-integrability

Abstract

A rigorous proof is given in the standard map (associated with a Frenkel-Kontorowa model) for the existence of chaotic trajectories with unbounded momenta for large enough coupling constant k > k 0. These chaotic trajectories (with finite entropy per site) are coded by integer sequences { m i } such that the sequence b i = |m i+1 + m i−1−2m i| be bounded by some integer b. The bound k 0 in k depends on b and can be lowered for coding sequences { m i } fulfilling more restrictive conditions. The obtained chaotic trajectories correspond to stationary configurations of the Frenkel-Kontorowa model with a finite (non-zero) photon gap (called gap parameter in dimensionless units). This property implies that the trajectory (or the configuration { u i }) can be uniquely continued as a uniformly continuous function of the model parameter k in some neighborhood of the initial configuration. A non-zero gap parameter implies that the Lyapunov coefficient is strictly positive (when it is defined). In addition, the existence of dilating and contracting manifolds is proven for these chaotic trajectories. “Exotic” trajectories such as ballistic trajectories are also proven to exist as a consequence of these theorems. The concept of anti-integrability emerges from these theorems. In the anti-integrable limit which can be only defined for a discrete time dynamical system, the coordinates of the trajectory at time i do not depend on the coordinates at time i - 1. Thus, at this singular limit, the existence of chaotic trajectories is trivial and the dynamical system reduces to a Bernoulli shift. It is well known that the KAM tori of symplectic dynamical originates by continuity from the invariant tori which exists in the integrible limit (under certain conditions). In a similar way, it appears that the chaotic trajectories of dynamical systems originate by continuity from those which exists at the anti-integrable limits (also under certain conditions).