Anti-integrability for Three-Dimensional Quadratic Maps

Abstract

We study the dynamics of the three-dimensional quadratic diffeomorphism using a concept first introduced 30 years ago for the Frenkel--Kontorova model of condensed matter physics: the anti-integrable (AI) limit. At the traditional AI limit, orbits of a map degenerate to sequences of symbols and the dynamics is reduced to the shift operator, a pure form of chaos. Under nondegeneracy conditions, a contraction mapping argument can show that infinitely many AI states continue to orbits of the deterministic map. For the 3D quadratic map, the AI limit that we study is a quadratic correspondence whose branches, a pair of one-dimensional maps, introduce symbolic dynamics on two symbols. The AI states, however, are nontrivial orbits of this correspondence. The character of these orbits depends on whether the quadratic takes the form of an ellipse, a hyperbola, or a pair of lines. Using contraction arguments, we find parameter domains for each case such that each symbol sequence corresponds to a unique AI state. In some parameter domains, sufficient conditions are then found for each such AI state to continue away from the limit becoming an orbit of the original 3D map. Numerical continuation methods extend these results, allowing computation of bifurcations to obtain orbits with horseshoe-like structures and intriguing self-similarity. We conjecture that pairs of periodic orbits in saddle-node and period-doubling bifurcations have symbol sequences that differ in exactly one position.