Georgia Institute of Technology College of Sciences

We give a comprehensive parameter study of the three-dimensional quadratic diffeomorphism to understand its attracting and chaotic dynamics. For large parameter values, we use a concept 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. For the 3D quadratic map, the AI limit that we study becomes a pair of one-dimensional maps, introducing symbolic dynamics on two symbols. Using contraction arguments, we find parameter domains such that each symbol sequence corresponds to a unique AI state. In some of these 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, and allowing us to obtain orbits with horseshoe-like structures and intriguing self-similarity.

For small parameter values, we focus on the dissipative, orientation preserving case to study the codimension-one and two bifurcations. Periodic orbits, born at resonant, Neimark-Sacker bifurcations, give rise to Arnold tongues in parameter space. Aperiodic attractors include invariant circles and chaotic orbits; these are distinguished by rotation number and Lyapunov exponents. Chaotic orbits include Hénon-like and Lorenz-like attractors, which can arise from period-doubling cascades, and those born from the destruction of invariant circles. The latter lie on paraboloids near the local unstable manifold of a fixed point.

Lastly, we present a generalized proof for the existence of AI states using similar contraction arguments to find larger parameter domains for the one-to-one correspondence of symbol sequences and AI states. We apply numerical continuation to these results to determine the persistence of low-period and heteroclinic AI states to the full, deterministic 3D map for a volume-contracting case. We find the corresponding AI state of a chaotic attractor and continue this state towards the full map. The numerical results show that the AI states continue to resonant and chaotic attractors along a 3D folded horseshoe that is similar to the classical 2D Hénon attractor.