The hidden complexity of a double-scroll attractor: Analytic proofs from a piecewise-smooth system

Double-scroll attractors are one of the pillars of modern chaos theory. However, rigorous computer-free analysis of their existence and global structure is often elusive. Here, we address this fundamental problem by constructing an analytically tractable piecewise-smooth system with a double-scroll attractor. We derive a Poincare return map to prove the existence of the double-scroll attractor and explicitly characterize its global dynamical properties. In particular, we reveal a hidden set of countably many saddle orbits associated with infinite-period Smale horseshoes. These complex hyperbolic sets emerge from an ordered iterative process that yields sequential intersections between different horseshoes and their preimages. This novel distinctive feature differs from the classical Smale horseshoes, directly intersecting with their own preimages. Our global analysis suggests that the structure of the classical Chua attractor and other figure-eight attractors might be more complex than previously thought.

Automated summary

This study constructs an analytically tractable piecewise-smooth system and derives a Poincare return map to prove the existence of a double-scroll attractor and characterize its global dynamical properties. The study also discovers a set of saddle orbits associated with infinite-period Smale horseshoes. These complex hyperbolic sets arise from an ordered iterative process of sequential intersections between different horseshoes and their preimages.