Identifying integrable dynamics remains a formidable challenge, and despite centuries of research, only a handful of cases are known. The discovery process often relies on human intuition and deep understanding of the underlying dynamical patterns. Key to these discoveries is recognizing symmetries in the system that are not immediately obvious, often leading to conserved quantities through Noether's theorem.
Modern computational tools and numerical methods have expanded our ability to study complex systems, but they do not easily reveal integrability. In this talk, we present two novel and distinct (by first principles) algorithms that enable the automated and systematic discovery of new integrable systems.
Our methods successfully rediscover some of the famous McMillan-Suris maps and ultradiscrete Painlevé equations, and reveal over 100 new integrable families. These include a novel category of planar tilings characterized by discrete symmetries emerging from the invertibility of transformations and intrinsically linked to integrability. Additionally, some newly discovered systems exhibit the peculiar behavior of "integrable diffusion," characterized by infinite and quasirandom hopping between tiles of periodicity while remaining confined to a set of invariant segments.
We will discuss various applications in physics and mathematics, including systems such as an accelerator lattice with a thin nonlinear lens, a kicked rotator (an oscillator subjected to periodically switched external forces), and tilings by polygons.