A Billiard Problem in Nonlinear Dissipative Systems

We propose three levels of modeling: a moving-boundary (MB) model, a particle model, and a discrete-time model.

This book addresses a new class of billiard problems, focusing on the motion of a self-propelling disk in a nonlinear dissipative system on a rectangular domain. Unlike classical billiards, which have been extensively studied in mathematics, this setting introduces unique dynamics inspired by experiments with camphor disks floating on water—a well-known phenomenon in nonlinear science. Laboratory observations reveal two striking properties: (i) the disk reflects without physical collision at the boundary, and (ii) the reflection angle exceeds the incidence angle, differing from the perfect elastic reflection of classical billiards. These features suggest that the behavior of a self-propelling disk is fundamentally distinct from classical billiard motion. The purpose of this book is to provide a mathematical understanding of such dynamics. We propose three levels of modeling: a moving-boundary (MB) model, a particle model, and a discrete-time model. To demonstrate that the MB model satisfies properties (i) and (ii), we derive the particle model for slow disk motion, describing its position and velocity. Numerical simulations indicate that although classical billiards in a rectangle are simple, the particle model exhibits complex behavior depending on the domain’s shape. To analyze this complexity, we construct discrete-time models that capture the evolution of reflection angles and positions. Using dynamical systems theory, bifurcation analysis, and complementary numerical methods, we show that a self-propelling disk can display intricate and varied billiard motions—even in a rectangular domain—due to angle interactions. This book emphasizes that the trajectory of a billiard disk in a nonlinear dissipative system is determined by inherent dynamics, unlike classical billiards, where outcomes depend heavily on player skill.