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Light propagation on a two-dimensional curved surface embedded in athree-dimensional space has attracted increasing attention as an analog modelof four-dimensional curved spacetime in laboratory. Despite recent developmentsin modern cosmology on the dynamics and evolution of the universe,investigation of nonlinear dynamics of light in non-Euclidean geometry is stillscarce and remains challenging. Here, we study classical and wave chaoticdynamics on a family of surfaces of revolution by considering its equivalentconformally transformed flat billiard, with nonuniform distribution ofrefractive index. This equivalence is established by showing how these twosystems have the same equations and the same dynamics. By exploring thePoincar\'{e} surface of section, the Lyapunov exponent and the statistics ofeigenmodes and eigenfrequency spectrum in the transformed inhomogeneous tablebilliard, we find that the degree of chaos is fully controlled by a singlegeometric parameter of the curved surface. A simple interpretation of ourfindings in transformed billiards, the "fictitious force", allows to extend ourprediction to other class of curved surfaces. This powerful analogy between twoa prior unrelated systems not only brings forward a novel approach to controlthe degree of chaos, but also provides potentialities for further studies andapplications in various fields, such as billiards design, optical fibers, orlaser microcavities.

Light chaotic dynamics in the transformation from curved to flat surfaces