Describing a universal critical behavior in a transition from order to chaos

We present a comprehensive discussion of a transition from integrability to nonintegrability in an oval billiard with a static boundary. This transition is controlled by a deformation parameter ε, which modifies the boundary shape from circular, corresponding to ε=0 and an integrable dynamics, to oval for ε≠0, where nonintegrability emerges. The deformation of the circular billiard gives rise to a chaotic layer that develops along a well-defined stripe in phase space. By introducing a set of transformations that isolate this chaotic stripe, we characterize the diffusive spreading of ensembles of trajectories and identify an observable, ω_{rms,sat}, which plays the role of an order parameter for the transition. For small deformations, the saturation value of the diffusion obeys the scaling law ω_{rms,sat}∝ε^{α[over ̃]}, with a critical exponent α[over ̃]=0.507(2), vanishing continuously as ε→0. The associated susceptibility, χ=dω_{rms,sat}/dε, diverges in the same limit, signaling the presence of critical behavior analogous to that observed in second-order (continuous) phase transitions in statistical mechanics.