Describing a Universal Critical Behavior in a transition from order to chaos

We present a comprehensive discussion of a transition from integrability to non-integrability in an oval billiard with a static boundary. This transition is controlled by a deformation parameter $\epsilon$, which modifies the boundary shape from circular, corresponding to $\epsilon=0$ and an integrable dynamics, to oval for $\epsilon\neq 0$, where non-integrability 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 characterise the diffusive spreading of ensembles of trajectories and identify an observable, $\omega_{rms,{\rm sat}}$, which plays the role of an order parameter for the transition. For small deformations, the saturation value of the diffusion obeys the scaling law $\omega_{rms,{\rm sat}}\propto\epsilon^{\tilde{\alpha}}$, with a critical exponent $\tilde{\alpha}=0.507(2)$, vanishing continuously as $\epsilon\rightarrow 0$. The associated susceptibility, $\chi=d\omega_{rms,{\rm sat}}/d\epsilon$, diverges in the same limit, signalling the presence of critical behavior analogous to that observed in second-order (continuous) phase transitions in statistical mechanics.