Universal Second-Order Phase Transition from Integrability to Chaos

We report a dynamical phase transition from integrability to non-integrability in a simple oval-like billiard with boundary $R(\theta)=1+\epsilon\cos(p\theta)$. For $\epsilon=0$, the phase space is {\it foliated} by invariant curves corresponding to periodic or quasiperiodic motion, whereas for small $\epsilon$ a thin chaotic layer separates rotational and librational trajectories. As $\epsilon$ increases, this layer grows according to a well-defined scaling law whose chaotic dispersion follows $\omega_{\rm rms,sat}\sim\epsilon^{\tilde{\alpha}}$, where the exponent $\tilde{\alpha}$ coincides with those of the Fermi-Ulam model, periodically corrugated waveguides, and a family of discrete mappings, revealing a universal mechanism for the onset of chaos in weakly perturbed integrable systems. The deviation of the reflection angle in the billiard, $\omega_{\rm rms,sat}$, acts as an order parameter: it vanishes continuously as $\epsilon\to 0$, signalling an ordered (integrable) phase, while its susceptibility $\chi=d\omega_{\rm rms,sat}/d\epsilon$ diverges, indicating a second-order phase transition. A symmetry breaking and an analytically solvable diffusion process complete the near-critical phenomenology. These results establish a unified framework for the emergence of chaos from integrability.