Universal Second-Order Phase Transition from Integrability to Chaos

Abstract

We report a dynamical phase transition from integrability to non-integrability in a simple oval-like billiard with boundary $R(θ)=1+ε\cos(pθ)$. For $ε=0$, the phase space is {\it foliated} by invariant curves corresponding to periodic or quasiperiodic motion, whereas for small $ε$ a thin chaotic layer separates rotational and librational trajectories. As $ε$ increases, this layer grows according to a well-defined scaling law whose chaotic dispersion follows $ω_{\rm rms,sat}\simε^{\tildeα}$, where the exponent $\tildeα$ 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, $ω_{\rm rms,sat}$, acts as an order parameter: it vanishes continuously as $ε\to 0$, signalling an ordered (integrable) phase, while its susceptibility $χ=dω_{\rm rms,sat}/dε$ 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.

Figures (3)

• Figure 1: (Color online) (a) Sketch of the angles describing the dynamics of the billiard. The boundary was constructed using $R=1+\epsilon\cos(p\theta)$ with $p=2$ (assumed constant in this Letter) and $\epsilon=0.1$. In the figure, $\theta_0$ denotes the polar angle at the initial collision, $\vec{T}_0$ is the tangent vector at $\theta_0$, and $\alpha_0$ is the angle of the trajectory measured with respect to $\vec{T}_0$. (b) Phase space for $\epsilon=0$ showing a symmetric and ordered dynamics. (c) Phase space for $\epsilon=0.05$, displaying a mixed structure with a chaotic layer near $\alpha=\pi/2$, periodic islands and invariant spanning curves. The periodic domain confined by the chaotic layer corresponds to librational motion, while the invariant spanning curves above and below it correspond to rotational motion (whispering-gallery orbits wisp0wispering). (d) Zoom of (c) showing the chaotic domain bounded by the upper and lower chaotic borders (continuous red curves).
• Figure 2: (Color online) (a) Phase space for $\epsilon=0.05$ under the transformation $\gamma=\alpha-\pi/2$. The phase space is now symmetric about $\gamma=0$, meaning that the positive part mirrors the negative part. (b) Plot of the square angle $\gamma^2$ vs. $\theta$ for the same control parameter of (b). (c) The chaotic stripe in panel (b) bounded by two limiting curves. (d) Chaotic layer after the transformation $\gamma^2-\gamma_{min}^2$. The bullet points correspond to a numerical approximation of the chaotic border.
• Figure 3: (Color online) (a) Plot of $\Omega~vs.~n$ for different control parameters, as labelled in the figure. (b) Plot of $\omega_{rms,sat}~vs.~\epsilon$. A power-law fit gives $\tilde{\alpha}=0.507(2)$. The error bars of the average measurements are also represented.