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

Abstract

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 $ε$, which modifies the boundary shape from circular, corresponding to $ε=0$ and an integrable dynamics, to oval for $ε\neq 0$, 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, $ω_{rms,{\rm sat}}$, which plays the role of an order parameter for the transition. For small deformations, the saturation value of the diffusion obeys the scaling law $ω_{rms,{\rm sat}}\proptoε^{\tildeα}$, with a critical exponent $\tildeα=0.507(2)$, vanishing continuously as $ε\rightarrow 0$. The associated susceptibility, $χ=dω_{rms,{\rm sat}}/dε$, diverges in the same limit, signalling the presence of critical behavior analogous to that observed in second-order (continuous) phase transitions in statistical mechanics.

Figures (12)

• Figure 1: Sketch of the angles describing the billiard dynamics. The boundary is constructed using $R=1+\epsilon\cos(p\theta)$ with $p=2$ (assumed constant throughout this work) and $\epsilon=0.05$. The variable $\theta_n$ denotes the polar angle at the $n$th collision, $\vec{T}_n$ is the tangent vector at $\theta_n$, and $\alpha_n$ is the angle of the trajectory measured with respect to the tangent vector $\vec{T}_n$.
• Figure 2: (Color online) Phase-space of the mapping (\ref{['B_eq4']}) for the control parameter $\epsilon=0.1$. A mixed phase-space structure is observed, with periodic islands surrounded by a chaotic sea bounded by a set of whispering gallery orbits (WGO), as indicated in the figure.
• Figure 3: (Color online) Plot of the phase-space for mapping (\ref{['B_eq4']}) for the control parameter: (a) $\epsilon=0$, where a set of regular orbits is observed; (b) $\epsilon=0.05$, where a mixed phase-space structure emerges, consisting of periodic islands surrounded by a chaotic sea bounded by whispering gallery orbits.
• Figure 4: Phase-space portraits of the mapping (\ref{['B_eq4']}) for the control parameter $\epsilon=0.01$ at different levels of magnification: (a) $\theta\in[0,2\pi]$ and $\alpha\in[0,\pi]$, showing the full phase space; (b) $\theta\in[0,2\pi]$ and $\alpha\in[\pi/2-0.4,\pi/2+0.4]$, highlighting the region near the islands and revealing a separatrix-like structure; (c) $\theta\in[\pi-0.2,\pi+0.2]$ and $\alpha\in[\pi/2-0.02,\pi/2+0.02]$, where the separatrix-like structure is seen to occupy a finite area; (d) $\theta\in[\pi-0.0075,\pi+0.0075]$ and $\alpha\in[\pi/2-0.002,\pi/2+0.002]$, showing a chaotic region forming a narrow chaotic stripe near the saddle point $(\theta,\alpha)=(\pi,\pi/2)$.
• Figure 5: (a) Phase-space portrait for $\epsilon=0.05$, showing a chaotic layer symmetric with respect to $\alpha=\pi/2$. (b) Same as (a) after the transformation $\gamma=\alpha-\pi/2$, revealing symmetry with respect to $\gamma=0$. (c) Same as (b) after the transformation $\gamma\rightarrow \gamma^2$, which folds the negative part of the phase space onto the positive one. (d) Envelopes of the stripe delimiting the chaotic layer in phase space.