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 $\epsilon$, which modifies the boundary shape from circular, corresponding to $\epsilon =0$ and an integrable dynamics, to oval for $\epsilon \ne 0$, 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, ${\omega }_{\mathrm{rms},\mathrm{sat}}$, which plays the role of an order parameter for the transition. For small deformations, the saturation value of the diffusion obeys the scaling law ${\omega }_{\mathrm{rms},\mathrm{sat}}\propto {\epsilon }^{\tilde{\alpha }}$, with a critical exponent $\tilde{\alpha }=0.507\left(2\right)$, vanishing continuously as $\epsilon \to 0$. The associated susceptibility, $\chi =d{\omega }_{\mathrm{rms},\mathrm{sat}}/d\epsilon$, diverges in the same limit, signaling the presence of critical behavior analogous to that observed in second-order (continuous) phase transitions in statistical mechanics.

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• Chaos
• Dynamical systems