Anne Ketri P. da Fonseca

arXiv2026

First-Order Transition to Chaos with Critical Slowing Down

Can the transition from integrability to chaos be discontinuous? We show that it can, and that the resulting first-order dynamical transition coexists with critical slowing down. Using an analytically tractable confined random walk and a deterministic stadium-like billiard, we find a finite jump of the stationary diffusive observable at the transition while the relaxation time diverges. Both systems display normal diffusion and the same exponent set $(α,β,z)=(0,1/2,-2)$. We trace this agreement to a common coarse-grained mechanism: diffusion in a finite accessible domain with a diffusion coefficient that vanishes quadratically with the perturbation. The results identify a discontinuous route from integrability to chaos and provide evidence for a broader universality class of first-order dynamical transitions.

nlin.CD↗

arXiv2026

First-order integrability-breaking phase transitions in dynamical systems

We investigate a discontinuous route from integrability to chaos using a confined stochastic random walk and a deterministic stadium-like billiard. In both systems, the stationary diffusive observable exhibits a finite jump at the transition: it vanishes at the unperturbed limit but approaches a finite, geometry-controlled value for arbitrarily small nonzero perturbations, providing the characteristic order-parameter signature of a first-order transition. Despite this discontinuity, the relaxation timescale diverges as the transition is approached, revealing critical slowing down. Both models exhibit normal diffusion with $β=1/2$, a perturbation-independent stationary state with $α=0$, and a crossover iteration scaling as $n_x\proptoλ^{-2}$, yielding $z=-2$, where $λ$ denotes the corresponding perturbation parameter. The common exponent set $(α,β,z)=(0,1/2,-2)$ originates from the same coarse-grained mechanism: normal diffusion within a finite accessible domain with a diffusion coefficient that vanishes quadratically at the transition. The agreement between stochastic transport and deterministic chaotic scattering provides strong evidence for a common class of discontinuous dynamical transitions and extends the statistical-mechanics description of phase transitions to integrability-breaking dynamics.

nlin.CD↗