Integrability to chaos in the standard map: a continuous dynamical transition with critical-like scaling

We investigate the onset of non-integrability in the conservative standard map within the framework of critical phenomena. We show that the transition from integrable to chaotic dynamics can be characterized by a geometrical order parameter given by the width of the stochastic layer surrounding the hyperbolic fixed point. This quantity vanishes continuously in the integrable limit and obeys the scaling law $(d\propto K^{\alpha})$, with critical exponent $(\alpha\simeq1/2)$. The associated susceptibility, defined as the response of the stochastic-layer width to the nonlinear perturbation, diverges as $(K\rightarrow0)$, providing a hallmark of critical-like behavior. We further identify the destruction of the continuous foliated phase-space structure as the symmetry-breaking mechanism underlying the transition and show that the elementary kick-induced action increments constitute the microscopic excitations responsible for chaotic transport. In addition, we discuss how stability islands, cantori, and sticky regions act as effective dynamical defects that govern long-time transport within the chaotic sea. The critical exponent $(\alpha\simeq1/2)$ coincides with values previously reported for Fermi-Ulam accelerators, corrugated waveguides, and oval billiards, suggesting a possible universality class for the onset of chaotic diffusion in weakly perturbed Hamiltonian systems. These results establish a unified framework connecting Hamiltonian chaos, transport theory, and the theory of continuous phase transitions.