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A scaling law for the time dependence of the dynamic aperture, i.e., the region of phase space where stable motion occurs, has been proposed in previous papers [M. Giovannozzi, W. Scandale, E. Todescoand , Part. Accel. 56, 195 (1996)PLACBD0031-2460; M. Giovannozzi, W. Scandale, and E. Todesco, in Proceedings of the 1997 Particle Accelerator Conference, edited by M. Comyn, M. K. Craddock, M. Reiser, and J. Thomson (IEEE Service Center, Piscataway, NJ, 1997), p. 1445; M. Giovannozzi, W. Scandale, and E. Todesco, Phys. Rev. E 57, 3432 (1998)PLEEE81063-651X10.1103/PhysRevE.57.3432]. This law, based on the analysis of numerical simulations data, is not entirely phenomenological, but motivated by some fundamental theorems of the theory of dynamical systems and indicates that the dynamic aperture has a logarithmic dependence on time. This result is used in turn as a basis for deriving a scaling law for the intensity evolution in hadron storage rings. This relationship is presented and discussed in detail in this paper. Furthermore, experimental data were compared to the predictions of this law and showed a remarkable agreement.