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We describe a basic framework for studying dynamic scaling that has roots indynamical systems and probability theory. Within this framework, we studySmoluchowski's coagulation equation for the three simplest rate kernels$K(x,y)=2$, $x+y$ and $xy$. In another work, we classified all self-similarsolutions and all universality classes (domains of attraction) for scalinglimits under weak convergence (Comm. Pure Appl. Math 57 (2004)1197-1232). Herewe add to this a complete description of the set of all limit points ofsolutions modulo scaling (the scaling attractor) and the dynamics on this limitset (the ultimate dynamics). The main tool is Bertoin's L\'{e}vy-Khintchinerepresentation formula for eternal solutions of Smoluchowski's equation (Adv.Appl. Prob. 12 (2002) 547--64). This representation linearizes the dynamics onthe scaling attractor, revealing these dynamics to be conjugate to a continuousdilation, and chaotic in a classical sense. Furthermore, our study of scalinglimits explains how Smoluchowski dynamics ``compactifies'' in a natural waythat accounts for clusters of zero and infinite size (dust and gel)

The Scaling Attractor and Ultimate Dynamics for Smoluchowski’s Coagulation Equations