A detailed description of fractional kinetics is given in connection to islands’ topology in the phase space of a system. The method of renormalization group is applied to the fractional kinetic equation in order to obtain characteristic exponents of the fractional space and time derivatives, and an analytic expression for the transport exponents. Numerous simulations for the web-map and standard map demonstrate different results of the theory. Special attention is applied to study the singular zone, a domain near the island boundary with a self-similar hierarchy of subislands. The birth and collapse of islands of different types are considered.

In this paper, the author compares behaviors of systems which can be described by fractional differential and fractional difference equations using the fractional and fractional difference Caputo standard α-Families of maps as examples. The author shows that properties of fractional difference maps (systems with falling factorial-law memory) are similar to the properties of fractional maps (systems with power-law memory). The similarities (types of attractors, power-law convergence of trajectories, existence of cascade of bifurcations and intermittent cascade of bifurcations type trajectories, and dependence of properties on the memory parameter α) and differences in properties of falling factorial- and power-law memory maps are investigated.

We modified the way in which the Universal Map is obtained in the regular dynamics to derive the Universal α-Family of Maps depending on a single parameter α>0, which is the order of the fractional derivative in the nonlinear fractional differential equation describing a system experiencing periodic kicks. We consider two particular α-families corresponding to the Standard and Logistic Maps. For fractional α<2 in the area of parameter values of the transition through the period doubling cascade of bifurcations from regular to chaotic motion in regular dynamics corresponding fractional systems demonstrate a new type of attractors—cascade of bifurcations type trajectories.

In this paper, we propose a method to calculate asymptotically period two sinks and define the range of stability of fixed points for a variety of discrete fractional systems of the order $0<\alpha <2$ . The method is tested on various forms of fractional generalizations of the standard and logistic maps. Based on our analysis, we make a conjecture that chaos is impossible in the corresponding continuous fractional systems.

We consider a model of a billiard-type system, which consists of two chambers connected through a hole. One chamber has a circle-shaped scatterer inside (Sinai billiard with infinite horizon), and the other one has a Cassini oval with a concave border. The phase space of the Cassini billiard contains islands, and its parameters are taken in such a way as to produce a self-similar island hierarchy. Poincaré recurrences to the left and to the right chambers are considered. It is shown that the corresponding distribution function does not reach “equipartition” even during the time ${10}^{10}$. The explanation is based on the existence of singularities in the phase space, which induces anomalous kinetics. The analogy to the Maxwell’s Demon model is discussed.