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Fractional Dynamical Systems
In this paper the author presents the results of the preliminary
investigation of fractional dynamical systems based on the results of numerical
simulations of fractional maps. Fractional maps are equivalent to fractional
differential equations describing systems experiencing periodic kicks. Their
properties depend on the value of two parameters: the non-linearity parameter,
which arises from the corresponding regular dynamical systems; and the memory
parameter which is the order of the fractional derivative in the corresponding
non-linear fractional differential equations. The examples of the fractional
Standard and Logistic maps demonstrate that phase space of non-linear
fractional dynamical systems may contain periodic sinks, attracting slow
diverging trajectories, attracting accelerator mode trajectories, chaotic
attractors, and cascade of bifurcations type trajectories whose properties are
different from properties of attractors in regular dynamical systems. The
author argues that discovered properties should be evident in the natural
(biological, psychological, physical, etc.) and engineering systems with
power-law memory.Comment: 6 pages, 4 figure
Quasistatic dynamical systems
We introduce the notion of a quasistatic dynamical system, which generalizes
that of an ordinary dynamical system. Quasistatic dynamical systems are
inspired by the namesake processes in thermodynamics, which are idealized
processes where the observed system transforms (infinitesimally) slowly due to
external influence, tracing out a continuous path of thermodynamic equilibria
over an (infinitely) long time span. Time-evolution of states under a
quasistatic dynamical system is entirely deterministic, but choosing the
initial state randomly renders the process a stochastic one. In the
prototypical setting where the time-evolution is specified by strongly chaotic
maps on the circle, we obtain a description of the statistical behaviour as a
stochastic diffusion process, under surprisingly mild conditions on the initial
distribution, by solving a well-posed martingale problem. We also consider
various admissible ways of centering the process, with the curious conclusion
that the "obvious" centering suggested by the initial distribution sometimes
fails to yield the expected diffusion.Comment: 40 page
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