Research and Recent Publications
3. Dynamical Systems and Chaos Theory
This project relates to two topics of Dynamical Systems and Chaos Theory: 1. Classification of periodic orbits for the continuous
endomorphisms in the interval. This topic originates from the celebrated result by Sharkovski (1964) on the coexistence
of periodic orbits of continuous maps on the interval, which presents
a surprisingly simple and elegant hierarchy of distribution of periodic orbits according to their periods;
2. Asymptotic behavior of parameter dependent continuous maps, chaos phenomena and
universal transition from periodic to chaotic behavior for nonlinear maps. This topic originates from the pioneering
work of Feigenbaum (1978) on the universal transition route from periodic
to chaotic behavior through period doubling bifurcations for the logistic type unimodal maps.
Recently my DSCT research team solved an open problem on the classification of second
minimal odd orbits of the continuous endomorphisms on the interval. It is proved that there are $4k3$ types of second minimal
$(2k+1)$orbits, each characterized with unique cyclic permutation and directed graph of
transitions with accuracy up to inverses. We then revealed a fascinating universal law of
distribution of periodic orbits in chaotic regime for oneparameter family of unimodal continuous
maps on the interval, and very deep connection between Sharkovski ordering and universality in chaos.
Here are some recent papers:
 Almas U. Abdulla, Rashad U. Abdulla, Ugur G. Abdulla, On the Minimal 2(2k+1)orbits of the Continuous Endomorphisms on the Real Line with Application in
Chaos Theory,
Journal of Difference Equations and Applications, Volume 19, 9(2013), 13951416.
 Ugur G. Abdulla, Rashad U. Abdulla, Muhammad U. Abdulla and Naveed H. Iqbal Second Minimal Orbits, Sharkovski Ordering and Universality in Chaos,
International Journal of Bifurcation and Chaos, Volume 27, Number 5, May 2017,
Arxiv:1610.00814.
 Ugur G. Abdulla, Rashad U. Abdulla, Muhammad U. Abdulla, Naveed H. Iqbal Classification of the
Second Minimal Odd Periodic Orbits in the Sharkovskii Ordering, 2017,
Arxiv:1701.02695
