## Research Program

### Dynamical Systems and Chaos Theory: Sharkovski Ordering and Universality in Chaos

Faculty Mentor: Dr. Ugur Abdulla

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.

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 $4k-3$ 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 one-parameter family of unimodal continuous maps on the interval, and very deep connection between Sharkovski ordering and universality in chaos.

In summer session of 2017 students will pursue research on outstanding open problems related to both topics.

#### References

• A. U. Abdulla, R. U. Abdulla, U. 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), 1395-1416.
• L. S. Block, W. A. Coppel, Dynamics in One Dimension, Springer-Verlag, Berlin, 1992.
• U.G. Abdulla, R.U. Abdulla, M.U. Abdulla and N.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.
• U.G. Abdulla, R.U. Abdulla, M.U. Abdulla, N.H. Iqbal Classification of the Second Minimal Odd Periodic Orbits in the Sharkovskii Ordering, SIAM Journal on Applied Dynamical Systems (Submitted) (2017), Arxiv:1701.02695