In a joint work with my undergraduate research students Miss Almas Abdulla (MIT, Class of 2014) and Rashad Abdulla (UPenn, Class of 2017) we revealed a new law of the distribution of periodic orbits in chaotic regime for the one parameter family of discrete dynamical systems. We presented a new constructive proof of the result on the structure of minimal $2(2k+1)$-orbits of the continuous endomorphisms $f:I \to I$, where $I$ is a nondegenerate interval on the real line. It is proved that there are four types of digraphs with accuracy up to inverse graphs. It is demonstrated that the first appearance, as the parameter increased, of the $2(2k+1)$-periodic window within the chaotic regime in the bifurcation diagram of the one-parameter family of logistic type unimodal continuous endomorphisms is always a minimal $2(2k+1)$-orbit with Type I digraph.

In summer session of 2014 students will pursue research on the fine classification of the periodic orbits of the continuous endomorphisms and analysis of the structure of the periodic windows within the chaotic regime of the bifurcation diagram for the unimodal continuous endomorphisms. This research requires creative combination of theoretical and numerical analysis.

#### 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.