Dynamical Systems and Ergodic 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, and 2. Asymptotic behavior of parameter-dependent continuous maps, chaos phenomena and universal transition from periodic to chaotic behavior for nonlinear maps. The first 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. This second 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 Dynamical Systems and Ergodic Theory 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. Recent papers are listed below:
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
Second minimal orbits, Sharkovski ordering and universality in chaos., Volume 27, Number 5, May 2017, Arxiv:1610.00814.
Classification of the Second Minimal Odd Periodic Orbits in the Sharkovskii Ordering, 2017, ArXiv:1701.02695