Albert C. J. Luo, Southern Illinois University Edwardsville, USA
(with Yu Guo, McCoy School of Engineering, Midwestern State University, USA)
It is not easy to find periodic motions to chaos in a pendulum system even though the periodically forced pendulum is one of the simplest nonlinear systems. However, the inherent complex dynamics of the periodically forced pendulum is much beyond our imaginations through the traditional thought of the linear dynamical systems. Until now, we did not know complex motions of pendulum yet. What are the mechanism and mathematics of such complex motions in the pendulum? The results presented herein give a new view of complex motions in the periodically forced pendulum. Thus, in this paper, periodic motions to chaos in a periodically forced pendulum are predicted analytically by a semi-analytical method. The method is based on discretization of differential equations of the dynamical system to obtain implicit maps. Using the implicit maps, mapping structures for specific periodic motions are developed, and the corresponding periodic motions can be predicted analytically through such mapping structures. Analytical bifurcation trees of periodic motions to chaos are obtained, and the corresponding stability and bifurcation analysis of periodic motions to chaos are carried out by eigenvalue analysis. From the analytical prediction of periodic motions to chaos, the corresponding frequency-amplitude characteristics are obtained for a better understanding of motions complexity in the periodically forced pendulum. Finally, numerical simulations of selected periodic motions are illustrated. The non-travelable and travelable periodic motions on the bifurcation trees are discovered. Through this investigation, the periodic motions to chaos in the periodically forced pendulums can be understood further. Based on the perturbation method, one cannot achieve the adequate solutions presented herein for periodic motions to chaos in the periodically forced pendulum.
J. A. Tenreiro Machado, Institute of Engineering, Polytechnic of Porto, Portugal
Application of Fractional Calculus in Engineering Sciences
Fractional Calculus (FC) started in 1695 when L'Hôpital wrote a letter to Leibniz asking for the meaning of Dny for n = 1/2.
Starting with the ideas of Leibniz many important mathematicians developed the theoretical concepts. During the thirties A. Gemant and O. Heaviside applied FC in the areas of mechanical and electrical engineering, respectively. Nevertheless, these important contributions were somehow forgotten and only during the eighties FC emerged associated with phenomena such as fractal and chaos and, consequently, in the modeling of dynamical systems.
In the last years FC become a 'new' tool for modeling and control dynamical systems. Based on the FC mathematical concepts, this lecture starts by introducing the FC fundamental historical and mathematical concepts. In a second part the lecture reviews the main approaches for implementing fractional operators and presents several applications in the areas of modeling and control, namely fractional PID, fractional electrical impedances, robotics, nonlinear system control, and finance dynamical analysis.
Yangjian Cai School of Physical Science and Technology, Soochow University, China
Partially coherent beams with prescribed beam properties
Partially coherent beams are preferred in many applications, such as free-space optical communications, optical imaging, optical manipulation and remote detection. Partially coherent beam with prescribed beam properties (i.e., prescribed phases and correlation functions) have advantages over conventional partially coherent beam in many applications, and such beam displays many extraordinary properties. In this talk, I will introduce recent development of partially coherent beams with prescribed beam properties.
Hari M. Srivastava, University of Victoria, Canada