Chaotic Dynamics of Distributed Vibrating Systems Containing Nonlinearities in Multidimensional Spaces
Chaos is one of the most complex and fascinating nonlinear phenomena omnipresent in all branches of engineering and natural and social sciences. Chaos, particularly chaotic vibrations, is the focus of this proposal. It is a request of renewal for the NPRP Project (Grant #4-1162-1-181) entitled â Chaotic Dynamics of Distributed Parameter Systems Containing Nonlinearitiesâ . The new emphasis will be on chaotic vibration phenomena in higher dimensional spaces for hyperbolic partial differential equations containing nonlinearities either in the boundary conditions or distributed in the interior of the domain. The methodology consists of a combination of analysis, modeling, and computation will be developed to study and visualize such chaotic phenomena, including the following: (1) Dimension-reduction mechanisms suitable for mathematical analysis; (2) Generalization of the method of characteristics to multidimensional settings; (3) Numerical schemes based on OpenFOAM to expedite numerical simulation and chaos visualization; (4) Bifurcation analysis for reduced-order equations and the exploration of its significance for the original multidimensional system. Intellectual merit: Significant new understanding will be gained on the nonlinear phenomena, bifurcation, chaos, control and anti-control of vibrations in partial differential equations for the challenging settings in multidimensional domains and geometries. Many interesting pattern formations and deterministic random motions can be analyzed, understood and visualized through modeling, computation and visualization, pushing the study of nonlinear science to new frontiers. Potential impact: The project seeks to integrate the research program with the training and education of higher education students and post-doctorals for the human capital development of Qatar in the important QNRS fields of nonlinear science, interdisciplinary studies, modeling and large-scale scientific computation.