Mathematical Sciences: Studies in Linear and Nonlinear Analysis

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9410557 Huang It is proposed to continue the study of Floqoet theory for differential delay equations and dynamics of coupled oscillators. In pratice, a system often depends not only on its present states but also on past states. As a result, we are lead to the study of differential delay equations. Linear periodic systems constitute a very important in the field of differential delay equations. Studying Floquet theory of such systems enables one to understand the behavior and structure of their solutions. We plan to extend our study of Floquet theory for linear differential delay equations to broader classes of equations. We would also like to investigate the conditions under which there are no small solutions. Models of coupled osscillators arise naturally from problems in physics and biology. Since in pratice, oscillators can be coupled in many different ways,we can get a variety of equations that describe their motion. To understand the dynamics of these systems, we try to generalize the known results for two coupled oscillators to them. We also would like to study the stabilities of the discrete rotating waves and cluster solutions for globally coupled oscillators. ***