Studies On Quadratic Generalized Hamiltonian System Defined On (?)o(3)~* And Dynamics Of Stokes Flows

(?)o(3)~*is the three-dimensional dual space of the third-order Lie algebra (?)o(3).The Lie algebra is composed of all 3 x 3 real antisymmetric matrices.The Lie group corre-sponding to (?)o(3)is (?)o(3)=(A ? GL(3;R):ATA = I3;det(A)= 1}(also known as the third-order special orthogonal matrix group).The generalized Hamiltonian system defined on (?)o(3)~* has a wide range of physical backgrounds,including Gyrostat attitude dynamic model with several rotors in torque-free motion,classical water molecular dy-namics model;optimal control problem on Lie group and incompressible fluid dynnamics model in fluid dynamics.In this master paper,we mainly studies the quadratic generalized Hamiltonian sys-tem of Lie algebra (?)o(3)~* and a class of steady confined Stokes flow model.First,based on the paper[7]for the classification of quadratic generalized Hamiltonian systems on (?)o(3),the dynamics of a generalized Hamiltonian system with three parameters are studied,and problems related to equilibra,stability and bifurcation properties are analyzed.All pos-sible phase portraits structure are obtained.Second,the general results obtained are combined with the perturbation theory and the Melnikov method to study the dynamic properties of the steady confined Stokes flow model.The existence of periodic orbits and homoclinic orbits for the Stokes flow model is discussed.Third,a special kind of STF flow is obtained for the Stokes flow model in the paper[8].This paper further discusses the orbital bifurcation and phase portraits of this type of STF flow.