Application Of Singularity Theory In Bifurcation Analysis Of Nonlinear Dynamics

With the development and progress of science and technology,bifurcation theory and methods of the dynamical systems are the important parts of nonlinear dynamics and widely applied in the eigneering fields.In these years,many methods for bifurcation analysis of the dynamical systems have been proposed.Singularity theory has been widely applied as qualitative bifurcation analysis method,we can use the uniform and clear method to deal with various complex bifurcation problem,and establish comprehensive links between system dynamic behavior and system parameters with singularity theory.But up to now,singularity theory is relatively perfect with one state variable and one bifurcation parameter,but singularity theory faces the challenge with multiple state variables and multiple bifurcation parameters.In this paper,in the case of not decreasing the state variables and bifurcation parameters,singularity theory is investigated with two state variables and three or more bifurcation parameters,and applied it to the actual engineering system.The main contents of this dissertation are as follows(1)Singularity theory is developed base on the present singularity theory.Firstly,Restricted tangent space is calculated from the bifurcation equation.Secondly,to truncate the bifurcation equation,get strong equivalent and the simplified form of polynomial,and simplified as normal form under the case of nondegenerate.In general,the linear terms about expression of restricted tangent space are missed in expression of normal form.Finally,the codimension of normal form is obtained in restricted tangent space,and the codimension of restricted tangent space is got in function space,based on all codimension of normal form in function space,the universal unfolding of normal form can be obtained.(2)Based on the theoretical results of the second chapter,the singularity theory is utilized to investigate the static bifurcation of a circular truss antenna structure with two detuning parameters and a thermal excitation under the case of 1:2 internal resonance.Firstly,the bifurcation equations of the circular truss antenna structure under the case of 1:2 internal resonance are introduced.Secondly,Restricted tangent space is calculated obtained from the bifurcation equation,and truncated the bifurcation equations,and got strong equivalent and the simplified form of polynomial,and simplified as normal form under the case of nondegenerate.Finally,the universal unfoldings of the bifurcation equations with the codimension-4 are obtained,and the transition sets in the 2d and 3d parameter plane are depicted.The results show that the number of solutions of the bifurcation equations is different in different regions of the plane and the stereogram of the transition set.(3)Based on the theoretical results of the second chapter,the singularity theory is utilized to investigate the static bifurcation of the symmetric cross-ply composite laminated plates with two detuning parameters,a in-plane excitations and a transverse excitation under the case of 1:1,1:2 and 1:3 internal resonance.Firstly,the bifurcation equations of the symmetric cross-ply composite laminated plates are introduced.Secondly,restricted tangent space is calculated obtained from the bifurcation equation,and truncated the bifurcation equation,got strong equivalent and the simplified form of polynomial,and simplified as normal form under the case of nondegenerate.Finally,the universal unfoldings of the bifurcation equations with the codimension-4 are obtained,and the transition sets in the 2d and 3d parameter plane are depicted.The results show that the number of solutions of the bifurcation equations is different in different regions of the plane and the stereogram of the transition set.The universal unfoldings of the bifurcation equations under the case of 1:1 and 1:2 internal resonance are different,and the universal unfolding of the bifurcation equations under the case of 1:1 internal resonance are included in the universal unfolding of the bifurcation equations under the case of 1:3 internal resonance.(4)Based on the theoretical results of the second chapter,the singularity theory is utilized to investigate the static bifurcation of the composite laminated piezoelectric rectangular plate structure with two detuning parameters and a transverse excitation under the case of 1:2 internal resonance.Firstly,the bifurcation equations of the circular truss antenna structure under the case of 1:2 internal resonance are introduced.Secondly,Restricted tangent space is calculated obtained from the bifurcation equations,and truncated the bifurcation equations,and got strong equivalent and the simplified form of polynomial,and simplified as normal form under the case of nondegenerate.Finally,the universal unfoldings of the bifurcation equations with the codimension-4 are obtained,and the transition sets in the 2d and 3d parameter plane are depicted.The results show that the number of solutions of the bifurcation equations is different in different regions of the plane and the stereogram of the transition set.