Research On The Stability Of Dynamics Using Compound Matrices And Symmetric Groups

The focus of this dissertation is to study issues related to stability of dynamicsystems on the base of compound matrices and symmetric groups, especially the bi-furcation problem. Compound matrices and symmetric groups are not only two ofthe most important concepts in algebra, but also are widely applied in the research ofstability of dynamic systems. Compound matrices are the wonderful tools to investi-gate the stability of matrices, and are the bridge between low and high dimensionaldynamic systems, which has the unique charm in dealing with some problems aboutthe global stability, the existence of periodic orbits and so on. Symmetric groups arethe theoretic foundation for studying symmetric dynamic systems. Based on the grouptheory, M. Golubitsky et al. obtained the Hopf bifurcation theory for the symmetricordinary differential equation (ODE). And J. Wu extended it to the symmetric delayeddifferential equation (DDE).The bifurcation problem is one of the best important subjects in dynamic sys-tems. Motivated by M. Li et al. who used compound matrices to judge the stability ofmatrices and the existence of Hopf bifurcations in continuous dynamic systems, weobtained the effective method to judge the Schur stability of matrices on the base ofspectral property of compound matrices, which can be used to judge the asymptoticalstability and the existence of Hopf bifurcation of discrete dynamic systems.The DDE describes the evolution systems depending on both the present stateand the past state, which has wide applications in ecology, physics, chemistry, engi-neering, information science, economics and physical science and so on. Since it isa very difficult task to research the infinite dimensional nature caused by the delays,the deep investigation of the bifurcation problem in DDE needs not only the theoryof the classical differential equations, but also the knowledge of algebra, functional,topology one. Some general theorems are available about the Hopf bifurcation forDDE, for example, the n dimensional Bendixson theorem, the symmetric Hopf bifur-cation theorem for DDE and so on. However,applications of these general theoremsto concrete dynamic models often involve the following difficult tasks: (I) calculationof the second additive compound matrices, the case of 3×3 and 4×4 matrices is easily resolved; (II) distribution of zeros in characteristic equations which are usu-ally transcendental and depend on parameters; (III) discussion on certain generalizedeigenspaces of the infinitesimal generator of continuous semigroups for a linearizedsystem; (IV) analysis on the symmetry of systems. To resolve some of the aboveproblems, the following issues are organized:(1) The simple and direct method to calculate the second additive compoundmatrix of any n×n matrix is obtained, which can be used to calculate the compoundequation of any n dimensional dynamic systems. Applying the calculation methodcoupled with n dimensional Bendixson theorem of Muldowney et al. and global Hopfbifurcation of J. Wu, a class of BAM neural network models with delays are discussed.The conclusion that there is no nonconstant periodic solutions with some special pe-riod is got. And global existence of periodic solutions are established.(2) The eigenvalues and eigenvectors of circle block matrix are exhibited, whichsolve the problem on generalized eigenspaces of linearized systems for all Dn-symmetric systems. Applying it coupled with symmetric Hopf bifurcation theoremof J. Wu, the stability and Hopf problem of a class of n-coupled BVP oscillators mod-els with delays and a class of coupled neural oscillators network models with delaysare investigated. Common Hopf bifurcations occurring at the zero equilibrium as thedelay increases are exhibited. The equivariant Hopf bifurcations are obtained andanalyzed, and their spatio-temporal patterns are demonstrated.(3) Using the presentation theory of symmetric groups, the symmetry of concretemodels discussed is investigated. Depicting the fixed point subspaces of isotropicsubgroups of symmetric groups, the corresponding periodic solutions patterns are de-scribed.(4) By means of space decomposition, the distribution of zeros of the character-istic equations is subtly discussed. Hence, sufficient conditions are derived to ensurethat the zero solution of models are asymptotically stable.