Normal Forms Of Dynamic Systems With Equilibrium Manifolds And Some Applied Problems

This thesis studies the normal form theory of systems with equilibrium manifolds,and does some research of related application in non-smooth systems.Equilibrium Manifolds in vector fields are manifolds composed of equilibria. Dy-namic systems with equilibrium manifolds are usually degenerative. Recent years somemodels possessing equilibrium manifolds have been found from various fields such asbiology and power markets. Comparing with an isolated equilibrium point, equilib-rium manifolds have some different properties. Particularly, stability of the systemmay change along the equilibrium manifold, leading to bifurcation without parameters.Bifurcation without parameters is a non-trivial extension of bifurcation with parame-ters, that has important applied background. Its classification problem may be reducedto computing the normal form of a point on the equilibrium manifold.In the beginning of this thesis, the general form of planar systems with an equi-librium curve is deduced. According to the linearized matrixes, the systems are clas-sified into three categories: non-degenerative, first-order degenerative and higher-orderdegenerative cases. For the non-degenerative case, the orbital structure near the equi-librium curve is directly derived, and the second order normal form is established. Forthe degenerative case, due to the zero linearized matrix the system cannot be treatedwith the traditional normal form theory. Therefore the simplest normal form theoryis introduced in this thesis. It can be regarded as generalization from the traditionalnormal form theory, by using Lie Algebra to solve the normal forms of ODEs. For thefirst-order degenerative case, the simplest normal forms are completely obtained. Thatmeans an approximate classification of structures near the equilibrium curve is attained.For some special higher-order degenerative cases, their simplest normal forms are alsoattainable, as illustrated in this thesis.The global dynamic problems of equilibrium manifolds are more complicated.Two applied problems in non-smooth systems related to equilibrium manifolds aregiven. The first one is the affection of transmission line limits of bidding problemin power market. Considering of the transmission line limits, the model of bidding pro- cess changes. The system may contain an equilibrium manifold in some circumstance.The system is transformed to an iteration process near the equilibrium manifold. Thesecond model is about differential automation, which is a hybrid dynamic system com-posed of ODEs whose right-sides are piecewise constant. This thesis extends the linearmodels to the non-linear ones, derives the existence theorem of the unique periodictrajectory. And finally a preliminary modeling process on memory problems is given.