The Normal Forms Of Bifurcations With Codimension And Their Fast And Slow Effects

The theory of normal forms,designed to simplify the analysis of nonlinear problems by an analytical perturbation technique due to Poincare,is a useful tool in the study of bifurcation behavior of nonlinear dynamical systems near equilibria.This thesis presents a computational method and computer programs for computing the normal forms of a general n-dimensional differential equation whose Jacobian matrix evaluated at an equilibrium involves a pair of double-zero and a pair of pure imaginary eigenvalues.Computer programs using a symbolic computer language Maple are developed to facilitate the application of the method.The iterative procedure,which can be automatically executed by a user without knowing computer algebra,is developed for finding expressions of the normal forms and associated nonlinear transformations up to a given order.This thesis then presents the unfolding of the normal form obtained in previous calculations.Loosely speaking,to unfolding a system of ordinary differential equations is to add parameters to the system,with the intention of accounting for the dynamical behavior of all possible systems close to the original one.The theory of unfolding has a wide range of applications in physical problems.Engineers typically unfold a system by adding damping terms and detuning parameters.Since the number of possible unfolding terms can be infinite,it is difficult to work with the unfolded problem.The behavior up to topological equivalence of the normal form with universal unfolding parameters is studied.Roughly speaking,the universal unfolding means the miniversal unfolding with respect to topological equivalence.The number of the universal unfolding terms is equal to the number of the codimension of the bifurcation.The method of asymptotic unfolding is used to compute the universal unfolding parameters of the normal form for a pair of double zero and a pair of pure imaginary eigenvalues.Many processes in nature are characterized by periodic bursts of activity alternating between quiescence and spiking states.The bursting oscillations occur in dynamical models in which variables evolve on two different timescales,i.e.,fast-slow systems.This thesis studies the bifurcations of the fast subsystem that lead to bursting can be collapsed to a single local bifurcation,generally of higher codimension.The main purpose in this thesis is to find the bursting oscillations which are recovered as the slow variables periodically trace a close path in the universal unfolding of the ordinary differential equations.In contrast to the geometric analysis method,the local approach used in this work based on the codimension and unfolding of the degenerate singularities and the fast slow structures in the systems to classify the pattern of bursting.Using the above approach,this thesis systematically studies the universal unfolding and behavior of bifurcations of codimension 3 zero-zero-Hopf bifurcation and Takens-Hopf bifurcation and the possible bursters.The study in this thesis shows that the local approach has several advantages over the geometrical method.In particular,the local theory provides the method to analyze the mechanism of bursting oscillations which can be found using numerical techniques.And moreover,the local theory provides a rational method of classification that indicates how complex a system needs to be in order for it to support bursters of given types.