Bifurcations In Two Ordinary Differential Equation Models

There exist many systems that evolve over time in both nature and human society,such as the motion of the planets in the solar system,the spread of infectious diseases among people,and the development of an economy during industrialization.Such evolutions can be described by differentials,and such systems can be modeled by equations containing differentials or derivatives of a function,i.e.,differential equations.For example,the Van der Pol’s equation modeling an electrical circuit with a triode valve,the SIR(Susceptible-InfectedRecovered)model describing the spread of infectious diseases,and the Lorenz equation for air convection.By investigating the properties of solutions of differential equations,we can explain some nature laws and social phenomena,predict the evolution trend according to current state,or process optimal or better strategies to control the development of a state.Approximation and omitting are generally unavoidable during the process of modeling complex real problems by ordinary differential equations.Thus,for the effectiveness of the model,ordinary differential equations is wanted to keep some properties under small perturbations.However,some ordinary differential equations can not keep their qualitative properties under small perturbations,for example,the ordinary differential equations with either nonhyperbolic equilibrium or nonhyperbolic periodic orbit.Therefore,we need to consider the bifurcations,which can display all possible situations for the real problem.This thesis study dynamical properties of two models of ordinary differential equations.The first model describe the aggregation of tuna around two FOBs(floating objects).The second model studies the influence of the environmental externalities and the industrialization of a two sector economy.The second chapter is devoted to methods for discussing global orbital structure of a planar autonomous system based on its local qualitative properties and some basics of bifurcation theory.In the third chapter,we study an ODE model about the distribution of tuna.Robert et.al.reduced the existence of equilibria to the distribution of zeros of a cubic polynomial and a quartic polynomial,found that the system have as most five equilibria,and discussed the stability of equilibria by their eigenvalues.Since it is difficult to compute eigenvalues by the complex expressions of zeros of polynomials,we utilize the monotonic intervals to determine the location of zeros and signs of eigenvalues qualitatively.Our method enables us to find two more equilibria than known results and complete the qualitative analysis of all equilibria effectively.Then,we display all bifurcations at equilibria,including saddle-node bifurcation,pitchfork bifurcation and two bifurcations of co-dimension 2.One bifurcation of co-dimension 2is one-dimensional cusp bifurcation.The other one is an incomplete unfolding of a one-dimensional degenerate system of co-dimension 4,of which only a part of co-dimension 2 can be unfolded within the system.Next,we prove nonexistence of closed orbits,homoclinic loops and heteroclinic loops,and present the phase portraits of the system.Finally,according to the dynamical properties,we find that the distribution of tuna under two FOBs will tend to and keep in equilibria,and present final distributions in all cases and the critical conditions for transitions of cases.In the fourth chapter,we study an ODE model which describes the industrialization of open economies.Because of the involved fractional powers,its interior equilibria are hardly found by solving a transcendental equation.Antoci et.al.discussed the existence of interior equilibria in some cases by upper and lower bounds of a extreme value of the transcendental equation,and studied the qualitative properties of hyperbolic equilibria by their eigenvalues.We polynomialize the expression of extreme value by eliminating the transcendental term,so that we can obtain the number of interior equilibria in all cases.However,it is more difficult to analyze qualitative properties of nonhyperbolic equilibria with unknown coordinates.We study qualitative properties of nonhyperbolic equilibria by discussing the stability on their center manifolds.For global properties,we prove that the system has two equilibria at infinity,and analyze their qualitative properties by Briot-Bouquet transformation.Moreover,the system is not defined at the origin,and the extended system is not differentiable at the origin.Since the origin of the extended system is an equilibrium,we analyze the orbital structure near the origin by the GNS method.Then we display all bifurcations of equilibria such as saddle-node bifurcation,transcritical bifurcation and a saddle-node-transcritical bifurcation of co-dimension 2.Furthermore,we prove nonexistence of closed orbits,homoclinic loops and heteroclinic loops,exhibit global orbital structure of the system,and presented global phase portraits.Finally,we simulate the orbits numerically to verify the the results obtained by qualitative analysis,and find that a sustainable way of development is available only if the carrying capacity of nature resource belongs to a bounded interval and the negative impact of industria production is lower than a critical value.