On The Qualitative Problems Of Some Dynamical Systems And Some Models

In this dissertation,we focus on the qualitative theories of ordinary differential equations,integrability theory,geometry singular perturbation theory and the appli-cations of these theories.Especially,we study the center-focus problems,the algebraic integrability and the analysis on some biological models.In details,first,we complete-ly classify the center of the planar quasi-homogeneous systems,and also provide an algorithm following which we can get all the quasi-homogeneous systems which has a center at the origin with a given degree.Then,we analyze the algebraic integrabili-ty of the FitzHugh-Nagumo system,and also characterize the global dynamics of the FitzHugh-Nagumo system with algebraic invariant surfaces.One typical model which describes the ion flows through the ion channel is the Poisson-Nernst-Planck system.By the geometry singular perturbation theory and qualitative analysis,we derive many interesting consequences on effects of large permanent charges,and explain the mech-anism of one well-known counter-intuitive phenomenon-declining property.The study of this dissertation contains three aspects.In the first part,we consider the center-focus problems of the quasi-homogeneous systems.In[86],the authors classify the center of the quasi-homogeneous systems of weight degree 1,2,3,4.In[56],according to the properties of the quasi-homogeneous systems,the authors provide an algorithm to get all the quasi-homogeneous systems with a given degree,and also get all the quasi-homogeneous systems of degree 2,3 by this algorithm.Then,In[13],[76],[123],the authors get all the quasi-homogeneous systems of degree 4,5 by the algorithm introduced in[56],and analyze the center conditions and the topological structures of these systems.According to these work,the quasi-homogeneous systems of degree 2,4 have no center,only one type of the quasi-homogeneous systems of degree 3 have center,and two types of the quasi-homogeneous systems of degree 5 have center.In the second chapter of this paper,we discuss the center conditions of the quasi-homogeneous systems with higher degree,that is to say,we consider a more general situation.For the any given degree n,first we prove that any quasi-homogeneous systems with even number have no center,and the former results that the quasi-homogeneous systems with degree 2 and 4 have no center is included in this conclusion.Then for the quasi-homogeneous systems with odd degree,we get two conclusions.On one hand,we get the form of the quasi-homogeneous systems which have center at the origin,and classify the center of the systems with the form,then the center of the quasi-homogeneous systems turns to be the centers of corresponding homogeneous systems.And this new strategy simplify the method of classifying the center of quasi-homogeneous systems.According to this result,the center conditions of quasi-homogeneous systems of degree 3 and 5 are the center conditions of linear systems,and by this strategy,we avoid using the complex methods in[13]and[123].On the other hand,we provide an simple algorithm to get all the quasi-homogeneous systems with any degree n,and take the quasi-homogeneous systems with degree 7 as an example to show the feasibility and simplification of this algorithm.In the second part,we focus on the algebraic integrability and the global topolog-ical structures of the three dimensional FitzHugh-Nagumo system.FitzHugh-Nagumo system is one of the simplest models for describing the propagation of nerve impulses along an axon and the excitation of neural membranes.In 1878,Darboux proposed a new method to analyze the algebraic integrability of the polynomial systems,and established the Darboux integrability theory,35,36].Based on the basic work of Brun-s[20],Poincare[111,112],the algebraic integrability of the polynomial systems turns to be the complete classification of the Darboux polynomials.And Poincare mentioned that there is no effective method to get Darboux polynomials.In the work of solving the algebraic integrability of the Lorenz system,Llibre and Zhang[96]proposed an method-characteristic curve method.For the three dimensional FitzHugh-Nagumo system,In the third chapter,we completely classify the Darboux polynomials of the FitzHugh-Nagumo system.However,the difficulty of this problem is the method.The character-istic method is not valid for the FitzHugh-Nagumo system.To overcome this difficulty,we provide a new method to get the Darboux polynomials,and generalize the char-acteristic curve method.We introduce an assistant system of the FitzHugh-Nagumo system,then use the characteristic curve method to get all the Darboux polynomials of the assistant system.At last,based on the relation between the FitzHugh-Nagumo system and the assistant system,the Darboux polynomials of the FitzHugh-Nagumo system has been classified.In the forth chapter,we characterize the global dynamics of the FitzHugh-Nagumo system with invariant algebraic surfaces.The analysis of the global topological struc-tures of the three dimensional systems is difficult.When the three dimensional sys-tems are restricted on the invariant surfaces,the dimension of the system is two.In the former works,the authors always analyze the topological structures of the two dimensional systems.However,in our work,we analyze the topological structures of the FitzHugh-Nagumo system in the Poincare ball,by the blow up technique and the three dimensional complification.When restricted the FitzHugh-Nagumo system on the invariant surfaces,the system is not analytic,combining the topological structure of the restricted system,the singular points of the original system and the topological structure of the invariant surfaces,we get the topological structures of the FitzHugh-Nagumo system.From the phase portraits,we get that there exits only one heteroclinic orbits,so for the PDE equation,there exits bounded travelling fronts.At the last part,we discuss one popular problem-ion channel problem.The excitability and conductibility in biological systems are related to the movements of the ions.And the ion channel provide the place for the ions.Poisson-Nernst-Planck(PNP)system is one of the typical model to describe the ion flows through the ion channel.In this part,we consider the one dimensional steady-state PNP system where the variables are the potential ?.the concentration of the k-th ion ck,and the flux of the k-th ion Jk,x=0,1 are the ends of the channel.The corresponding boundary values are Compared to other models,this model contains the effects of the permanent charge and the boundary values.Notice that the permanent charge distributes on the channel,and it plays an important role on the ion flows.The only data we get from the biological experiments is the current.The changes of the current reflects the changes of the movements of the ions.Indeed,the current I and the fluxes Jk satisfies Then,in this work,we focus on the analysis of the fluxes Jk to explain the reason why the current changes.Eisenberg and Liu in[42]discuss the solutions of PNP model by the geometry singular perturbation theory.Based on this result,there are lots of works on the analysis on the ion channel,such as the authors talk about the effects of the small permanent charge on the ion channel in[68].In fact,compared to the concentration of the ions at the boundary condition Lk,Rk,the permanent charge is large.Then in this work we discuss the effects of the large permanent charges on the ion channel.In the fifth chapter,we get the solutions of the system with the large permanent charge,and by analyzing the solutions,we get some properties of the ion channel,such as the saturation of the current,and the effects of the boundary values.This is the first time to get the property of the saturation of the current.At the last chapter,we explain one intrinsic property of the ion channel-declining phenomenon.This method is observed in the biological experiments.Up to now,there has been no work on analyzing the mechanism of this property.Our results reflects this phenomenon,and we also analyze the mechanism of this property.And this result has its practical significance.