The Research Of Solving Solutions And Stability Of Some Kinds Of Nonlinear Differential Systems

With the development of computer technology,more accurate mathematical models have been established in the fields of physics,biology,economics,medicine,psychology and population.The mathematical model includes time delay differential system,differential difference system,partial differential system and ordinary differential system.The delay differential system is generally divided into retarted type time delay differential system,advanced type time delay differential system,hybrid type delay differential system and neutral delay differential system.The solution of mathematical model and the study of stability are of great value to explain the theoretical and practical significance of mathematical model.In this paper,we give the auxiliary equation with the method of combining the function transformation,via the aid of symbolic computation system of Mathematica(or of Maple),through several steps,we studied the solution of the several kinds of nonlinear differential systems,the phase diagram analysis and stability problems,and we get some new conclusions.1.We study the solution and stability problems of hybrid multi-delay differential system with forced items..Step1.We give the exact solution of the hyperbolic(trigonometric)function type auxiliary equation.Step2.By used the properties of hyperbolic functions and trigonometric functions,the form solution of a hybrid multi-time-delay differential system with forced entry is selected.Step3.Through the hyperbolic(triangle)and functional auxiliary equation form solution,the solving solutions problem of a hybrid multi-delay differential system with forced items is changed to the solving solutions problem of nonlinear overdetermined differential equations.Step4.By used the symbolic computation system Mathematica,the solution of the nonlinear overdetermined differential equations are solved,and constructed the exact solution of the hybrid multi-time-delay differential system with forced items.Step5,Mathematica is used to analyze the stability of the system.2.Used the method of Lyapunov,the singularity classification,phas ediagram analysis and stability problem of two kinds of nonlinear differential equations are analyzed.Through the functions,the generalized modified Dullin-Gottwald-Holm(D-G-H)equation and the Degasperis-Procesi(D-P)equation are changed into ordinary differential equations.On this basis,the following two tasks are carried out.(1)Used dynamic method,we analyze the singularity classification,phase diagram analysis and stability problem of two kinds of nonlinear differential equations.(2)The solving solution problem of two nonlinear differential equations is studied by the method combining the first integral with the auxiliary equation.Step1.Used the first integral,the solving solution of the generalized modified D-G-H equation and D-P equation is changed into the solving solution of several solved nonlinear ordinary differential equations.Step2.By the conclusions of Backlund transformation and the related nonlinear ordinary differential equations,the new exact solutions of infinite series consisting of the Riemann ? function,Jacobi elliptic function,hyperbolic function and trigonometric functions of two kinds of nonlinear differential equations are constructed.3.Used the conclusions of one of the nonlinear second order ordinary differential equations,two kinds of solutions of any order nonlinear evolution equations,and the stability problem are researched.Step1.Used the method of combining function transformation with the first integration,the new solutions,the Backlund transformation and the nonlinear superposition formula of one kind of nonlinear second order ordinary differential equations are given.Step2.Through function transformation,the solving solution problems of the generalized Camassa-Holm equation and the generalized Fitzhu-Nagumo equation are transformed into solving solution problems of nonlinear ordinary differential equations.Step3.By choosing the form solution of the nonlinear ordinary differential equation and a kind of nonlinear second order ordinary differential equation,the solving solution problem of the nonlinear ordinary differential equation is changed into the solving solution problem ofnonlinear algebraic equations.Step4.By using the symbolic computation system Mathematica,solve the solution of the nonlinear algebraic equations.Step5.According to the step 1 to step4,constructs many kinds of new infinite sequences solutions consisting of Jacobi elliptic function,Riemann ?function,hyperbolic functions,trigonometric functions and rational function of the generalized Camassa-Holm equation and generalized Fitzhugh-Nagumo equation.These solutions include the optical soliton solution,peak soliton solution and the tight soliton solutions.Step6.Mathematica is used to analyze the properties of the solutions.