The Research On Oscillation Of Two Kinds Of Nonlinear Delay Differential Equations

At present,the reason why the solutions and properties of nonlinear delay differential equations solutions are hot issues that many phenomena in different sciences may be described by these equations,for example,the existence,the global stability and attractivity,oscillations,hopf bifurcation,chao and so on.Some researchers devoted to studies whether the numerical solutions of nonlinear delay differential equations preserve its behaviors.In this paper,we mainly consider the oscillation of solutions on two kinds of significant equations,one of which has been generally used to describe the population dynamics and tumour growth called Gompertz equation.Another is a nonlinear delay Lotka-Volterra differential equation generalized from a single species population model.In the first chapter,some studies on the oscillation of solutions of nonlinear differential equations since 2000 are introduced.As well as we have general introduction of relevant researches related on the two kinds of equations.The second chapter of this paper concerns with the oscillation of analytical solutions and numerical solutions of a nonlinear delay Gompertz differential equation with one delay.By invariant transformation for oscillation,it can turn the nonlinear equation into linear equality so that the corresponding discrete equation with the linear ?-method follows.Then some suffificient conditions for the numerical solutions of the equation were derived from the theoretical analysis.The non-oscillatory behaviors of numerical solutions are also analyzed and demonstrated the positive non-oscillatory numerical solutions tend to the equilibrium.Lastly,numerical examples are given to test our theoretical results.In the third chapter,a further discussion on the oscillation of two nonlinear delay Gompertz differential equations was proposed.We linearized the nonlinear delay Gompertz differential equations based on the Taylor formula.Moreover,some suffificient conditions under which the numerical solutions of the equation were acquired by linear ?-method and the oscillation theory.We give some numerical experiments to verify the theoretical results above.In the last chapter,we mainly consider the oscillation of a nonlinear delay Lotka-Volterra differential equation.Firtsly,we obatained the matching difference schemes applying the explicit Euler method,the implicit Euler method and the linear ?-method to linearizd differential equation,respectively.On one hand,by analyzing the several matching difference schemes,several suffificient conditions were given.On the other hand,we proved that its positive non-oscillatory numerical solutions are asymptotically stable.Finally,to verified obatained resultes we conduct corresponding numerical experiments.