Study On Periodic Solutions Of Coupled Time-delay Nonlinear Systems Based On Multi-frequency Homotopy Analysis

With the development of science and technology and the deepening of theoretical research,nonlinear problems with time-delay have attracted the attention of researchers in many fields such as mathematics,physics and mechanical engineering.However,it is very difficult and extremely important to solve such problems.At present,the methods for studying time-delay nonlinearity mainly include incremental harmonic method,multi-scale method,homotopy perturbation method and homotopy analysis method.Among them,the homotopy analysis method is widely used in recent years.This method fundamentally overcomes the dependence of perturbation theory on small parameter and has a wide range of applications.In addition,the homotopy analysis method provides a simple method to ensure that the obtained series solution converges,thus obtaining a sufficiently accurate analytical approximate solution.The homotopy analysis method opens up a new idea for solving the nonlinear time-delay problem.In this paper,the multi-frequency homotopy analysis method is used to study the two-degree-of-freedom coupled Duffing system with time-delay,and the approximate solution of the system is obtained.In the first chapter,the research background of time-delay and the research status of nonlinear time-delay systems and homotopy analysis methods are introduced.In the second chapter,at first,the detailed steps of applying the multifrequency homotopy analysis method to solve the general two-degree-of-freedom coupled Duffing system with time delay are given.Then,by solving the constructed nonlinear algebraic equation,the single periodic and period-doubling solutions of the system are obtained.Further,comparing the periodic solution obtained by the multi-frequency homotopy analysis method and the numerical solution obtained by the numerical method,it is found that the approximate solution is in good agreement with the numerical solution.Finally,we apply the Euler method to discretize the time-delay coupled Duffing system,and draw the bifurcation diagram of the discretized system.It is found that the system exist period-doubling solution.In the third chapter,based on the research in the second chapter,we choose different convergence parameter values to discuss the influence of convergence control parameter on the approximate solutions of nonlinear time-delay systems.Therefore,we find that when is greater than a certain value,the deviation between the approximate solution and the exact solution gradually increases.Finally,two research prospects are proposed.First,based on the research of time-delay coupled Duffing system,we can further consider the multi-steady behavior of the system.Second,we only discuss the effects of different convergence control parameter values on the approximate solutions of nonlinear system.We can consider the processing method of functional extremum problem to discuss the influence of convergence control parameter and convergence control function on the approximate solution of nonlinear system.