Finding Periodic Solutions Of Delayed Differential Equations By Particle Swarm Optimization

In recent years,the theory of time-delay differential equations has made great progress.It has achieved important results in many aspects such as the basic theory of solution,stability theory,periodic solution theory,and operator theory.The general form of differential equations with delay is a functional differential equation.Its initial condition is no longer a state at a certain moment,but a state within a certain period of time,that is,its solution is defined in an initial function (?)(?),(-????0).Therefore,the periodic solution of the delay differential equation should satisfy x_T((?))=(?),where T is the period.It is very important to find an effective initial function to locate the periodic solution.However,the traditional method to calculate the periodic solution of the delay differential equation usually requires a large amount of calculation.Newton method is an effective method to solve delay differential equations.However,in order to obtain better calculation accuracy,Newton's method needs to spend a lot of calculations,especially when the solution domain of the periodic solution is small or the periodic solution is unstable.In addition,the application of Newton's method also requires a priori better estimate of the initial function to ensure convergence.In order to solve these problems,the particle swarm algorithm is used to approximate the periodic solution of the delay differential equation,which can expand the scope of the solution and obtain high accuracy.The main content of this article is as follows:The first chapter briefly introduces the history of differential equations and the origin,development and research status of the differential equations with delays.It focuses on the introduction of several conventional methods for solving periodic solutions of delay differential equations.The definition of the differential equations for delays is also introduced,the nature and the difference with the ordinary differential equations,and the simple nature of periodic solutions of delay differential equations.It is emphatically described that the delay differential equation is an infinite dimensional system,and the influence of time delay on the dynamic properties of the system.In the second chapter,the source and implementation steps of the traditional particle swarm optimization algorithm and the specific process of applying the particle swarm algorithm to solve the periodic solution of the delay differential equation are introduced.It is pointed out that the traditional particle swarm optimization algorithm has the advantages of easy implementation,high precision,fast convergence and so on.The example shows that traditional particle swarm optimization has significant advantages compared with Newton's method and overcomes the disadvantage of Newton's high search range for the initial function.Even for a given search range,high-precision periodic solutions can be obtained.Even in cycles Unknown circumstances can still get better results.In the third chapter,the origin and steps of orthogonal particle swarm optimization are introduced,and periodic solutions of delay differential equations are calculated by orthogonal particle swarm optimization.It is pointed out that the orthogonal particle swarm optimization algorithm has a search compared to the traditional particle swarm optimization algorithm.With fewer algebras and faster evolution,orthogonal particle swarm algorithms can perform well even in networks with time-delay.