Unstable Periodic Solutions In Chaotic Systems

In this paper,we use Taylor expansion to discretize the dynamical system in order to get the pseudo periodic solution by simulating this discrete system.Then we optimize the results by the fastest descent method,and get the periodic solution of the system.We find many interesting numerical results such as periodic doubling phenomenon,which reflect the rich traversal characteristics of dynamical system towards chaos.We choose two dynamical systems including Lorenz system and Duffiing system to search the periodic orbits.Three tasks need to be done in order to accomplish this goal in Lorenz system.The first step is to discretize the system and transform the continuous Lorenz system into the discrete dynamic system.Expanding the Lorenz system to three order derivative,we get the iterative form of the Lorenz system.The second step is to iterate over the initial value in a certain range for above iterative form.If one of the values obtained through the iteration satisfies the conditions we required,we construct a vector formed by initial point and a series of points iterated from this initial point as following(x1,y1,z1,,x2,y2,z2,…,xn,yn,zn)T.This vector is chosen as a pseudo periodic solution.The third step is to optimize the pseudo periodic solution.We set a restrictive condition and construct a target function for the pseudo periodic solution.Then we choose the fastest descent method to optimize the pseudo cycle solution when the program is terminated until the target function satisfied restrictive condition.In this time,the new pseudo periodic solution obtained by optimization is identified as periodic solution.We calculate the periodic solution through the above three steps.Compared with other methods,our method is more stable and more applicable.Through the above method,we calculate the simple periodic solution,the super large periodic solution and the data of the multiple periodic solution for two systems,and show the rich characteristics of chaos,which helps us research and understand the chaos.