An algorithm for computing 2D stable and unstable manifolds of hyperbolic fixed points of discrete dynamical systems is shown. With the fact that Jacobian transports derivative along orbit of an invariant manifold, an algorithm for computing 1D manifold is proposed. The mesh point is located with a Prediction-Correction scheme which reduces searching time and at the same time gives rise to a simplified accuracy condition. 2D manifold is computed by covering it with orbits of 1D sub-manifold. A generalized Foliation condition is used to guarantee that 2D manifold is growing equally along orbits of 1D sub-manifold in different directions. Performance of the algorithm is demonstrated with hyper chaotic 3D Hénon map and Lorenz system.

We present a deep improved particle filtering theory which selects samples from comparison of both weight increasing trend and weight.It decides fission and reproduction of samples through statistic analysis of samples.Though approaches increase computing time for trend judging,computing accuracy is improved.

In order to balance growth rate of manifold in all directions and construct global manifold structure of a dynamical system,a radial control factor is adopted to normalize the original dynamical system.Taking radius component of the tangent vector as a standard,this method controls manifold expanding at same speed in all directions.Theoretical analysis and example calculation demonstrate that manifolds before and after normalization have same orbit with the original one,which means their global manifold structures are consistent.Lorenz and Duffing systems are taken for examples to demonstrate effectiveness of the proposed approach.It indicates that the method not only get same effect as geodesic process but also present manifold in discrete flow way,which avoids many complicated boundary value problems.

With definition of DHT(distinguished hyperbolic trajectory) and existing measure function in phase space,a measure function in extended phase space is presented.Existing algorithms with constant accuracy parameters is laborious as high precision is required.In order to overcome this shortage,a variable-step convergence algorithm is proposed.The main idea is to estimate initial region with the help of ISP(instantaneous stagnation points) and adopt variable-step grids to increase efficiency.With theoretical analysis and numerical calculation,an optimal range of key parameter is given.Two-dimensional and three-dimensional Duffing systems are used to test the performance.It shows that convergence route gained by the developed measure function is smooth and stable.And the variable-step convergence algorithm is more efficient.

With embedding theorem proposed by Takens,we study methods for choosing proper delay time in phase space reconstruction of chaotic time series.A united method to incorporate advantages of average displacement and mutual information is put forward.In mutual information calculation we employ binary tree coding to divide and mark grids which makes it implemented easily.We determine layer numbers according to percentages of sparse grid.Numerical experiments of R ssler and Lorenz systems verifies accaracy of the method.