Computation Of Invariant Measure Based On Meshfree Method

Many scientific and engineering problems often can be studied by the properties of discrete dy-namical systems and these systems often have regularity in the sense of statistics. Some large-scale statistics, such as invariant measure, have an important role for understanding discrete dynamical systems.The maximum entropy method is one of the main methods to calculate the invariant measure. A maximum entropy method based on piecewise linear function to calculate the invariant measure for one dimensional nonlinear transformation S from I to I was proposed by Ding Jiu etc. The-oretical analysis and numerical experiments show that this maximum entropy method is fast and effective. The maximum entropy method was extended to two-dimensional space to calculate the invariant measure by Xu Chunwei etc. They chose the piecewise linear functions on triangular ele-ment as the moment functions in maximum entropy method for calculating the invariant measure in two-dimensional space. In this paper, we choose the basis function based on meshfree method as the moment function in maximum entropy method for calculating invariant measure.The main results of this article are following:(1) For two-dimensional map, we choose the basis function based on meshfree method as the moment functions of the maximum entropy method to calculate the invariant measure. Because these moment functions satisfy the property of partition of unit and approximately locally support property, the Jacobian matrix of the nonlinear equations obtained by the maximum entropy method is band matrix and positive-definite. These properties guarantee that the nonlinear equations have only one solution and can be solved efficiently.(2) The algorithm and details of the maximum entropy method based on meshfree method are presented. Numerical results show that the new maximum entropy method is convergent and effective.(3) The numerical results of the new method are compared with those based on piecewise linear functions. The approximate density function of the new method based on meshfree method has higher accuracy.