Two-dimensional Invariant Measure Computation Based On Maximum Entropy Method

Many scientific and engineering problems often can be studied by the propertiesof discrete dynamical systems and these systems often have regularity in the sense ofstatistic. Some large-scale statistics, such as invariant measure, have an important rolefor understanding discrete dynamical systems.The maximum entropy method is one of the main methods to calculate theinvariant measure. Professor Ding Jiu ect. proposed a maximum entropy methodbased on piecewise linear function to calculate the invariant measure for nonlineartransformation S from [0,1] to [0,1]. Theoretical analysis and numerical experimentsshow that this maximum entropy method is fast and effective.The main results of this paper are listed in the following：(1) The maximum entropy method is extended to two-dimensional space tocalculate the invariant measure. It has certain application potential to do thispromotion, because in science and engineering issues we often meet two-dimensionalor higher dimensional chaotic dynamical systems.(2) Base on finite element method, we define the piecewise linear functions ontriangular element as the moment functions in maximum entropy method forcalculating the invariant measure in two-dimensional space. We prove that suchmoment functions satisfy the property of partition of unit and locally support propertyin the two-dimensional space. These properties ensure that the system of nonlinearequations obtained from maximum entropy method can be solved efficiently. Becausewhen we apply Newton’s iteration method the Jacobian matrices are band matricesand they are positive-definite.(3) We can also find that if we use triangular element, the integral part of thenon-linear equations can be calculated exactly, which is presented in this paper.However, if rectangular element is used, the integral part can only be calculated bynumerical integral method.(4) We also give the convergence analysis and our numerical results show that the new maximum entropy method is efficient for calculating invariant measure intwo-dimensional space.