Symplectic-Structure-Preserving Uncertain Differential Equations

Xiuling Yin,Yanfeng Shen,Xiulian Gao,Yanqin Liu,Jinchan Wang

doi:10.3390/sym13081424

Abstract

Uncertain differential equations are important mathematical models in uncertain environments. This paper investigates uncertain multi-dimensional and multiple-factor differential equations. First, the solvability of the equations is analyzed. The α-path distributions and expected values of solutions are given. Then, structure preserving uncertain differential equations, uncertain Hamiltonian systems driven by Liu processes, which possess a kind of uncertain symplectic structures, are presented. A symplectic scheme with six-order accuracy and a Yao-Chen algorithm are applied to design an algorithm to solve uncertain Hamiltonian systems. At last, numerical experiments are given to investigate four uncertain Hamiltonian systems, which highlight the efficiency of our algorithm.

Highlights

An uncertain differential equation is a kind of mathematical model to describe physical processes in uncertain environments [1,2,3]
Probability theory is suitable for big data analysis, while uncertain theory is appropriate for the situation of fewer data information
In this paper, based on the Liu process, we present uncertain Hamiltonian system models

Summary

Introduction

An uncertain differential equation (abbreviated as UDE) is a kind of mathematical model to describe physical processes in uncertain environments [1,2,3]. Since stochastic Hamiltonian systems possess symplectic geometric structures and symmetries, they are used to describe approximately conservative physical processes in some cases. In this paper, based on the Liu process, we present uncertain Hamiltonian system models. We present a kind of symplectic-structure-preserving UDE, an uncertain Hamiltonian system. It is a good supplement to the model of stochastic Hamiltonian systems; We analyze the well-posedness and α−path distributions of general 2m−dimension and n−factor UDEs. Naturally, under the Lipschitz continuity and linear growth condition, the uncertain Hamiltonian system is well-posed. An algorithm is given to compute α-path solutions and expected values of uncertain Hamiltonian systems. In Appendix A, we list a paradox of stochastic Hamiltonian systems in one case

Multi-Dimensional and Multiple-Factor UDE

Numerical Results

Conclusions