Symplectic-Structure-Preserving Uncertain Differential Equations

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 alpha-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.