Difference Schemes for Differential Equations with a Polynomial Right-Hand Side, Defining Birational Correspondences

Abstract

This paper explores the numerical intergator of ODE based on combination of Appelroth’s quadratization of dynamical systems with polynomial right-hand sides and Kahan’s discretization method. Utilizing Appelroth’s technique, we reduce any system of ordinary differential equations with a polynomial right-hand side to a quadratic form, enabling the application of Kahan’s method. In this way, we get a difference scheme defining the one-to-one correspondence between the initial and final positions of the system (Cremona map). It provides important information about the Kahan method for differential equations with a quadratic right-hand side, because we obtain dynamical systems with a quadratic right-hand side that have movable branch points. We analyze algebraic properties of solutions obtained through this approach, showing that (1) the Kahan scheme describes the branch points as poles, significantly deviating from the behavior of the exact solution of the problem near these points, and (2) it disrupts algebraic invariant variety, in particular integral relations describing the relationship between old and Appelroth’s variables. This study advances numerical methods, emphasizing the possibility of designing difference schemes whose algebraic properties differ significantly from those of the initial dynamical system.

Keywords

• finite difference method
• dynamical systems
• computer algebra
• birational map
• Cremona transformation