Birational maps from polarization and the preservation of measure and integrals

Abstract

The main result of this paper is the discretization of second-order Hamiltonian systems of the form ẍ = –K ∇W (x), where K is a constant symmetric matrix and W : ℝn → ℝ is a polynomial of degree d = 4 in any number of variables n. The discretization uses the method of polarization and preserves both the energy and the invariant measure of the differential equation, as well as the dimension of the phase space. This generalises earlier work for discretizations of first order systems with d = 3, and of second order systems with d = 4 and n = 1.

Measure preservation and integrals for Lotka–Volterra tree-systems and their Kahan discretisation

“…Example 6. The expression (12) is a Darboux function of the map (28) which has cofactor J, the Jacobian determinant (31).…”