On exact discretization of the cubic-quintic Duffing oscillator

TL;DR

This paper develops exact discretizations for the cubic and quintic Duffing oscillators, preserving their Hamiltonian structure, using intersection theory to construct finite-difference equations that maintain integrability.

Contribution

It introduces a novel application of intersection theory to derive exact finite-difference schemes for classical integrable systems like the Duffing oscillator.

Findings

Exact discretizations preserve Hamiltonian structure.

Finite-difference equations share the form of the original Hamiltonian system.

Method applicable to other integrable systems.

Abstract

Application of intersection theory to construction of n-point finite-difference equations associated with classical integrable systems is discussed. As an example, we present a few exact discretizations of one-dimensional cubic and quintic Duffing oscillators sharing form of Hamiltonian and canonical Poisson bracket up to the integer scaling factor.