Structure-preserving algorithms for the Duffing equation

In this paper, the dissipative and the forced terms of the Duffing equation are considered as the perturbations of nonlinear Hamiltonian equations and the perturbational effect is indicated by parameter \varepsilon . Firstly, based on the gradient-Hamiltonian decomposition theory of vector fields, by using splitting methods, this paper constructs structure-preserving algorithms (SPAs) for the Duffing equation. Then, according to the Liouville formula, it proves that the Jacobian matrix determinants of the SPAs are equal to that of the exact flow of the Duffing equation. However, considering the explicit Runge--Kutta methods, this paper finds that there is an error term of order p+1 for the Jacobian matrix determinants. The volume evolution law of a given region in phase space is discussed for different algorithms, respectively. As a result, the sum of Lyapunov exponents is exactly invariable for the SPAs proposed in this paper. Finally, through numerical experiments, relative norm errors and absolute energy errors of phase trajectories of the SPAs and the Heun method (a second-order Runge--Kutta method) are compared. Computational results illustrate that the SPAs are evidently better than the Heun method when \varepsilon is small or equal to zero.