Long-time accuracy for approximate slow manifolds in a finite-dimensional model of balance (Q2644286)

The paper is concerned with the slow singular limit for the equations of a single charged particle in a planar anharmonic oscillator potential under the influence of an external magnetic field described as \[ \varepsilon\ddot{q}-J\dot{q}+\nabla V(q)=0,\tag{1} \] where \(q:\mathbb{R\rightarrow R}^{2}\) and \[ J=\left( \begin{matrix} 0 & -1\\ 1 & 0 \end{matrix} \right) \] is the canonical symplectic matrix. The model (1) has been used by the authors as a toy model for exploring variational high-order approximations to the slow dynamics in rotating fluids. It is characterized by the presence of a fast, nearly harmonic oscillatory motion through a balance between inertia and magnetic terms, \[ \varepsilon\ddot{q}_{\varepsilon}=J\dot{q}_{\varepsilon}, \] and a slow, generically anharmonic motion through a balance between magnetic and potential terms, \[ J\dot{q}_{\varepsilon}=\nabla V(q_{\varepsilon}). \] Since one observes the motion on two separate time scales without explicit separation of fast and slow subsystems, construction of a slow manifold is far from being trivial. Addressing the question of whether the dynamics on the slow manifold can shadow trajectories of the parent system over very long times, the authors demonstrate that the first-order reduced model remains \(O\left( \varepsilon\right) \) accurate over a long \(\varepsilon^{-1}\) time scale. The paper concludes with a brief discussion of the result in the broader context of the authors' original motivation and comments on restrictions of its generalization to large systems.