Geometric Numerical Integration of Liénard Systems via a Contact Hamiltonian Approach

Abstract

Starting from a contact Hamiltonian description of Liénard systems, we introduce a new family of explicit geometric integrators for these nonlinear dynamical systems. Focusing on the paradigmatic example of the van der Pol oscillator, we demonstrate that these integrators are particularly stable and preserve the qualitative features of the dynamics, even for relatively large values of the time step and in the stiff regime.

The flow method for the Baker-Campbell-Hausdorff formula: exact results

“…In particular, we have focused on Lie algebras that can be expressed as algebras of contact Hamiltonian functions endowed with the Jacobi bracket. Our original motivation for this investigation was to improve the error analysis of contact splitting numerical integrators [14,48], which we presented in the last section for the special example of the damped harmonic oscillator. Having explicit modified Hamiltonians allows fine estimates on the numerical errors and a precise a priori comparison of the performance of different numerical algorithms.…”

“…Moreover, the analysis and use of symplectic and contact integrators is a very active and prolific field e.g. in mechanics [3,14,17,25,47,48], general relativity [43,44] and plasma physics [34,35], and thus we consider that our results can be helpful in these contexts as well.…”

“…We prove the usefulness of our method using some interesting examples arising from the recent literature in contact geometry and Hamiltonian systems, namely, the Heisenberg algebra [26], the contact Heisenberg algebra [12], the quadratic symplectic and the quadratic contact algebras, and the complexification of su (2). Furthermore, we also include an analysis of contact splitting numerical integrators [14,48] for the damped harmonic oscillator [11]. More precisely, in this example our method will be used in order to obtain a priori estimates of the numerical errors for different splitting integrators.…”

Preface to: Differential Geometry: Structures on Manifolds and Their Applications

“…One can distinguish several topics that have been investigated in the papers of this Special Issue. In addition to the proposed keywords we have already presented in the beginning, we also have some specific ones that appeared in the papers [1][2][3][4][5][6][7][8][9][10][11] that we emphasize in Table 2.…”

Symplectic Dynamics and Simultaneous Resonance Analysis of Memristor Circuit Based on Its van der Pol Oscillator

“…Then, Qin proposed a multi-step symplectic method and applied it to the numerical analysis of wave problems [30], Forest and Ruth constructed the symplectic maps for nonlinear motion of particles in accelerators [31], Yoshida modified non-existence of first integral by symplectic integration methods [32], Cieśli ński compares several discretizations of the simple pendulum equation in a series of numerical experiments and puts forward a new numerical scheme of improved discrete gradient method [33]. Subsequently, the development, analysis and use of various numerical solution algorithms for differential equations concerned in geometric numerical integration [34][35][36][37][38][39]. These algorithms preserve the geometric or qualitative properties of the exact solutions, such as an integral or symmetry, or preservation of a differential invariant such as symplecticity or phase space volume.…”