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.
Highlights
Liénard systems are a class of two-dimensional nonlinear dynamical systems that exhibit a stable limit cycle
The paper is organised as follows: in Section 2 we provide a Hamiltonian formulation of Liénard systems based on contact Hamiltonian dynamics, and in Section 3 we introduce a new class of explicit geometric integrators for these systems that are naturally derived by splitting the Hamiltonian
In this work we have proposed a novel approach to the geometric numerical integration of an important class of nonlinear dynamical systems, that is, Liénard systems
Summary
Liénard systems are a class of two-dimensional nonlinear dynamical systems that exhibit a stable limit cycle. In [6], the classical Bateman trick for the harmonic oscillator was extended to the van der Pol oscillator and further generalised to all Liénard systems with a quadratic potential Both these approaches involve a four-dimensional phase–space, and in both the authors have focused on the perturbation theory and have not explored the consequences of the Hamiltonisation for the numerical integration. From yet another perspective, in [7] the authors have presented various splitting schemes for “conditionally linear systems” (these include Liénard systems) which, not geometric, are based on the standard splitting schemes for symplectic Hamiltonian systems, and showed good qualitative and quantitative results. For additional information on the classical approach to the analysis of Liénard systems we refer to [23]
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