Geometric Numerical Integration of Liénard Systems via a Contact Hamiltonian Approach

Round 1

Reviewer 1 Report

Report on the paper Geometric numerical integration of Linnard systems via a contact Hamiltonian approach written by

F. Zadra, A. Bravetti and M. Seri

The paper under review is a very nice contribution towards better under-
standing the properties of some non-linear dynamical systems, more precisely
the Lienard systems and its basic example the van der Pol oscillator. The
diculties become from the nonlinear nature of the problem and the lack of
geometric structures. The authors propose a treatment of the Lienard sys-
tems from the point of view of contact Hamiltonian systems. They apply the
Hamilton  ows on contact manifolds to geometrically integrate the Lienard
systems which become a new approach to attack the problem.

In conclusion, I think that the paper is new, very interesting and opens
tractable new areas for possible investigations towards deeper understanding the analysis and geometry of the non-linear dynamical systems, the Lienard systems. I admit that the investigations and the results in the presented paper could have further essential applications of great interests in mathematics and physics. I believe that the proofs are correct and in my opinion the originality and clarity of this work should make this paper very attractive for the readers of the Mathematics. By no means, I strongly recommend accepting it for publication in Mathematics.

Reviewer 2 Report

The topic of the Lienard 2D nonlinear dynamical systems investigated by the authors is not new. However, they present new families of explicit geometric integrators for such systems, while it is shown that these integrators are stable (even when the system considered is stiff) and preserve the qualitative features of the limit cycle of the dynamics. The properties of these new integrators are analytically as well as numerically demonstrated at the example of the van der Pol oscillator. The text is well and clearly written and the conclusions are consistent with the core argumentation of the paper. This is a nice paper on Lienard systems and their geometric numerical integration based on Hamiltonian flows on 3D contact manifolds. The references are up-to-date, while the presentation is sound and clear. The paper can be published.