Non-integrability of the Armbruster-Guckenheimer-Kim quartic Hamiltonian through Morales-Ramis theory

We show the non-integrability of the three-parameter Armburster-Guckenheimer-Kim quartic Hamiltonian using Morales-Ramis theory, with the exception of the three already known integrable cases. We use Poincar\'e sections to illustrate the breakdown of regular motion for some parameter values.Comment: Accepted for publication in SIAM Journal on Applied Dynamical Systems. Adapted version for arxiv with 19 pages and 11 figure

Differential equations with natural matrices coefficients

In this paper we present the main results of the master thesis in applied mathematics of the second named author, which was supervised by the first named author. Such results, and for instance this paper, concerns to some differential and algebraic results involving natural matrices. The problem of solving differential equations is very ancient and is very important to get explicit solutions of differential equations to be applied in physics and other areas. In this paper, as well in the master thesis, we study the differential and algebraic structure of linear differential equations with natural matrix coefficients and generalizations. These results are original and important for researchers interested in differential algebra and applications of differential equations

Semiclassical quantification of some two degree of freedom potentials: a Differential Galois approach

In this work we explain the relevance of the Differential Galois Theory in the semiclassical (or WKB) quantification of some two degree of freedom potentials. The key point is that the semiclassical path integral quantification around a particular solution depends on the variational equation around that solution: a very well-known object in dynamical systems and variational calculus. Then, as the variational equation is a linear ordinary differential system, it is possible to apply the Differential Galois Theory to study its solvability in closed form. We obtain closed form solutions for the semiclassical quantum fluctuations around constant velocity solutions for some systems like the classical Hermite/Verhulst, Bessel, Legendre, and Lam\'e potentials. We remark that some of the systems studied are not integrable, in the Liouville-Arnold sense

Some tastings in Morales-Ramis theory

In this paper we present a short material concerning to some results in Morales-Ramis theory, which relates two different notions of integrability: Integrability of Hamiltonian systems through Liouville Arnold theorem and integrability of linear differential equations through differential Galois theory. As contribution, we obtain the abelian differential Galois group of the variational equation related to a bi-parametric Hamiltonian system

Differential galois groups and representation of quivers for seismic models with constant hessian of square of slowness

The trajectory of energy is modeled by the solution of the Eikonal equation, which can be solved by solving a Hamiltonian system. This system is amenable of treatment from the point of view of the theory of differential algebra. In particular, by Morales-Ramis theory, it is possible to analyze integrable Hamiltonian systems through the abelian structure of their variational equations. In this paper, we obtain the abelian differential Galois groups and the representation of the quiver, that allow us to obtain such abelian differential Galois groups, for some seismic models with constant Hessian of square of slowness, proposed in [20], which are equivalent to linear Hamiltonian systems with three uncoupled harmonic oscillators. © 2017 Pushpa Publishing House, Allahabad, India