We study bifurcations of homoclinic orbits to hyperbolic saddle equilibria in a class of four-dimensional systems which may be Hamiltonian or not. Only one parameter is enough to treat these types of bifurcations in Hamiltonian systems but two parameters are needed in general systems. We apply a version of Melnikov's method due to Gruendler to obtain saddle-node and pitchfork types of bifurcation results for homoclinic orbits. Furthermore we prove that if these bifurcations occur, then the variational equations around the homoclinic orbits are integrable in the meaning of differential Galois theory under the assumption that the homoclinic orbits lie on analytic invariant manifolds. We illustrate our theories with an example which arises as stationary states of coupled real Ginzburg-Landau partial differential equations, and demonstrate the theoretical results by numerical ones.

In this work, we unfold some differential algebraic aspects of Darboux first integrals of polynomial vector fields. An interesting improvement is that our approach can be applied both to autonomous and non-autonomous vector fields. We give a sufficient and necessary condition for the existence of a Darboux first integral of a specific form for a polynomial vector field with some known algebraic invariant hypersurfaces. For the autonomous case, the classical result of Darboux is obtained as a corollary. For the non-autonomous case our characterization improves a known criterium.

In this paper we develop a differential Galois theory for algebraic LieVessiot systems in algebraic homogeneous spaces. Lie-Vessiot systems are non autonomous vector fields that are linear combinations with timedependent coefficients of fundamental vector fields of an algebraic Lie group action. Those systems are the building blocks for differential equations that admit superposition of solutions. Lie-Vessiot systems in algebraic homogeneous spaces include the case of linear differential equations. Therefore, the differential Galois theory for Lie-Vessiot systems is an extension of the classical Picard-Vessiot theory. In particular, algebraic Lie-Vessiot systems are solvable in terms of Kolchin's strongly normal extensions. Therefore, strongly normal extensions are geometrically interpreted as the fields of functions on principal homogeneous spaces over the Galois group. Finally we consider the problem of integrability and solvability of automorphic differential equations. Our main tool is a classical method of reduction, somewhere cited as Lie reduction. We develop and algebraic version of this method, that we call Lie-Kolchin reduction. Obstructions to the application are related to Galois cohomology.

This paper is dedicated to the differential Galois theory in the complex analytic context for Lie-Vessiot systems. Those are the natural generalization of linear systems, and the more general class of differential equations adimitting superposition laws, as recently stated in [5]. A Lie-Vessiot system is automatically translated into a equation in a Lie group that we call automorphic system. Reciprocally an automorphic system induces a hierarchy of Lie-Vessiot systems. In this work we study the global analytic aspects of a classical method of reduction of differential equations, due to S. Lie. We propose an differential Galois theory for automorphic systems, and explore the relationship between integrability in terms of Galois theory and the Lie's reduction method. Finally we explore the algebra of Lie symmetries of a general automorphic system. Summary of resultsThroughout the first section we review the concept of Lie-Vessiot system, and we state the global version of Lie's superposition theorem; it is a recallment of results contained in previous works [5] and [4]. Second section is devoted to the notion of automorphic system. We study its geometry and the relationship with general Lie-Vessiot systems: it is introduced the concept of Lie-Vessiot hierarchy, that relates an automorphic system with a family of Lie-Vessiot systems induced in homogeneous spaces. In the third section we introduce Lie's reduction method (Theorem 3.5) in the complex analytic context. We note that Lie's reduction method is local in the time parameter, and then we explore the obstructions to the existence of a reduction method global in time (Proposition 3.9). In the fourth section we propose a Galois theory for automorphic systems, based on the Lie reduction method. In our theory, the Galois group is the smallest group

No abstract

Abstract. We study a Sturm-Liouville type eigenvalue problem for second-order differential equations on the infinite interval (−∞, ∞). Here the eigenfunctions are nonzero solutions exponentially decaying at infinity. We prove that at any discrete eigenvalue the differential equations are integrable in the setting of differential Galois theory under general assumptions. Our result is illustrated with three examples for a stationary Schrödinger equation having a generalized Hulthén potential; a linear stability equation for a traveling front in the Allen-Cahn equation; and an eigenvalue problem related to the Lamé equation.

L'accès aux articles de la revue « Annales de la faculté des sciences de Toulouse Mathématiques » (http://afst.centre-mersenne.org/) implique l'accord avec les conditions générales d'utilisation (http://afst.centre-mersenne.org/legal/). Les articles sont publiés sous la license CC-BY 4.0.

Abstract. The aim of this article is to study rational parallelisms of algebraic varieties by means of the transcendence of their symmetries. The nature of this transcendence is measured by a Galois group built from the Picard-Vessiot theory of principal connections.