Nous nous proposons, dans ce travail, d'étudier certaines propriétés géométriques telles que diverses symétries et diverses concavités radiales, directionnelles, etc., pour des équations completement non linéaires (...).

Cascade second order ODEs on manifolds are defined. These objects are locally represented by coupled second order ODEs such that any solution of one of them can represent an external force for the other one. A generic saddle-node bifurcation theorem for 1-parameter families of cascade second order ODEs is proved.

An isotypic Kronecker web is a family of corank m foliations ${\left\{{ℱ}_t\right\}}_{t\in ℝP^1}$ such that the curve of annihilators t ↦ (T x F t)⊥ ∈ Grm(T x* M) is a rational normal curve in the Grassmannian Grm(T x*M) at any point x ∈ M. For m = 1 we get Veronese webs introduced by I. Gelfand and I. Zakharevich [Gelfand I.M., Zakharevich I., Webs, Veronese curves, and bi-Hamiltonian systems, J. Funct. Anal., 1991, 99(1), 150–178]. In the present paper, we consider the problem of local classification of isotypic Kronecker webs...

The geometry of second-order systems of ordinary differential equations represented by $2$-connections on the trivial bundle $errorℝ\times M\to ℝ$ is studied. The formalism used, being completely utilizable within the framework of more general situations (partial equations), turns out to be of interest in confrontation with a traditional approach (semisprays), moreover, it amounts to certain new ideas and results. The paper is aimed at discussion on the interrelations between all types of connections having to do with...

We compute cohomology spaces of Lie algebras that describe differential invariants of third order ordinary differential equations. We prove that the algebra of all differential invariants is generated by 2 tensorial invariants of order 2, one invariant of order 3 and one invariant of order 4. The main computational tool is a Serre-Hochschild spectral sequence and the representation theory of semisimple Lie algebras. We compute differential invariants up to degree 2 as application.

We prove Gronwall-type estimates for the distance of integral curves of smooth vector fields on a Riemannian manifold. Such estimates are of central importance for all methods of solving ODEs in a verified way, i.e., with full control of roundoff errors. Our results may therefore be seen as a prerequisite for the generalization of such methods to the setting of Riemannian manifolds.