Jean-Marie Strelcyn

Around Jouanolou non-integrability theorem☆

Abstract We generalize the well-known Jouanolou non-integrability theorem concerning the system of ordinary differential equations: x ′ 1 = x s 2 , x ′ 2 = x s 3 , x ′ 3 = x s 1 , s ϵ N , s ϵ N , s ≥ 2 to an arbitrary prime number n ≥ 3 of variables and arbitrary integer exponent s ≥ 3. Our proof is completely elementary.

On the algebraic non-integrability of Halphen system

Abstract It is proved that the Halphen system of ordinary differential equations has no non-trivial rational first integrals.

Nonintegrability of Bianchi VIII Hamiltonian system

In this paper we study the Bianchi VIII cosmological dynamical system. Our aim is to show that this system is nonintegrable. To show this we use an extension of Ziglin theory made by Morales-Ruiz and Ramis.

Generators of rings of constants for some diagonal derivations in polynomial rings

Abstract Let K be a field of characteristic zero. We show that if n ⩾ 3, given r ⩾ 0 there exists a diagonal K -derivation of K [ x 1 , …, x n ] such that the minimal number of generators over K of the ring of constants is equal to r .

The Euler equations on Lie algebra so(4): An elementary approach to integrability condition

We give a new derivation of the Manakov and the product case integrability conditions of the Euler equations on Lie algebra so(4). Fourth first integral functionally independent of the three already known integrals is obtained explicitly for all values of the parameters by an algorithmic approach.

NUMERICAL INTEGRATION OF HAMILTONIAN SYSTEMS IN THE PRESENCE OF ADDITIONAL INTEGRALS: APPLICATION OF THE OBSERVER METHOD

The observer method introduced by Busvelle et al. (1993), is a simple idea that can be used to improve reliable numerical integration of a system of differential equations in the presence of first integrals. In the cited paper the observer method was already tested on some important examples like the Kepler problem and the Gavrilov-Shilnikov system.

An Elementary approach to Integrability Condition for the Euler Equations on Lie Algebra so(4)

The completely elementary derivation of the Manakov integrability condition of the Euler equations on the Lie algebra so(4) is given. Four functionally independent first integrals are written explicitly for all values of parameters. Some simple cases of partial integrability are found.