Galoisian obstruction to the integrability of general dynamical systems

The Galoisian approach to study the integrability of classical Hamiltonian systems, the so-called Morales–Ramis theory, has been proved to be useful and powerful by many applications. Here, two analogous forms of the Morales–Ramis theory for general dynamical systems both in vector field and mapping forms are given. Galois groups of the corresponding variational equations are studied, and some necessary conditions of the system to possess a certain number of integrals are presented. Several applications are given at last to illustrate our results.