Differential Galois theory and non-integrability of planar polynomial vector fields

Abstract

We study a necessary condition for the integrability of the polynomials fields in the plane by means of the differential Galois theory. More concretely, by means of the variational equations around a particular solution it is obtained a necessary condition for the existence of a rational first integral. The method is systematic starting with the first order variational equation. We illustrate this result with several families of examples. A key point is to check wether a suitable primitive is elementary or not. Using a theorem by Liouville, the problem is equivalent to the existence of a rational solution of a certain first order linear equation, the Risch equation. This is a classical problem studied by Risch in 1969, and the solution is given by the "Risch algorithm". In this way we point out the connection of the non integrablity with some higher transcendent functions, like the error function.

Nonintegrability of nearly integrable dynamical systems near resonant periodic orbits

“…Using this expression, we write the kth-order VE of (3.2) along the periodic orbit (x, ε) = (x(t), 0) as 1) , y (2) ) + 2 f (1,1) (y (1) , χ (2) ) + 2 f (1,1) (y (1) , χ (2) ) + 2 f (0,2) (χ (1) , χ (2) )…”

“…where such terms as (y(1)) 0 and (χ (1) ) 0 = 1 have been eliminated and the summation in the last equation has been taken over all integers j, l, r, s, i 1 , . .…”

“…with y = y (k) and χ = k! (χ (1) ) k . We regard (3.5) as a linear differential equation on a Riemann surface, again.…”

Nonintegrability of the unfoldings of codimension-two bifurcations

“…In Section 2 we give the key result to reduce the problems of (1.1) and (1.2) to those of (1.6) and (1.7), respectively. In Section 3 we review a necessary part of Acosta-Humánez et al [2] for nonintegrability of planar polynomial vector fields and extend their discussion to give the other key result to analyze (1.6) and (1.7). The proof of Theorem 1.3 is provided in Section 4, and the proofs of Theorems 1.4 and 1.5 are provided in Section 5. whereD ⊂ C × C m is a region containing the m-dimensional y-plane {(0, y) ∈ C × C m | θ ∈ C, y ∈ C m }, and R :D → C,g :D → C m and Θ :D → R are analytic.…”

“…Moreover, if deg(κ 1d ) ≤ deg(κ 1n ), then the VE 1 and consequently the LVE k have an irregular singularity at infinity for k ≥ 2. Rational nonintegrability of (3.1) in this situation was extensively discussed in [2].…”