Inverse problems for multiple invariant curves

Abstract

Abstract. Planar polynomial vector fields which admit invariant algebraic curves, Darboux integrating factors or Darboux first integrals are of special interest. In the present paper we solve the inverse problem for invariant algebraic curves with a given multiplicity and for integrating factors, under generic assumptions regarding the (multiple) invariant algebraic curves involved. In particular we prove, in this generic scenario, that the existence of a Darboux integrating factor implies Darboux integrability. Furthermore we construct examples where the genericity assumption does not hold and indicate that the situation is different for these.

Inverse Problems in Darboux’ Theory of Integrability

“…, d r ) of F. We will call these the trivial vector fields admitting the given integrating factor. Nontrivial vector fields exist for certain exponents, as the next result shows (see [26], [10] and [12]). …”

“…But this works only if additional nondegeneracy conditions hold for the stationary points at infinity, see [13] and [10], and will fail in general. However, one can instead consider the degree δ y with respect to y, and start from the assumption f j = γ j y n j + terms of smaller degree in y, with constants γ j = 0, for all j.…”

“…The more general type of integrating factor in part (b) arises from integrating factors of type (3) by coalescence of curves; see e.g. [10]. The result in part (b) is essentially due to Singer [24], who showed that a system with a Liouvillian first integral admits an integrating factor whose logarithmic differential is a closed rational 1-form.…”

“…These inverse problems on the affine plane have been discussed systematically in recent work by the authors; see [10], [11], [12]. Since the original papers contain (unavoidably, it seems) quite technical sections, it is sensible to present the straightforward basic ideas in a survey, and to illustrate them by examples.…”

Inverse Approach in Ordinary Differential Equations: Applications to Lagrangian and Hamiltonian Mechanics

“…Erugin ideas were developed in particular in [17]. We observe that such kind of problem has recently been developed in R 2 or C 2 mainly restricted to polynomial differential equations (see for instance [5,6,7,27,39,41,42,43]). …”