Necessary and sufficient conditions for the existence of invariant algebraic curves

Maria Demina

doi:10.14232/ejqtde.2021.1.48

Abstract

We present a set of conditions enabling a polynomial system of ordinary differential equations in the plane to have invariant algebraic curves. These conditions are necessary and sufficient. Our main tools include factorizations over the field of Puiseux series near infinity of bivariate polynomials generating invariant algebraic curves. The set of conditions can be algorithmically verified. This fact gives rise to a method, which is able not only to find some irreducible invariant algebraic curves, but also to perform their classification. We study in details the problem of classifying invariant algebraic curves in the most difficult case: we consider differential systems with infinite number of trajectories passing through infinity. As an example, we find necessary and sufficient conditions such that a general polynomial Lienard differential system has invariant algebraic curves. We present a set of all irreducible invariant algebraic curves for quintic Lienard differential systems with a linear damping function. It is supposed in scientific literature that the degrees of their irreducible invariant algebraic curves are bounded by 6. While we derive irreducible invariant algebraic curves of degree 9.

Highlights

Performing the complete classification of trajectories contained in algebraic curves or surfaces for a given polynomial system of ordinary differential equations is a very difficult problem
Due to the invariance of the polynomial F(x, y) with respect to permutations of the Puiseux series y1(x), . . . , yN(x) and the structure of recurrence relations satisfied by coefficients of a Puiseux series solving an algebraic first-order ordinary differential equation, we conclude that the polynomial F(x, y) inherits the invariance with respect to the permutations of c(m1), . . . , c(mM)
Since we have considered all possible combinations of the Puiseux series from the field C∞{x} that solve equation (3.1), we conclude that other irreducible invariant algebraic curves cannot exist

Summary

Introduction

Performing the complete classification of trajectories contained in algebraic curves or surfaces for a given polynomial system of ordinary differential equations is a very difficult problem. Let us mention that the problem of finding all irreducible invariant algebraic curves of differential systems (1.1) with infinite number of trajectories passing through infinity was not considered in articles [5, 6, 10] This case turns out to be the most difficult. Let us note that the case n = 2m + 1 is in certain sense degenerate and the problem of classifying invariant algebraic curves for n = 2m + 1 is very complicated This degeneracy can be explained analyzing properties of Puiseux series satisfying an algebraic first-order ordinary differential equation of the form (1.2) related to associated Liénard differential systems. In the Appendix, an algorithm of finding Puiseux series solving an algebraic first-order ordinary differential equation is described

Computational aspects of the Puiseux series method

Invariant algebraic curves for Liénard differential systems

Examples

Conclusion