On the Poincaré problem and Liouvillian integrability of quadratic Liénard differential equations

Abstract

We present the complete classification of irreducible invariant algebraic curves of quadratic Liénard differential equations. We prove that these equations have irreducible invariant algebraic curves of unbounded degrees, in contrast with what is wrongly claimed in the literature. In addition, we classify all the quadratic Liénard differential equations that admit a Liouvillian first integral.

On the equations of Poizat and Liénard

“…Numerous authors attempted to generalize Odani's results on invariant curves [66,41]. Many recent works utilize the results of Odani and generalizations to characterize Liouvillian first integrals of Liénard equations in various special cases [37,38,7,39,9,12,11]. Many of the special cases considered make assumptions about the degrees of f (x), g(x) in equation 6, while others make detailed assumptions not unlike the criteria employed by Brestovski [4].…”

“…, x n are not independent over some differential field k extending C, then there is a differential polynomial in two variables (of order zero or one) with coefficients in C such that p(x i , x j , x ′ j ) = 0. 12 In this section, we go farther, showing that in our case p can be taken to be a polynomial relation between x i and x j not involving any derivative. Then in the following section, we give a precise characterization of what the possible polynomial relations between solutions are in terms of basic invariants of the rational function appearing in Equation (⋆) (e.g.…”

Invariant Algebraic Curves of Generalized Liénard Polynomial Differential Systems

“…In fact, the characterization of the invariant algebraic curves of system (1) for this case is not complete. Recently, cases m = 1 and n = 2 have been solved (see [18]).…”

Integrability and solvability of polynomial Liénard differential systems

“…In fact, the degree with respect to y of the polynomial F 2 (x, y) can be an arbitrary natural number. This fact was established in article [26]. Our next step is to investigate the existence of non-autonomous Darboux-Jacobi last multipliers.…”

“…2. classical Lie symmetry analysis [7,8] and λ symmetries [9]; 3. local [10][11][12][13][14] and non-local transformations [15][16][17][18][19]; 4. differential Galois theory [20]; 5. extended Prelle-Singer method [21] and Darboux theory of integrability [17,19,[22][23][24][25][26][27][28].…”