Invariant algebraic curves for Liénard dynamical systems revisited

Abstract

A novel algebraic method for finding invariant algebraic curves for a polynomial vector field in C 2 is introduced. The structure of irreducible invariant algebraic curves for Liénard dynamical systems x t = y, y t = −f (x)y − g(x) with deg g = deg f + 1 is obtained. It is shown that there exist Liénard systems that possess more complicated invariant algebraic curves than it was supposed before. As an example, all irreducible invariant algebraic curves for the Liénard differential system with deg f = 2 and deg g = 3 are obtained. All these results seem to be new.

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

“…(3.1)There exists the one-to-one correspondence between irreducible invariant algebraic curves f (x, y) = 0 of Liénard dynamical system (1.2) and irreducible invariant algebraic curves G(s, z) of system (3.1). The following theorem was proved in article[5].Theorem 3.1. Let G(s, z) ∈ C[s, z] \ C be an irreducible invariant algebraic curve of dynamical system (3.1).…”

Integrability Properties of Cubic Liénard Oscillators with Linear Damping

“…Proof. Substituting λ(y, w) = λ 0 (y)w l , F(y, w) = µ(y)w N with l, N ∈ N ∪ {0}, 0 ≤ l ≤ 2 into Equation (19) and balancing the highest-order terms, we conclude that µ(y) ∈ C, l = 0, and N ∈ N. This means that cofactors of Darboux polynomials do not depend on w and there are no Darboux polynomials independent on w. In addition, we observe that the highest-order coefficient (with respect to w) of F(y, w) is a constant. Without loss of generality we set µ = 1.…”

“…Now let us perform the classification of Puiseux series near the point y = ∞ that satisfy Equation (21). For this aim we shall use the Painlevé methods and the power geometry [19,20]. There exists only one dominant balance producing asymptotics near the point y = ∞.…”

“…. It is known that starting from power asymptotics it is possible to derive asymptotic series possessing these asymptotics as leading-order terms [19,20].…”

“…Calculating the polynomial part in expression (24) with s 1 = 1, s 2 = 0 and s 1 = 0, s 2 = 1 and s 1 = 1, s 2 = 1, we find the Darboux polynomials. Substituting the results into (19), we get restrictions on the parameters of the original system and explicit expressions of the cofactors. If s 1 = 1, s 2 = 1, then N = 1 and λ 1 = −y + A 0 .…”