Algebraic invariant curves and first integrals for Riccati polynomial differential systems

In many problems appearing in applied mathematics in the nonlinear ordinary differential systems, as in physics, chemist, economics, etc., if we have a differential system on a manifold of dimension, two of them having a first integral, then its phase portrait is completely determined. While the existence of first integrals for differential systems on manifolds of a dimension higher than two allows to reduce the dimension of the space in as many dimensions as independent first integrals we have. Hence, to know first integrals is important, but the following question appears: Given a differential system, how to know if it has a first integral? The symmetries of many differential systems force the existence of first integrals. This paper has two main objectives. First, we study how to compute first integrals for polynomial differential systems using the so-called Darboux theory of integrability. Furthermore, second, we show how to use the existence of first integrals for finding limit cycles in piecewise differential systems.

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Centers and limit cycles for a generalized kukles differential systems of degree 8

10.21203/rs.3.rs-170528/v1 ◽

2021 ◽

Author(s):

Loubna Damene ◽

Rebiha Benterki

Keyword(s):

Limit Cycles ◽

Main Tool ◽

Upper Bounds ◽

First Integrals ◽

Averaging Theory ◽

Phase Portraits ◽

Differential Systems ◽

Polynomial Differential Systems

Abstract In this paper we provide all the global phase portraits of the generalized kukles differential systems x= y; y = x + ax8 + bx6y2 + cx4y4 + dx2y6 + ey8; symmetric with respect to the x{axis, with a2 + b2 + c2 + d2 + e2 6= 0, and by using the averaging theory up to seven order, we give the upper bounds of limit cycles which can bifurcate from its center when we perturb it inside the class of all polynomial differential systems of degree 8. The main tool used for proving these results is based in the first integrals of the systems which form the discontinuous piecewise differential systems.

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Liouvillian First Integrals for Generalized Riccati Polynomial Differential Systems

Advanced Nonlinear Studies ◽

10.1515/ans-2015-0411 ◽

2015 ◽

Vol 15 (4) ◽

Cited By ~ 2

Author(s):

Jaume Llibre ◽

Claudia Valls

Keyword(s):

First Integrals ◽

Differential Systems ◽

Polynomial Differential Systems

AbstractWe study the existence and non-existence of Liouvillian first integrals for the generalized Riccati polynomial differential systems of the form xʹ = y, yʹ = a(x)y

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Liouvillian first integrals for a class of generalized Liénard polynomial differential systems

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210515000906 ◽

2016 ◽

Vol 146 (6) ◽

pp. 1195-1210 ◽

Cited By ~ 1

Author(s):

Jaume Llibre ◽

Clàudia Valls

Keyword(s):

Algebraic Curve ◽

First Integrals ◽

Differential Systems ◽

Polynomial Differential Systems ◽

Invariant Algebraic Curve

We study the existence of Liouvillian first integrals for the generalized Liénard polynomial differential systems of the form xʹ = y, yʹ = –g(x) – f(x)y, where f(x) = 3Q(x)Qʹ(x)P(x) + Q(x)2Pʹ(x) and g(x) = Q(x)Qʹ(x)(Q(x)2P(x)2 – 1) with P,Q ∈ ℂ[x]. This class of generalized Liénard polynomial differential systems has the invariant algebraic curve (y + Q(x)P(x))2 – Q(x)2 = 0 of hyperelliptic type.

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