On the integrability of Liénard systems with a strong saddle

Giné, Jaume; Llibre, Jaume

doi:10.1016/j.aml.2017.03.004

Cited by 7 publications

(4 citation statements)

References 17 publications

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“…The Liénard second order differential equation (1) can be written as the equivalent planar differential system When F (x) and g(x) are polynomial functions in the variable x, the Liénard differential system (2) is called the generalized polynomial Liénard differential system has been studied extensively, see [14,38,41] for center conditions, [13,16,21,22,27,37] for the number of limit cycles, [1,40] for the amplitude of limit cycles, [15,26] for integrability conditions, [38,39] for isochronous conditions, and [2,3,17] for global phase portraits and bifurcation diagrams.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Phase portraits of piecewise linear continuous differential systems with two zones separated by a straight line

Llibre

2019

Journal of Differential Equations

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Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Phase portraits of piecewise linear continuous differential systems with two zones separated by a straight line

Llibre

2019

Journal of Differential Equations

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“…We do not know if there are nonlinear integrable cases in systems (2). Later on in [5] appears explicitly the following:…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

On a conjecture on the integrability of Liénard systems

Llibre

Murza

Valls

2019

Rend. Circ. Mat. Palermo, II. Ser

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We consider the Liénard differential systemsin C 2 where F (x) is an analytic function satisfying F (0) = 0 and F ′ (0) ̸ = 0. Then these systems have a strong saddle at the origin of coordinates. It has been conjecture that if such systems have an analytic first integral defined in a neighborhood of the origin, then the function F (x) is linear, i.e. F (x) = ax. Here we prove this conjecture, and show that when F (x) is linear and system (1) has an analytic first integral this is a polynomial.

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“…Let us enumerate the most important studies: 1.Local analysis 2–6 and formal first integrals 7–9 ; 2.Classical Lie symmetry analysis 10,11 and λ symmetries 12 ; …”

Section: Introductionmentioning

confidence: 99%

“…1. Local analysis [2][3][4][5][6] and formal first integrals [7][8][9] ; 2. Classical Lie symmetry analysis 10,11 and 𝜆 symmetries 12 ; 3.…”

Section: Introductionmentioning

confidence: 99%

Integrability and solvability of polynomial Liénard differential systems

Demina

2022

Stud Appl Math

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We provide the necessary and sufficient conditions of Liouvillian integrability for Liénard differential systems describing nonlinear oscillators with a polynomial damping and a polynomial restoring force. We prove that Liénard differential systems are not Darboux integrable excluding subfamilies with certain restrictions on the degrees of the polynomials arising in the systems. We demonstrate that if the degree of a polynomial responsible for the restoring force is greater than the degree of a polynomial producing the damping, then a generic Liénard differential system is not Liouvillian integrable with the exception of linear Liénard systems. However, for any fixed degrees of the polynomials describing the damping and the restoring force we present subfamilies possessing Liouvillian first integrals. As a by‐product of our results, we find a number of novel Liouvillian integrable subfamilies. In addition, we study the existence of nonautonomous Darboux first integrals and nonautonomous Jacobi last multipliers with a time‐dependent exponential factor.

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On the integrability of Liénard systems with a strong saddle

Abstract: Està subjecte a una llicència de Reconeixement-NoComercial-SenseObraDerivada 4.0 de Creative Commons

Cited by 7 publications

References 17 publications

Phase portraits of piecewise linear continuous differential systems with two zones separated by a straight line

Phase portraits of piecewise linear continuous differential systems with two zones separated by a straight line

On a conjecture on the integrability of Liénard systems

Integrability and solvability of polynomial Liénard differential systems

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