Liouvillian integrability of a general Rayleigh-Duffing oscillator

Giné, Jaume; Valls, Clàudìa

doi:10.1080/14029251.2019.1591710

Cited by 22 publications

(4 citation statements)

References 12 publications

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“…The tool relies on geometric characteristics is called the classification of phase portraits in the Poincaré disk. A significant number of papers regarding limit cycles, first integrals and invariants curves [9,16,18,20,22,24,31] has been published, where the main goal was studying the qualitative behavior of these solutions.…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

A Class of Non-Linear Oscillators with Non-Autonomous First Integrals and Algebraic Limit Cycles

Belattar¹,

Cheurfa²,

Bendjeddou³

et al. 2023

Preprint

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Section: Introduction and The Main Resultsmentioning

confidence: 99%

A Class of Non-Linear Oscillators with Non-Autonomous First Integrals and Algebraic Limit Cycles

Belattar¹,

Cheurfa²,

Bendjeddou³

et al. 2023

Preprint

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“…It is known that Darboux polynomials are of great importance if one studies the integrability problem and wants to derive all independent first integrals that are Darboux or Liouvillian functions [11,12]. An effective method of finding and classifying Darboux polynomials is the method of Puiseux series introduced in articles [10,17]. The main idea of this method is to use the representation of Darboux polynomials in the field of Puiseux series.…”

Section: Darboux Polynomialsmentioning

confidence: 99%

“…Therefore, the main aim of this work is to study various aspects of the integrability of (1) and their interconnections and find new completely integrable cases of (1). There are different approaches for studying integrability of nonlinear oscillators like point symmetries, algebraic integrability, local and nonlocal equivalence problems (see, e.g., [6][7][8][9][10]) and references therein). In this work we apply two of them to study (1).…”

Section: Introductionmentioning

confidence: 99%

Integrability Properties of Cubic Liénard Oscillators with Linear Damping

Demina

Sinelshchikov

2019

Symmetry

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We consider a family of cubic Liénard oscillators with linear damping. Particular cases of this family of equations are abundant in various applications, including physics and biology. There are several approaches for studying integrability of the considered family of equations such as Lie point symmetries, algebraic integrability, linearizability conditions via various transformations and so on. Here we study integrability of these oscillators from two different points of view, namely, linearizability via nonlocal transformations and the Darboux theory of integrability. With the help of these approaches we find two completely integrable cases of the studied equation. Moreover, we demonstrate that the equations under consideration have a generalized Darboux first integral of a certain form if and only if they are linearizable.

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“…Equations from family (1.1) often appear in numerous applications in mechanics, physics and so on [1,2]. Therefore, various aspects of integrability of (1.1) have been studied in a number of works (see, e.g., [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]). For example, in [5,7,9,10] authors considered applications of several linearizing transformations and λ-symmetries for finding first integrals of equations from family (1.1).…”

Section: Introductionmentioning

confidence: 99%

On linearizability via nonlocal transformations and first integrals for second-order ordinary differential equations

Sinelshchikov

2020

Preprint

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Nonlinear second-order ordinary differential equations are common in various fields of science, such as physics, mechanics and biology. Here we provide a new family of integrable second-order ordinary differential equations by considering the general case of a linearization problem via certain nonlocal transformations. In addition, we show that each equation from the linearizable family admits a transcendental first integral and study particular cases when this first integral is autonomous or rational. Thus, as a byproduct of solving this linearization problem we obtain a classification of second-order differential equations admitting a certain transcendental first integral. To demonstrate effectiveness of our approach, we consider several examples of autonomous and non-autonomous second order differential equations, including generalizations of the Duffing and Van der Pol oscillators, and construct their first integrals and general solutions. We also show that the corresponding first integrals can be used for finding periodic solutions, including limit cycles, of the considered equations.

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Liouvillian integrability of a general Rayleigh-Duffing oscillator

Abstract: We give a complete description of the Darboux and Liouville integrability of a general Rayleigh-Duffing oscillator through the characterization of its polynomial first integrals, Darboux polynomials and exponential factors.

Cited by 22 publications

References 12 publications

A Class of Non-Linear Oscillators with Non-Autonomous First Integrals and Algebraic Limit Cycles

A Class of Non-Linear Oscillators with Non-Autonomous First Integrals and Algebraic Limit Cycles

Integrability Properties of Cubic Liénard Oscillators with Linear Damping

On linearizability via nonlocal transformations and first integrals for second-order ordinary differential equations

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