Integrability Properties of Cubic Liénard Oscillators with Linear Damping

Demina, Maria V.; Sinelshchikov, Dmitry I.

doi:10.3390/sym11111378

Cited by 10 publications

(8 citation statements)

References 21 publications

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“…The proof of this lemma is analogous to the proof of Theorem 7.5. Concluding this section let us note that autonomous and non-autonomous Darboux first integrals of Liénard differential systems (1.1) satisfying the conditions deg f = 1, deg g = 3 and deg f = 2, deg g = 5 were classified in the non-resonant case in articles [17,19].…”

Section: 84)mentioning

confidence: 99%

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Integrability and solvability of polynomial Liénard differential systems

Demina¹

2021

Preprint

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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 polynomials arising in the systems. It is demonstrated that if the degree of a polynomial responsible for the restoring force is higher than the degree of a polynomial producing the damping, then a generic Liénard differential system is not Liouvillian integrable with the exception for a linear Liénard system. However, for any fixed degrees of the polynomials describing the damping and the restoring force there always exist subfamilies possessing Liouvillian first integrals. A number of novel Liouvillian integrable subfamilies parameterized by an arbitrary polynomial are presented. In addition, we study the existence of non-autonomous Darboux first integrals and non-autonomous Jacobi last multipliers with a time-dependent exponential factor.

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Section: 84)mentioning

confidence: 99%

“…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].…”

Section: Introductionmentioning

confidence: 99%

Integrability and solvability of polynomial Liénard differential systems

Demina¹

2021

Preprint

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“…">4.Differential Galois theory 24 ; 5.Extended Prelle–Singer method 25 and the Darboux theory of integrability 19,22,26–32 …”

Section: Introductionmentioning

confidence: 99%

“…Extended Prelle-Singer method 25 and the Darboux theory of integrability. 19,22,[26][27][28][29][30][31][32] A collection of integrable and solvable subfamilies of Liénard differential systems is presented by Polyanin and Zaitsev. 33 The transformation 𝑦(𝑥) = 1∕𝑤(𝑥) brings Abel differential equations of the second kind (4) to Abel differential equations of the first kind 𝑤 𝑥 = 𝑔(𝑥)𝑤 3 + 𝑓(𝑥)𝑤 2 .…”

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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“…However, it is known that there are interesting from an applied point of view nonlinear oscillators that can be linearized via (1.2) only if F z = 0 (see, e.g. [22]). Therefore, in this work we consider the full linearization problem for (1.1) and find all equations from family (1.1) that can be linearized with the help of (1.2) with F z = 0.…”

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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Integrability Properties of Cubic Liénard Oscillators with Linear Damping

Cited by 10 publications

References 21 publications

Integrability and solvability of polynomial Liénard differential systems

Integrability and solvability of polynomial Liénard differential systems

Integrability and solvability of polynomial Liénard differential systems

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

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