On an integrability criterion for a family of cubic oscillators

Dmitry Sinelshchikov; Dmitry Sinelshchikov

doi:10.3934/math.2021745

AIMS Mathematics

2021, Volume 6, Issue 11: 12902-12910. doi: 10.3934/math.2021745

Previous Article Next Article

Research article

On an integrability criterion for a family of cubic oscillators

Dmitry Sinelshchikov ^,

Department of Applied Mathematics, HSE University, 34 Tallinskaya Street, Moscow 123458, Russian Federation

Received: 22 July 2021 Accepted: 07 September 2021 Published: 10 September 2021
MSC : 34A05, 34A25

In this work we consider a family of cubic, with respect to the first derivative, nonlinear oscillators. We obtain the equivalence criterion for this family of equations and a non-canonical form of Ince Ⅶ equation, where as equivalence transformations we use generalized nonlocal transformations. As a result, we construct two integrable subfamilies of the considered family of equations. We also demonstrate that each member of these two subfamilies possesses an autonomous parametric first integral. Furthermore, we show that generalized nonlocal transformations preserve autonomous invariant curves for the equations from the studied family. As a consequence, we demonstrate that each member of these integrable subfamilies has two autonomous invariant curves, that correspond to irreducible polynomial invariant curves of the considered non-canonical form of Ince Ⅶ equation. We illustrate our results by two examples: An integrable cubic oscillator and a particular case of the Liénard (4, 9) equation.
- cubic oscillators,
- nonlocal transformations,
- parametric first integrals,
- invariant curves
Citation: Dmitry Sinelshchikov. On an integrability criterion for a family of cubic oscillators[J]. AIMS Mathematics, 2021, 6(11): 12902-12910. doi: 10.3934/math.2021745

Related Papers:

Abstract

In this work we consider a family of cubic, with respect to the first derivative, nonlinear oscillators. We obtain the equivalence criterion for this family of equations and a non-canonical form of Ince Ⅶ equation, where as equivalence transformations we use generalized nonlocal transformations. As a result, we construct two integrable subfamilies of the considered family of equations. We also demonstrate that each member of these two subfamilies possesses an autonomous parametric first integral. Furthermore, we show that generalized nonlocal transformations preserve autonomous invariant curves for the equations from the studied family. As a consequence, we demonstrate that each member of these integrable subfamilies has two autonomous invariant curves, that correspond to irreducible polynomial invariant curves of the considered non-canonical form of Ince Ⅶ equation. We illustrate our results by two examples: An integrable cubic oscillator and a particular case of the Liénard (4, 9) equation.

References

[1]	A. A. Andronov, A. A. Vitt, S. E. Khaikin, Theory of oscillators, New York: Dover Publications, 2011.
[2]	S. Ghosh, D. S. Ray, Chemical oscillator as a generalized Rayleigh oscillator, J. Chem. Phys., 139 (2013), 164112. doi: 10.1063/1.4826169
[3]	A. L. Kazakov, P. A. Kuznetsov, A. A. Lempert, On a heat wave for the nonlinear heat equation: An existence theorem and exact solution, In: G. Demidenko, E. Romenski, E. Toro, M. Dumbser, Continuum mechanics, applied mathematics and scientific computing: Godunov's legacy, Cham: Springer, 2020,223–228.
[4]	C. Muriel, J. L. Romero, Second-order ordinary differential equations and first integrals of the form $A(t, x)\dot{x} + B(t, x)$, J. Nonlinear Math. Phys., 16 (2009), 209–222.
[5]	W. Nakpim, S. V. Meleshko, Linearization of second-order ordinary differential equations by generalized sundman transformations, Symmetry, Integr. Geom. Methods Appl., 6 (2010), 051.
[6]	M. C. Nucci, K. M. Tamizhmani, Lagrangians for dissipative nonlinear oscillators: The method of Jacobi last multiplier, J. Nonlinear Math. Phys., 17 (2010), 167–178.
[7]	C. Muriel, J. L. Romero, Second-order ordinary differential equations with first integrals of the form $C(t) + 1/(A(t, x)\dot{x} + B(t, x))$, J. Nonlinear Math. Phys., 18 (2011), 237–250.
[8]	Y. Y. Bagderina, Invariants of a family of scalar second-order ordinary differential equations for Lie symmetries and first integrals, J. Phys. A Math. Theor., 49 (2016), 155202. doi: 10.1088/1751-8113/49/15/155202
[9]	A. Ruiz, C. Muriel, On the integrability of Liénard I-type equations via $\lambda$-symmetries and solvable structures, Appl. Math. Comput., 339 (2018), 888–898.
[10]	D. I. Sinelshchikov, N. A. Kudryashov, On the Jacobi last multipliers and Lagrangians for a family of Liénard-type equations, Appl. Math. Comput., 307 (2017), 257–264.
[11]	M. V. Demina, Novel algebraic aspects of Liouvillian integrability for two-dimensional polynomial dynamical systems, Phys. Lett. A, 382 (2018), 1353–1360. doi: 10.1016/j.physleta.2018.03.037
[12]	J. Giné, C. Valls, Liouvillian integrability of a general Rayleigh-Duffing oscillator, J. Nonlinear Math. Phys., 26 (2019), 169–187.
[13]	J. Giné, C. Valls, On the dynamics of the Rayleigh-Duffing oscillator, Nonlinear Anal. Real World Appl., 45 (2019), 309–319. doi: 10.1016/j.nonrwa.2018.07.007
[14]	M. R. Cândido, J. Llibre, C. Valls, Non-existence, existence, and uniqueness of limit cycles for a generalization of the Van der Pol-Duffing and the Rayleigh-Duffing oscillators, Phys. D Nonlinear Phenom., 407 (2020), 132458. doi: 10.1016/j.physd.2020.132458
[15]	D. I. Sinelshchikov, Linearizability conditions for the Rayleigh-like oscillators, Phys. Lett. A, 384 (2020), 126655. doi: 10.1016/j.physleta.2020.126655
[16]	D. I. Sinelshchikov, On linearizability via nonlocal transformations and first integrals for second-order ordinary differential equations, Chaos, Solitons Fractals, 141 (2020), 110318. doi: 10.1016/j.chaos.2020.110318
[17]	E. L. Ince, Ordinary differential equations, Dover, New York, 1956.
[18]	F. J. Bureau, Differential equations with fixed critical points, Ann. Mat. Pur. Appl., 64 (1964), 229–364. doi: 10.1007/BF02410054
[19]	X. Zhang, Integrability of dynamical systems: Algebra and analysis, Vol. 47, Singapore: Springer, 2017.
[20]	M. Demina, D. Sinelshchikov, Integrability properties of cubic Liénard oscillators with linear damping, Symmetry, 11 (2019), 1378. doi: 10.3390/sym11111378
[21]	M. V. Demina, D. I. Sinelshchikov, On the integrability of some forced nonlinear oscillators, Int. J. Nonlin. Mech., 121 (2020), 103439. doi: 10.1016/j.ijnonlinmec.2020.103439
[22]	M. V. Demina, D. I. Sinelshchikov, Darboux first integrals and linearizability of quadratic-quintic Duffing-van der Pol oscillators, J. Geom. Phys., 165 (2021), 104215. doi: 10.1016/j.geomphys.2021.104215
[23]	M. V. Demina, Liouvillian integrability of the generalized Duffing oscillators, Anal. Math. Phys., 11 (2021), 1–18. doi: 10.1007/s13324-020-00437-5

Reader Comments

Your name:*

Email:*
© 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)