Integrability Properties of Cubic Liénard Oscillators with Linear Damping
Abstract
:1. Introduction
2. Linearization via Nonlocal Transformations
3. Darboux Polynomials
- 1.
- if,, then the polynomialis of degree at most two (with respect to y) and
- 2.
- if,, where either,or,, then the polynomialtakes the form
- 3.
- if,, then the polynomialtakes the form
- 4.
- if, then the polynomialtakes the form
4. Conclusions and Discussion
Author Contributions
Funding
Conflicts of Interest
References
- Estevez, P.G.; Gordoa, P.R. Painleve analysis of the generalized Burgers-Huxley equation. J. Phys. A Math. Gen. 1990, 23, 4831–4837. [Google Scholar] [CrossRef]
- Yefimova, O.Y.; Kudryashov, N.A. Exact solutions of the Burgers-Huxley equation. J. Appl. Math. Mech. 2004, 68, 413–420. [Google Scholar] [CrossRef]
- Gudkov, V.V. A family of exact travelling wave solutions to nonlinear evolution and wave equations. J. Math. Phys. 1997, 38, 4794–4803. [Google Scholar] [CrossRef]
- Ervin, V.J.; Ames, W.F.; Adams, E. Nonlinear waves in pellet fusion. In Wave Phenomena: Modern Theory and Applications; Rogers, C., Moodie Bryant, T., Eds.; North Holland: Amsterdam, The Netherland, 1984; pp. 199–210. [Google Scholar]
- Chisholm, J.S.R.; Common, A.K. A class of second-order differential equations and related first-order systems. J. Phys. A Math. Gen. 1987, 20, 5459–5472. [Google Scholar] [CrossRef]
- Sinelshchikov, D.I.; Kudryashov, N.A. On the Jacobi last multipliers and Lagrangians for a family of Liénard-type equations. Appl. Math. Comput. 2017, 307, 257–264. [Google Scholar] [CrossRef]
- Bagderina, Y.Y. Invariants of a family of scalar second-order ordinary differential equations for Lie symmetries and first integrals. J. Phys. A Math. Theor. 2016, 49, 155202. [Google Scholar] [CrossRef]
- Giné, J.; Valls, C. On the dynamics of the Rayleigh–Duffing oscillator. Nonlinear Anal. Real World Appl. 2019, 45, 309–319. [Google Scholar] [CrossRef]
- Giné, J.; Valls, C. Liouvillian integrability of a general Rayleigh–Duffing oscillator. J. Nonlin. Math. Phys. 2019, 26, 169–187. [Google Scholar] [CrossRef]
- Demina, M.V. Novel algebraic aspects of Liouvillian integrability for two-dimensional polynomial dynamical systems. Phys. Lett. A 2018, 382, 1353–1360. [Google Scholar] [CrossRef]
- Singer, M. Liouvillian first integrals of differential systems. Trans. Am. Math. Soc. 1992, 333, 673–688. [Google Scholar] [CrossRef]
- Christopher, C.J. Invariant algebraic curves and conditions for a centre. Proc. R. Soc. Edinburgh Sect. A 1994, 124, 1209–1229. [Google Scholar] [CrossRef]
- Duarte, L.G.S.; Moreira, I.C.; Santos, F.C. Linearization under nonpoint transformations. J. Phys. A Math. Gen. 1994, 27, L739–L743. [Google Scholar] [CrossRef]
- Nakpim, W.; Meleshko, S.V. Linearization of Second-Order Ordinary Differential Equations by Generalized Sundman Transformations. Symmetry Integr. Geom. Methods Appl. 2010, 6, 1–11. [Google Scholar] [CrossRef]
- Kudryashov, N.A.; Sinelshchikov, D.I. On the criteria for integrability of the Liénard equation. Appl. Math. Lett. 2016, 57, 114–120. [Google Scholar] [CrossRef]
- Sinelshchikov, D.I.; Kudryashov, N.A. Integrable Nonautonomous Liénard-Type Equations. Theor. Math. Phys. 2018, 196, 1230–1240. [Google Scholar] [CrossRef]
- Demina, M.V. Invariant algebraic curves for Liénard dynamical systems revisited. Appl. Math. Lett. 2018, 84, 42–48. [Google Scholar] [CrossRef]
- Walker, R.J. Algebraic Curves; Springer: New York, NY, USA, 1978. [Google Scholar]
- Bruno, A.D. Power Geometry in Algebraic and Differential Equations; Elsevier Science (North–Holland): Amsterdam, The Netherland, 2000. [Google Scholar]
- Bruno, A.D. Asymptotic behaviour and expansions of solutions of an ordinary differential equation. Russ. Math. Surv. 2004, 59, 429–481. [Google Scholar] [CrossRef]
- Prelle, M.J.; Singer, M.F. Elementary first integrals of differential equations. Trans. Am. Math. Soc. 1983, 279, 215–229. [Google Scholar] [CrossRef]
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Demina, M.; Sinelshchikov, D. Integrability Properties of Cubic Liénard Oscillators with Linear Damping. Symmetry 2019, 11, 1378. https://doi.org/10.3390/sym11111378
Demina M, Sinelshchikov D. Integrability Properties of Cubic Liénard Oscillators with Linear Damping. Symmetry. 2019; 11(11):1378. https://doi.org/10.3390/sym11111378
Chicago/Turabian StyleDemina, Maria, and Dmitry Sinelshchikov. 2019. "Integrability Properties of Cubic Liénard Oscillators with Linear Damping" Symmetry 11, no. 11: 1378. https://doi.org/10.3390/sym11111378
APA StyleDemina, M., & Sinelshchikov, D. (2019). Integrability Properties of Cubic Liénard Oscillators with Linear Damping. Symmetry, 11(11), 1378. https://doi.org/10.3390/sym11111378