ABSTRACT
We consider a family of the Liénard–type equations, which members appear in a vast range of applications. We study connections, given by the generalized Sundman transformations, between this family of equations and type II Painlevé–Gambier equations. As a result, we find a new condition that allow us to construct the closed–form general solution for each member of the corresponding family of the Liénard–type equations. We illustrate our results by providing two new examples of the integrable Liénard–type equations.
- 1. A.A. Andronov, A.A. Vitt, S.E. Khaikin, Theory of Oscillators, Dover Publications, New York, 2011. Google Scholar
- 2. M. Lakshmanan, S. Rajasekar, Nonlinear dynamics : integrability, chaos, and patterns, Springer, Heidelberg, 2003. Google ScholarCrossref
- 3. V.K. Chandrasekar, M. Senthilvelan, M. Lakshmanan, On the complete integrability and linearization of certain second-order nonlinear ordinary differential equations, Proc. R. Soc. A Math. Phys. Eng. Sci. 461 (2005) 2451–2476. https://doi.org/10.1098/rspa.2005.1465, Google ScholarCrossref
- 4. A.K. Tiwari, S.N. Pandey, M. Senthilvelan, M. Lakshmanan, Lie point symmetries classification of the mixed Liénard-type equation, Nonlinear Dyn. 82 (2015) 19531968. https://doi.org/10.1007/s11071-015-2290-z, Google ScholarCrossref
- 5. L.G.S. Duarte, I.C. Moreira, F.C. Santos, Linearization under nonpoint transformations, J. Phys. A. Math. Gen. 27 (1994) L739–L743. https://doi.org/10.1088/0305-4470/27/19/004, Google ScholarCrossref
- 6. W. Nakpim, S.V. Meleshko, Linearization of Second-Order Ordinary Differential Equations by Generalized Sundman Transformations, Symmetry, Integr. Geom. Methods Appl. 6 (2010) 1–11. Google Scholar
- 7. 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 https://doi.org/10.1142/S1402925110000696, Google ScholarCrossref
- 8. N.A. Kudryashov, D.I. Sinelshchikov, On the connection of the quadratic Liénard equation with an equation for the elliptic functions, Regul. Chaotic Dyn. 20 (2015) 486–496. https://doi.org/10.1134/S1560354715040073, Google ScholarCrossref
- 9. N.A. Kudryashov, D.I. Sinelshchikov, On the criteria for integrability of the Liénard equation, Appl. Math. Lett. 57 (2016) 114–120. https://doi.org/10.1016/j.aml.2016.01.012, Google ScholarCrossref
- 10. N.A. Kudryashov, D.I. Sinelshchikov, On connections of the Liénard equation with some equations of Painlevé–Gambier type, sumitted to J. Math. Anal. Appl. Google Scholar
- 11. E.L. Ince, Ordinary differential equations, Dover, New York, 1956. Google Scholar
Please Note: The number of views represents the full text views from December 2016 to date. Article views prior to December 2016 are not included.
