The Lie symmetry group of the general Liénard-type equation. (English) Zbl 1436.34030
Summary: We consider the general Liénard-type equation \(\ddot{u}=\sum_{k=0}^n f_k\dot{u}^k\) for \(n\geq4\). This equation naturally admits the Lie symmetry \(\frac{\partial}{\partial t}\). We completely characterize when this equation admits another Lie symmetry, and give an easily verifiable condition for this on the functions \(f_0,\dots, f_n\). Moreover, we give an equivalent characterization of this condition. Similar results have already been obtained previously in the cases \(n = 1\) or \(n = 2\). That is, this paper handles all remaining cases except for \(n = 3\).
MSC:
| 34C14 | Symmetries, invariants of ordinary differential equations |
| 34A26 | Geometric methods in ordinary differential equations |
| 34C20 | Transformation and reduction of ordinary differential equations and systems, normal forms |
Keywords:
second-order ordinary differential equation; Liénard-type equation; Levinson-Smith-type equation; Lie group; symmetry analysis; Lie algebra of infinitesimal symmetriesReferences:
| [1] | Mickens, R. E., An Introduction to Nonlinear Oscillations, 224 (1981), Cambridge etc.: Cambridge University Press, Cambridge etc. · Zbl 0459.34002 |
| [2] | Levinson, N., On the existence of periodic solutions for second order differential equations with a forcing term, J. Math. Phys., Mass. Inst. Techn, 22, 41-48 (1943) · Zbl 0061.18909 |
| [3] | Levinson, N.; Smith, O. K., A general equation for relaxation oscillations, Duke Math. J, 9, 382-403 (1942) · Zbl 0061.18908 · doi:10.1215/S0012-7094-42-00928-1 |
| [4] | Strutt, J. W., The Theory of Sound, New York: Dover Publications, New York · JFM 05.0535.01 |
| [5] | Polyanin, A. D.; Zaitsev, V. F., Handbook of Exact Solutions for Ordinary Differential Equations (2003), Boca Raton, FL: CRC Press, Boca Raton, FL · Zbl 1015.34001 |
| [6] | van der Pol, B., On relaxation-oscillations, The Lond, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 2, 978-992 (1927) · JFM 52.0450.05 · doi:10.1080/14786442608564127 |
| [7] | van der Pol, B.; van der Mark, J., The heartbeat considered as a relaxation oscillation, and an electrical model of the heart, The Lond, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 6, 763-775 (1928) · doi:10.1080/14786441108564652 |
| [8] | Fitzhugh, R., Impulses and physiological states in theoretical models of nerve membrane, Biophys J, 1, 6, 445-466 (1961) · doi:10.1016/S0006-3495(61)86902-6 |
| [9] | Nagumo, J.; Arimoto, S.; Yoshizawa, S., An active pulse transmission line simulating nerve axon, Proc. IRE, 50, 2061-2070 (1962) · doi:10.1109/JRPROC.1962.288235 |
| [10] | Jones, D. S.; Plank, M.; Sleeman, B. D., Differential Equations and Mathematical Biology (2009), Chapman and Hall/CRC · Zbl 1298.92003 |
| [11] | Nucci, M. C.; Sanchini, G., Lagrangians and Conservation Laws of an Easter Island Population Model, Symmetry, 7, 3, 1613-1632 (2015) · Zbl 1375.37148 · doi:10.3390/sym7031613 |
| [12] | Faria, J.R., Cuestas, J.C. and Gil-Alana, L.A., Unemployment and entrepreneurship: a cyclical relation? (Discussion papers. Nottingham Trent University, Bottingham Business School, Economics Division, 2008/2). · Zbl 1179.91146 |
| [13] | Goodwin, R. M.; Feinstein, C. H., A growth cycle, Socialism, Capitalism and Economic Growth, 54-58 (1975), Cambridge University Press: Cambridge University Press, Cambridge |
| [14] | Kaldor, N., A model of the trade cycle, The Economic Journal, 50, 197, 78-92 (1940) · doi:10.2307/2225740 |
| [15] | Kalecki, M., A theory of the business cycle, The Review of Economic Studies, 4, 2, 77-97 (1937) · doi:10.2307/2967606 |
| [16] | Bocharov, A.V., Chetverikov, V.N., Duzhin, S.V., Khorꞌkova, N.G., Krasilꞌshchik, I.S., Samokhin, A.V., Torkhov, Y.N., Verbovetsky, A.M. and Vinogradov, A.M., Symmetries and conservation laws for differential equations of mathematical physics, Translations of Mathematical Monographs, Vol. 182 (American Mathematical Society, Providence, RI, 1999). Edited and with a preface by Krasilꞌshchik and Vinogradov, Translated from the 1997 Russian original by Verbovetsky [A. M. Verbovetskiĭ] and Krasil shchik. |
| [17] | Olver, P. J., Applications of Lie Groups to Differential Equations, 107 (1993), Springer-Verlag: Springer-Verlag, New York · Zbl 0785.58003 |
| [18] | Edwards, M.; Nucci, M. C., Application of Lie group analysis to a core group model for sexually transmitted diseases, J. Nonlinear Math. Phys, 13, 2, 211-230 (2006) · Zbl 1117.34040 · doi:10.2991/jnmp.2006.13.2.6 |
| [19] | Nucci, M. C., The role of symmetries in solving differential equations, Mathl. Comput. Modelling, 25, 8-9, 181-193 (1997) · Zbl 0884.76075 · doi:10.1016/S0895-7177(97)00068-X |
| [20] | Nucci, M. C.; Leach, P. G. L., Singularity and symmetry analyses of mathematical models of epidemics, South African Journal of Science, 105, 136-146 (2009) |
| [21] | Pandey, S. N.; Bindu, P. S.; Senthilvelan, M.; Lakshmanan, M., A group theoretical identification of integrable cases of the Liénard-type equation ẍ + f (x)ẋ + g(x) = 0. I: Equations having nonmaximal number of Lie point symmetries, J. Math. Phys, 50, 8, 082702 (2009) · Zbl 1328.34031 · doi:10.1063/1.3187783 |
| [22] | Pandey, S. N.; Bindu, P. S.; Senthilvelan, M.; Lakshmanan, M., A group theoretical identification of integrable equations in the Liénard-type equation ẍ + f (x)ẋ + g(x) = 0. II: Equations having maximal Lie point symmetries, J. Math. Phys, 50, 10, 102701 (2009) · Zbl 1283.34036 · doi:10.1063/1.3204075 |
| [23] | Tiwari, A. K.; Pandey, S. N.; Senthilvelan, M.; Lakshmanan, M., Classification of Lie point symmetries for quadratic Liénard type equation ẍ + f (x)ẋ^2 + g(x) = 0, J. Math. Phys, 54, 5, 053506 (2013) · Zbl 1295.34049 · doi:10.1063/1.4803455 |
| [24] | Tiwari, A. K.; Pandey, S. N.; Senthilvelan, M.; Lakshmanan, M., Erratum: “Classification of Lie point symmetries for quadratic Liénard type equation ẍ + f (x)ẋ^2 + g(x) = 0”, J. Math. Phys, 55, 5, 059901 (2014) · Zbl 1318.34052 · doi:10.1063/1.4871778 |
| [25] | Tiwari, A. K.; Pandey, S. N.; Senthilvelan, M.; Lakshmanan, M., Lie point symmetries classification of the mixed Liénard-type equation, Nonlinear Dynamics, 82, 4, 1953-1968 (2015) · Zbl 1437.34039 · doi:10.1007/s11071-015-2290-z |
| [26] | Paliathanasis, A.; Leach, P. G.L., Comment on “Classification of Lie point symmetries for quadratic Liénard type equation ẍ + f (x)ẋ^2 + g(x) = 0” [J. Math. Phys. 54, 053506 (2013)] and its erratum [J. Math. Phys. 55, 059901 (2014)], J. Math. Phys., 57, 2 (2016) · Zbl 1337.34041 · doi:10.1063/1.4940692 |
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