Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
planar polynomial systems; Kukles systems; generalized Liénard systems; non-liouvillian first integrals
planar polynomial systems; Kukles systems; generalized Liénard systems; non-liouvillian first integrals
Summary:
We show that a transformation method relating planar first-order differential systems to second order equations is an effective tool for finding non-liouvillian first integrals. We obtain explicit first integrals for a subclass of Kukles systems, including fourth and fifth order systems, and for generalized Liénard-type systems.
References:
[1] M. Bronstein and Sébastien Lafaille: Solutions of linear ordinary differential equations in terms of special functions. In: Proceedings of the 2002 International Symposium on Symbolic and Algebraic Computation, , , 2002, pp. 23–28. MR 2035229
[2] L. A. Čerkas: Conditions for a center for a certain Liénard equation. Differencial’nye Uravnenija 12 (1976), 292–298, 379. MR 0404755
[3] J. Chavarriga, I. A. García, and J. Giné: On Lie’s symmetries for planar polynomial differential systems. Nonlinearity 14 (2001), 863–880. DOI 10.1088/0951-7715/14/4/313 | MR 1837644
[4] E. S. Cheb-Terrab: Non-Liouvillian solutions for second order linear ODEs. Proceedings of the Workshop on Group Theory and Numerical Analysis, CRM, Montreal, 2003. MR 2182813
[5] G. Darboux: Mémoire sur les équations différentielles algébraiques du premier ordre et du premier degré (Mélanges). Bull. Sci. Math. 2 (1878), 60–96, 123–144, 151–200.
[6] H. Giacomini, J. Giné, and M. Grau: Integrability of planar polynomial differential systems through linear differential equations. Rocky Mountain J. Math. (2004), . MR 2234815
[7] J. Giné: Conditions for the existence of a center for the Kukles homogeneous system. Comput. Math. Appl. 43 (2002), 1261–1269. DOI 10.1016/S0898-1221(02)00098-6 | MR 1906353
[8] I. S. Gradshteyn, I. M. Ryzhik: Tables of Series, Products and Integrals. Academic Press, San Diego, 1994. MR 1243179
[9] A. Goriely: Integrability and Nonintegrability of Dynamical Systems. World Scientific Publishing, Singapore, 2001. MR 1857742 | Zbl 1002.34001
[10] C. Grosche, F. Steiner: How to solve path integrals in quantum mechanics. J. Math. Phys. 36 (1995), 2354–2385. DOI 10.1063/1.531043 | MR 1329257
[11] M. Hayashi: On the local center of generalized Liénard-type systems. Far East J. Math. Sci. (FJMS) 10 (2003), 265–283. MR 2008415 | Zbl 1052.34035
[12] D. Hilbert: Mathematical problems. Reprinted from Bull. Amer. Math. Soc. 8 (1902), 437–479. DOI 10.1090/S0002-9904-1902-00923-3 | MR 1557926
[13] P. D. Lax: Integrals of nonlinear equations of evolution and solitary waves. Comm. Pure Appl. Math. 21 (1968), 467–490. DOI 10.1002/cpa.3160210503 | MR 0235310 | Zbl 0162.41103
[14] Z. Liu, E. Sáez, and I. Szántó: A cubic system with an invariant triangle surrounding at least one limit cycle. Taiwanese J. Math. 7 (2003), 275–281. DOI 10.11650/twjm/1500575064 | MR 1978016
[15] R. Milson: Liouville transformation and exactly solvable Schrödinger equations. Internat. J. Theor. Phys. 37 (1998), 1735–1752. DOI 10.1023/A:1026696709617 | MR 1646396 | Zbl 0929.34075
[16] M. J. Prelle, M. F. Singer: Elementary first integrals of differential equations. Trans. Am. Math. Soc. 279 (1983), 215–229. DOI 10.1090/S0002-9947-1983-0704611-X | MR 0704611
[17] M. F. Singer: Liouvillian first integrals of differential equations. Trans. Amer. Math. Soc. 333 (1992), 673–688. DOI 10.1090/S0002-9947-1992-1062869-X | MR 1062869 | Zbl 0756.12006
[18] J.-M. Strelcyn, S. Wojciechowski: A method of finding integrals for three-dimensional dynamical systems. Phys. Lett. A 133 (1988), 207–212. DOI 10.1016/0375-9601(88)91018-3 | MR 0969750
[19] A. G. Ushveridze: Quasi-exactly Solvable Models in Quantum Mechanics. Institute of Physics Publishing, London, 1994. MR 1329549 | Zbl 0834.58042
[20] F. J. Wright: Recognising and solving special function ODEs. Preprint (2000).
[21] C.-D. Zhao, Q.-M. He: On the center of the generalized Liénard system. Czechoslovak Math. J. 52 (127) (2002), 817–832. DOI 10.1023/B:CMAJ.0000027236.52900.55 | MR 1940062 | Zbl 1021.34023
Search
Partner of
