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Linearizability conditions for Lotka-Volterra planar complex quartic systems having homogeneous nonlinearities. (English) Zbl 1217.34058
Summary: We investigate the linearizability problem for the two-dimensional Lotka-Volterra complex quartic systems which are linear systems perturbed by fourth degree homogeneous polynomials, i.e., we consider systems of the form x ' =x(1-a 30 x 3 -a 21 x y -a 12 xy 2 -a 0 3y 3 ), y ' =-y(1-b 30 x 3 -b 21 x 2 y-b 12 xy 2 -b 03 y 3 . The necessary and sufficient conditions for the linearizability of this system are found. From them the conditions for isochronicity of the corresponding real system can be derived.
34C20Transformation and reduction of ODE and systems, normal forms
34C05Location of integral curves, singular points, limit cycles (ODE)
37G05Normal forms
[1]Chavarriga, J.; Giacomini, H.; Giné, J.; Llibre, J.: On the integrability of two-dimensional flows, J. differential equations 157, 163-182 (1999) · Zbl 0940.37005 · doi:10.1006/jdeq.1998.3621
[2]Romanovski, V. G.; Shafer, D. S.: The center and cyclicity problems: A computational algebra approach, (2009)
[3]L. Feigenbaum, The center of oscillation versus the textbook writers of the early 18th century. in: From ancient omens to statistical mechanics, Acta Hist. Sci. Nat. Med. Edidit Bibl. Univ. Haun., vol. 39, Univ. Lib. Copenhagen, Copenhagen, 1987, pp. 193–202.
[4]Liapunov, A. M.: Stability of motion, (1966)
[5]Bibikov, Y. N.: Local theory of nonlinear analytic ordinary differential equations, Lecture notes in mathematics 702 (1979) · Zbl 0404.34005
[6]Giné, J.: Isochronous foci for analytic differential systems, Internat. J. Bifur. chaos appl. Sci. engrg. 13, No. 6, 1617-1623 (2003) · Zbl 1069.34046 · doi:10.1142/S0218127403007400
[7]Giné, J.; Grau, M.: Characterization of isochronous foci for planar analytic differential systems, Proc. R. Soc. Edinburgh sect. A 135, No. 5, 985-998 (2005) · Zbl 1092.34014 · doi:10.1017/S0308210500004236
[8]Zhang, G. Y.: From an iteration formula to Poincaré’s isochronous center theorem for holomorphic vector fields, Proc. amer. Math. soc. 135, No. 9, 2887-2891 (2007) · Zbl 1129.37027 · doi:10.1090/S0002-9939-07-08802-8
[9]Chavarriga, J.; Giné, J.; García, I. A.: Isochronous centers of a linear center perturbed by fourth degree homogeneous polynomial, Bull. sci. Math. 123, 77-96 (1999) · Zbl 0921.34032 · doi:10.1016/S0007-4497(99)80015-3
[10]Chavarriga, J.; Giné, J.; García, I. A.: Isochronous centers of a linear center perturbed by fifth degree homogeneous polynomial, J. comput. Appl. math. 126, 351-368 (2001) · Zbl 0978.34028 · doi:10.1016/S0377-0427(99)00364-7
[11]Lin, Y. P.; Li, J. B.: The normal form of a planar autonomous system and critical points of the period of closed orbits, Acta math. Sin. 34, 490-501 (1991) · Zbl 0744.34041
[12]Liu, Y.; Huang, W.: A new method to determine isochronous center conditions for polynomial differential systems, Bull. sci. Math. 127, No. 2, 133-148 (2003) · Zbl 1034.34032 · doi:10.1016/S0007-4497(02)00006-4
[13]Liu, Y.; Li, J.: Theory of values of singular point in complex autonomous differential systems, Sci. China ser. A 33, No. 1, 10-24 (1990) · Zbl 0686.34027
[14]Loud, W. S.: Behavior of the period of solutions of certain plane autonomous systems near centers, Contrib. differ. Equ. 3, 21-36 (1964) · Zbl 0139.04301
[15]Lukashevich, N. A.: The isochronism of a center of certain systems of differential equations, Differencial nye uravn. 1, 295-302 (1965) · Zbl 0178.43301
[16]Pleshkan, I.: A new method of investigating the isochronicity of a system of two differential equations, Differ. equ. 5, 796-802 (1969) · Zbl 0252.34034
[17]Romanovski, V. G.; Xingwu, Chen; Zhaoping, Hu: Linearizability of linear systems perturbed by fifth degree homogeneous polynomials, J. phys. A, math. Theor. 40, No. 22, 5905-5919 (2007) · Zbl 1127.34020 · doi:10.1088/1751-8113/40/22/010
[18]Romanovski, V. G.; Robnik, M.: The centre and isochronicity problems for some cubic systems, J. phys. A, math. Gen. 34, No. 47, 10267-10292 (2001) · Zbl 1014.34028 · doi:10.1088/0305-4470/34/47/326
[19]Xingwu, Chen; Romanovski, V. G.; Zhang, Weinian: Linearizability conditions of time-reversible quartic systems having homogeneous nonlinearities, Nonlinear anal. 69, No. 5–6, 1525-1539 (2008) · Zbl 1159.34025 · doi:10.1016/j.na.2007.07.009
[20]Lloyd, N. G.; Pearson, J. M.; Sáez, E.; Szántó, I.: Limit cycles of a cubic Kolmogorov system, Appl. math. Lett. 9, No. 1, 15-18 (1996) · Zbl 0858.34023 · doi:10.1016/0893-9659(95)00095-X
[21]Lloyd, N. G.; Pearson, J. M.; Sáez, E.; Szántó, I.: A cubic Kolmogorov system with six limit cycles, Comput. math. Appl. 44, No. 3–4, 445-455 (2002) · Zbl 1210.34048 · doi:10.1016/S0898-1221(02)00161-X
[22]Ye, Y.; Ye, W.: Cubic Kolmogorov differential system with two limit cycles surrounding the same focus, Ann. differential equations 1, No. 2, 201-207 (1985) · Zbl 0597.34020
[23]Coleman, C. S.: Hilbert’s 16th problem: how many cycles?, Differential equations models, vol. 1 1, 279-297 (1978)
[24]Chicone, C.; Jacobs, M.: Bifurcation of limit cycles from quadratic isochrones, J. differential equations 91, No. 2, 268-326 (1991) · Zbl 0733.34045 · doi:10.1016/0022-0396(91)90142-V
[25]Christopher, C.; Mardešić, P.; Rousseau, C.: Normalizable, integrable and linearizable saddle points for complex quadratic systems in C2, J. dyn. Control syst. 9, 311-363 (2003) · Zbl 1022.37035 · doi:10.1023/A:1024643521094
[26]Christopher, C.; Rousseau, C.: Nondegenerate linearizable centres of complex planar quadratic and symmetric cubic systems in C2, Publ. mat. 45, 95-123 (2001) · Zbl 0984.34023 · doi:10.5565/PUBLMAT_45101_04 · doi:http://mat.uab.es/pubmat/v45(1)/45101_04.pdf
[27]Giné, J.; Maza, S.: Orbital linearization in the quadratic Lotka–Volterra systems around singular points via Lie symmetries, J. nonlinear math. Phys. 16, No. 4, 455-464 (2009) · Zbl 1194.34065 · doi:10.1142/S1402925109000480
[28]Mardešić, P.; Rousseau, C.; Toni, B.: Linearization of isochronous centers, J. differential equations 121, 67-108 (1995) · Zbl 0830.34023 · doi:10.1006/jdeq.1995.1122
[29]Dolićanin, D.; Milovanović, G. V.; Romanovski, V. G.: Linearizability conditions for a cubic system, Appl. math. Comput. 190, No. 1, 937-945 (2007) · Zbl 1133.34331 · doi:10.1016/j.amc.2007.01.084
[30]Mardešić, P.; Moser-Jauslin, L.; Rousseau, C.: Darboux linearization and isochronous centers with a rational first integral, J. differential equations 134, 216-268 (1997) · Zbl 0881.34041 · doi:10.1006/jdeq.1996.3212
[31]Cox, D.; Little, J.; O’shea, D.: Ideals, varieties, and algorithms, (1992) · Zbl 0756.13017
[32]G. Pfister, W. Decker, H. Schönemann, S. Laplagne, primdec.lib. A singular 3.0 library for computing primary decomposition and radical of ideals, 2005.
[33]G.M. Greuel, G. Pfister, H. Schönemann, Singular 3.0 A computer algebra system for polynomial computations, Centre for Computer Algebra, University of Kaiserslautern, 2005. http://www.singular.uni-kl.de.
[34]Gianni, P.; Trager, B.; Zacharias, G.: Gröbner bases and primary decomposition of polynomials, J. symbolic comput. 6, 146-167 (1988) · Zbl 0667.13008 · doi:10.1016/S0747-7171(88)80040-3
[35]Wang, P. S.; Guy, M. J. T.; Davenport, J. H.: P-adic reconstruction of rational numbers, SIGSAM bull. 16, No. 2, 2-3 (1982) · Zbl 0489.68032 · doi:10.1145/1089292.1089293
[36]V.F. Edneral, Computer evaluation of cyclicity in planar cubic system, in: Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation, ACM Press, New York, 1997, pp. 305–309. · Zbl 0916.65078
[37]Fronville, A.; Sadovski, A. P.; Żoład̨ek, H.: The solution of the 1:-2 resonant center problem in the quadratic case, Fund. math. 157, 191-207 (1998) · Zbl 0943.34018
[38]Gravel, T.; Thibault, P.: Integrability and linearizability of the Lotka–Volterra system with a saddle point with rational hyperbolicity ratio, J. differential equations 184, 20-47 (2002) · Zbl 1054.34049 · doi:10.1006/jdeq.2001.4128
[39]Liu, Y.: The generalized focal values and bifurcation of limit cycles for quasi quadratic systems, Acta math. Sin. 45, 671-682 (2002) · Zbl 1027.34037
[40]Sibirskii, K. S.: Algebraic invariants of differential equations and matrices, (1976) · Zbl 0334.34014