×

The period function of the generalized Lotka-Volterra centers. (English) Zbl 1147.34034

The author studies the period function of the origin for the quadratic system
\[ \dot x=-y-bx^2-cxy+by^2,\quad \dot y= x+xy, \]
where \(b\) and \(c\) are real parameters. Although it is believed that this function will be increasing, this is yet an open question. He proves that for every value of the parameters \((b,c)\) other than \((-1/2,0),\) for which the center is isochronous, the period function of the origin is monotonous increasing near the outer boundary of the period annulus. In addition: (i) there is no bifurcation of critical periods from the outer boundary of the period annulus at the parameters inside \({\mathbb R}^2\setminus \{\Gamma_1\cup\Gamma_2\};\) (ii) if \((a,b)\in \Gamma_1\) then the period function is globally monotone increasing. Here \(\Gamma_1=\{ (b,c)\in {\mathbb R}^2 : b(b+1)(c^2+4b(b+1))=0 \}\) and \(\Gamma_2=\{(b,c)\in {\mathbb R}^2 : b\in(-1,0) ,\;c=0\}.\)
Notice that all these results confirm the expected property and that the sets \(\Gamma_j, j=1,2,\) essentially correspond to bifurcations in the phase portraits of the system. The proof of the first two results is quite technical and uses a previous result of the author and collaborators that gives an asymptotic expansion of the time function associated to the passage trough a hyperbolic saddle for some meromorphic vector fields. The proof of (ii) uses that for these cases the corresponding planar vector fields can be transformed into Hamiltonian systems having Hamiltonian functions of the form \(F(u)+G(v).\)

MSC:

34C25 Periodic solutions to ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
37C27 Periodic orbits of vector fields and flows
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Abramowitz, M.; Stegun, I. A., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (1992), Dover Publications: Dover Publications New York · Zbl 0515.33001
[2] Chicone, C., The monotonicity of the period function for planar Hamiltonian vector fields, J. Differential Equations, 69, 310-321 (1987) · Zbl 0622.34033
[3] Chicone, C., Geometric methods of two-point nonlinear boundary value problem, J. Differential Equations, 72, 360-407 (1988) · Zbl 0693.34014
[4] Chicone, C.; Dumortier, F., A quadratic system with a nonmonotonic period function, Proc. Amer. Math. Soc., 102, 706-710 (1988) · Zbl 0651.34043
[5] Chicone, C.; Dumortier, F., Finiteness for critical periods of planar analytic vector fields, Nonlinear Anal., 20, 315-335 (1993) · Zbl 0773.34032
[6] Chicone, C.; Jacobs, M., Bifurcation of critical periods for plane vector fields, Trans. Amer. Math. Soc., 312, 433-486 (1989) · Zbl 0678.58027
[7] Christopher, C. J.; Devlin, C. J., Isochronous centers in planar polynomial systems, SIAM J. Math. Anal., 28, 162-177 (1997) · Zbl 0881.34057
[8] Cima, A.; Gasull, A.; Mañosas, F., Period function for a class of Hamiltonian systems, J. Differential Equations, 168, 180-199 (2000) · Zbl 0991.37039
[9] Cima, A.; Mañosas, F.; Villadelprat, J., Isochronicity for several classes of Hamiltonian systems, J. Differential Equations, 157, 373-413 (1999) · Zbl 0941.34017
[10] Coppel, W. A.; Gavrilov, L., The period function of a Hamiltonian quadratic system, Differential Integral Equations, 6, 1357-1365 (1993) · Zbl 0780.34023
[11] Dumortier, F.; Roussarie, R.; Rousseau, C., Elementary graphics of cyclicity 1 and 2, Nonlinearity, 7, 1001-1043 (1994) · Zbl 0855.58043
[12] Freire, E.; Gasull, A.; Guillamon, A., First derivative of the period function with applications, J. Differential Equations, 204, 139-162 (2004) · Zbl 1063.34025
[13] Gasull, A.; Guillamon, A.; Mañosa, V., An explicit expression of the first Liapunov and period constants with applications, J. Math. Anal. Appl., 211, 190-212 (1997) · Zbl 0882.34040
[14] Gasull, A.; Guillamon, A.; Villadelprat, J., The period function for second-order quadratic ODEs is monotone, Qual. Theory Dyn. Syst., 5, 201-224 (2004)
[15] Gasull, A.; Mañosa, V.; Villadelprat, J., On the period of the limit cycles appearing in one-parameter bifurcations, J. Differential Equations, 213, 255-288 (2005) · Zbl 1081.34037
[16] Loud, W. S., Behaviour of the period of solutions of certain plane autonomous systems near centers, Contrib. Differential Equations, 3, 21-36 (1964) · Zbl 0139.04301
[17] 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
[18] Mardešić, P.; Marín, D.; Villadelprat, J., On the time function of the Dulac map for families of meromorphic vector fields, Nonlinearity, 16, 855-881 (2003) · Zbl 1034.34037
[19] Mardešić, P.; Marín, D.; Villadelprat, J., The period function of reversible quadratic centers, J. Differential Equations, 224, 120-171 (2006) · Zbl 1092.34020
[20] Marín, D.; Villadelprat, J., On the return time function around monodromic polycycles, J. Differential Equations, 228, 226-258 (2006) · Zbl 1110.34021
[21] Mourtada, A., Cyclicité finie des polycycles hyperboliques de champs de vecteurs du plan: mise sous forme normale, (Bifurcations of Planar Vector Fields. Bifurcations of Planar Vector Fields, Luminy, 1989. Bifurcations of Planar Vector Fields. Bifurcations of Planar Vector Fields, Luminy, 1989, Lecture Notes in Math., vol. 1455 (1990), Springer: Springer Berlin), 272-314 · Zbl 0719.58031
[22] Rothe, F., The periods of the Volterra-Lokta system, J. Reine Angew. Math., 355, 129-138 (1985) · Zbl 0547.92011
[23] Rothe, F., Remarks on periods of planar Hamiltonian systems, SIAM J. Math. Anal., 24, 129-154 (1993) · Zbl 0769.34032
[24] Roussarie, R., On the number of limit cycles which appear by perturbation of separatrix loop of planar vector fields, Bol. Soc. Brasil. Mat., 17, 67-101 (1986) · Zbl 0628.34032
[25] Rousseau, C.; Toni, B., Local bifurcations of critical periods in the reduced Kukles system, Canad. J. Math., 49, 338-358 (1997) · Zbl 0885.34033
[26] Saavedra, M., Asymptotic expansion of the period function, J. Differential Equations, 193, 359-373 (2003) · Zbl 1048.34084
[27] Schaaf, R., Global behaviour of solution branches for some Neumann problems depending on one or several parameters, J. Reine Angew. Math., 346, 1-31 (1984) · Zbl 0513.34033
[28] Schaaf, R., A class of Hamiltonian systems with increasing periods, J. Reine Angew. Math., 363, 96-109 (1985) · Zbl 0565.34037
[29] Schlomiuk, D., Algebraic and geometric aspects of the theory of polynomial vector fields, (Bifurcations and Periodic Orbits of Vector Fields. Bifurcations and Periodic Orbits of Vector Fields, Montreal, PQ, 1992 (1993), Kluwer Acad. Publ.: Kluwer Acad. Publ. Dordrecht), 429-467 · Zbl 0790.34031
[30] Schlomiuk, D., Algebraic particular integrals, integrability and the problem of the center, Trans. Amer. Math. Soc., 338, 799-841 (1993) · Zbl 0777.58028
[31] Smoller, J.; Wasserman, A., Global bifurcation of steady-state solutions, J. Differential Equations, 39, 269-290 (1981) · Zbl 0425.34028
[32] Waldvogel, J., The period in the Lotka-Volterra system is monotonic, J. Math. Anal. Appl., 114, 178-184 (1986) · Zbl 0588.92018
[33] Zevin, A. A.; Pinsky, M. A., Monotonicity criteria for an energy-period function in planar Hamiltonian systems, Nonlinearity, 14, 1425-1432 (2001) · Zbl 1001.37054
[34] Zhao, Y., The monotonicity of period function for codimension four quadratic system \(Q_4\), J. Differential Equations, 185, 370-387 (2002) · Zbl 1047.34024
[35] Zhao, Y., The period function for quadratic integrable systems with cubic orbits, J. Math. Anal. Appl., 301, 295-312 (2005) · Zbl 1068.34030
[36] Zhao, Y., On the monotonicity of the period function of a quadratic system, Discrete Contin. Dyn. Syst., 13, 795-810 (2005) · Zbl 1089.34038
[37] Żoła̧dek, H., Quadratic systems with center and their perturbations, J. Differential Equations, 109, 223-273 (1994) · Zbl 0797.34044
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.