Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-22T21:20:12.891Z Has data issue: false hasContentIssue false
  • English
  • Français

Global Phase Portraits for the Abel Quadratic Polynomial Differential Equations of the Second Kind With Z2-symmetries

Published online by Cambridge University Press:  20 November 2018

Jaume Llibre  and
Claudia Valls
Jaume Llibre
Affiliation:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain, e-mail: [email protected]
Claudia Valls
Affiliation:
Departamento de Matemática, Instituto Superior Técnico, Universidade Técnica de Lisboa, Av. Rovisco Pais 1049–001, Lisboa, Portugal, e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We provide normal forms and the global phase portraits on the Poincaré disk for all Abel quadratic polynomial diòerential equations of the second kind with ${{\mathbb{Z}}_{2}}$-symmetries.

Keywords

37J3537K10Abel polynomial differential system of the second kindvector fieldphase portrait
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 61 , Issue 1 , 01 March 2018 , pp. 149 - 165
Copyright
Copyright © Canadian Mathematical Society 2018

References

[1] Artes, J. C., Kooij, R. E., and Llibre, J., Structuraüy stähle quadratic vectorfields. Mem. Amer. Math. Soc. 134(1998), no. 639. http://dx.doi.org/10.1090/memo/0639.Google Scholar
[2] Artes, J. C. and Llibre, J., Phase portraits for quadratic Systems having a focus and one antisaddle. Rocky Mountain J.Math. 24 (1994), 875–889. http://dx.doi.org/10.1216/rmjm/1181072378.Google Scholar
[3] Artes, J. C. and Llibre, J., Quadratic Hamiltonian vectorfields. J. Differential Equations 107 (1994), 80–95. http://dx.doi.Org/1 0.1006/jdeq.1 994.1004.Google Scholar
[4] Artes, J. C. and Llibre, J., Corrigendum: A correction to the paper “Quadratic Hamiltonian vectorfields” [J. Differential Equations 107 (1994), 80-95], J. Differential Equations 129 (1996), 559–560. http://dx.doi.Org/10.1006/jdeq.1996.01 27.Google Scholar
[5] Artes, J. C. and Llibre, J., Quadratic vectorfields with a weak focus ofthird order, Publ. Mat. 41 (1997), 7–39. http://dx.doi.Org/10.5565/PUBLMAT_41197_02.Google Scholar
[6] Artes, J. C., Llibre, J., Rezende, A. C., Schlomiuk, D., and Vulpe, N., Global configurations of singularities for quadratic differential Systems with exactly twofinite singularities of total multiplicity four. Electron. J. Qual. Theory Differ. Equ. 60 (2014), 1–43.Google Scholar
[7] Artes, J. C., Llibre, J., and Schlomiuk, D., The geometry of quadratic differential Systems with a weak focus of second order. Internat. J. Bifur. Chaos Appl. Sei. Engrg. 16 (2006), 3127–3194. http://dx.doi.Org/10.1142/S021812740601 6720.Google Scholar
[8] Artes, J. C., Llibre, J., and Schlomiuk, D., The geometry of quadratic differential Systems with a weak focus and an invariant straight line. Internat. J. Bifur. Chaos Appl. Sei. Engrg. 20 (2010), 3627–3662. http://dx.doi.Org/10.1142/S021812741002791XGoogle Scholar
[9] Artes, J. C., Llibre, J., Schlomiuk, D., and Vulpe, N., Geometrie configurations of singularities for quadratic differential Systems with three distinet real simplefinite singularities. J. Fixed Point Theory Appl. 14 (2013), 555–618. http://dx.doi.Org/10.1007/s11784-014-0175-2.Google Scholar
[10] Artes, J. C., Llibre, J., Schlomiuk, D., and Vulpe, N., Geometrie configurations of singularities for quadratic differential Systems with total finite multiplicity nif = 2. Electron. J. of Differential Equations (2014), no. 159.Google Scholar
[11] Artes, J. C., Llibre, J., Schlomiuk, D., and Vulpe, N., Global configurations of singularities for quadratic differential Systems with total finite multiplicity three and at most two real singularities. Qual. Theory Dyn. Syst. 13 (2014), 305–351. http://dx.doi.Org/10.1007/s12346-014-011 9-7.Google Scholar
[12] Artes, J. C., Llibre, J., Schlomiuk, D., and Vulpe, N., From topological to geometric equivalence in the classification of singularities at infinity for quadratic vectorfields, Rocky Mountain J. Math. 45 (2015), 29–113. http://dx.doi.Org/10.1216/RMJ-2O1 5-45-1-29.Google Scholar
[13] Artes, J. C., Llibre, J., Schlomiuk, D., and Vulpe, N., Global configurations of singularities for quadratic differential Systems with exactly three finite singularities of total multiplicity four. Electron. J. Qual. Theory Differ. Equ. (2015), no. 49.Google Scholar
[14] Artes, J. C., Llibre, J., and Vulpe, N., Quadratic Systems with an integrable saddle: a complete classification in the coefficient Space R12. Nonlinear Anal. 75 (2012), 5416–5447. http://dx.doi.Org/10.101 6/j.na.2O12.04.043.Google Scholar
[15] Cairö, L. and Llibre, J., Phase portraits of quadratic polynomial vectorfields having a rational first integral ofdegree 2. Nonlinear Anal. 67 (2007), 327–348. http://dx.doi.Org/10.1016/j.na.2006.04.021.Google Scholar
[16] Coll, B., Ferragut, A., and Llibre, J., Phase portraits of the quadratic Systems with a polynomial inverse integratingfactor. Internat. J. Bifur. Chaos Appl. Sei. Engrg. 19 (2009), 765–783. http://dx.doi.org/10.1142/S0218127409023299.Google Scholar
[17] Coll, B. and Llibre, J., Limit cyclesfor a quadratic System with an invariant straight line and some evolution of phase portraits. In: Qualitative theory of differential equations. Colloq. Math. Soc. Jänos Bolyai, 53. Bolyai Institut, Szeged, Hungria, 1988, pp. 111–123.Google Scholar
[18] Coll, B., Gasull, A., and Llibre, J., Quadratic Systems with a uniquefinite rest point. Publ. Mat. 32 (1988), 199–259. http://dx.doi.Org/10.5565/PUBLMAT_32288_O8.Google Scholar
[19] Coll, B., Gasull, A., and Llibre, J., Some theorems on the existence, uniqueness, and nonexistence oflimit cyclesfor quadratic Systems. J. Differential Equations 67 (1987), 372–399. http://dx.doi.Org/10.1016/0022-0396(87)90133-1.Google Scholar
[20] Coppel, W. A., Some quadratic Systems with at most one limit cycle. Dynam. Report. Ser. Dynam. Systems Appl. Vol. 2. Wiley, Chichester, 1989, pp. 61–88.Google Scholar
[21] Date, T., Classification and analysis of two-dimensional real homogeneous quadratic differential equation Systems. J. Differential Equations 32 (1979), 311–334. http://dx.doi.Org/10.1016/0022-0396(79)90037-8.Google Scholar
[22] Dumortier, F. and Li, C., Quadratic Lienard equations with quadratic damping. J. Differential Equations 139 (1997), 41–59. http://dx.doi.org/10.1006/jdeq.1997.3291.Google Scholar
[23] Dumortier, F., Llibre, J., and Artes, J. C., Qualitative theory ofplanar differential Systems. Springer-Verlag, Berlin, 2006.Google Scholar
[24] Garcia, B., Llibre, J., and Perez del Rio, J. S., Phaseportraits ofthe quadratic vectorfields with a polynomialfirst integral. Rend. Circ. Mat. Palermo 55 (2006), 420–440. http://dx.doi.org/10.1007/BF02874780.Google Scholar
[25] Gasull, A., Li-Ren, S., and Llibre, J., Chordal quadratic Systems. Rocky Mountain J. Math. 16 (1986), 751–782. http://dx.doi.org/10.1216/RMJ-1986-16-4-751.Google Scholar
[26] Gasull, A. and Prohens, R., Quadratic and cubic Systems with degenerate infinity. J. Math. Anal. Appl. 198 (1996), 25–34. http://dx.doi.Org/10.1006/jmaa.1 996.0065.Google Scholar
[27] Gonzalez, E. A. Generic properties of polynomial vector fields at infinity. Trans. Amer. Math. Soc. 143 (1969), 201–222. http://dx.doi.org/10.1090/S0002-9947-1969-0252788-8.Google Scholar
[28] de Jager, P., Phase portraits for quadratic Systems with a higher order singularity with two zero eigenvalues. J. Differential Equations 87 (1990), 169–204. http://dx.doi.Org/10.1016/0022-0396(90)90021-CGoogle Scholar
[29] Jarque, X., Llibre, J., and Shafer, D. S., Structurally stähle quadratic foliations. Rocky Mountain J. Math. 38 (2008), 489–530. http://dx.doi.org/10.1216/RMJ-2008-38-2-489.Google Scholar
[30] Kaiin, Y. F. and Vulpe, N. I., Affine-invariant conditions for the topological discrimination of quadratic Hamiltonian differential Systems. Differential Equations 34 (1998), 297–301.Google Scholar
[31] Lamb, J. and Robert, M., Reversible equivariant linear Systems. J. Differential Equations 159 (1999), 239–278. http://dx.doi.Org/1 0.1006/jdeq.1 999.3632.Google Scholar
[32] Li, C., Two problems of planar quadratic Systems. Scientia Sinica Ser. A 26 (1983), 471–481.Google Scholar
[33] Li, J. and Liu, Y., Global bifurcation in a perturbed cubic System with *-symmetry. Acta Math. Appl. Sinica 8 (1992), 131–143. http://dx.doi.org/10.1007/BF02006149.Google Scholar
[34] Li, W., Llibre, J., Nicolau, M., and Zhang, X., On the differentiability of first integrals oftwo dimensionalflows. Proc. Amer. Math. Soc. 130 (2002), 2079–2088. http://dx.doi.org/10.1090/S0002-9939-02-06310-4.Google Scholar
[35] Li, C. Z., Llibre, J., and Zhang, Z. F., Weakfocus, limit cycles, and bifurcations for bounded quadratic Systems. J. Differential Equations 115 (1995), 193–223. http://dx.doi.Org/10.1006/jdeq.1995.1012.Google Scholar
[36] Liu, Y. and Li, J., Z2-equivariant cubic System which yields 13 limit cycles. Acta Math. Appl. Sinica Engl. Ser.30 (2014), 781–800. http://dx.doi.org/10.1007/s10255-014-0420-x.Google Scholar
[37] Llibre, J. and Medrado, J. C., Darboux integrability and reversible quadratic vector fields. Rocky Mountain J. Math. 35 (2005), 1999–2057. http://dx.doi.Org/10.1216/rmjm/11 81069627.Google Scholar
[38] Llibre, J. and Oliveira, R. D. S., Phase portraits of quadratic polynomial vector fields havinga rational first integral ofdegree 3. Nonlinear Anal. 70 (2009), 3549–3560. http://dx.doi.Org/10.1016/j.na.2008.07.012.Google Scholar
[39] Llibre, J. and Oliveira, R. D. S., Erratum to “Phase portraits of quadratic polynomial vector fields having a rational first integral ofdegree 3” [Nonlinear Anal. 70 (2009), 3549-3560]. Nonlinear Anal. 71 (2009), 6378–6379. http://dx.doi.Org/10.1016/j.na.2009.06.002.Google Scholar
[40] Llibre, J. and Schlomiuk, D., The geometry of quadratic differential Systems with a weakfocus of third order. Canad. J. Math. 56 (2004), 310–343. http://dx.doi.Org/10.4153/CJM-2004-015-2.Google Scholar
[41] Llibre, J. and Zhang, X., Topological phase portraits of planar semi-linear quadratic vector fields. Houston J. Math. 27 (2001), 247–296.Google Scholar
[42] Lupan, M. and Vulpe, N., Classification of quadratic Systems with a symmetry center and simple infinite Singularpoints. Bul. Acad. §tiine Repub. Mold. Mat. (2003), 102-119.Google Scholar
[43] Markus, L., Global structure of ordinary differential equations in the plane. Trans. Amer. Math Soc. 76 (1954), 127–148. http://dx.doi.org/10.1090/S0002-9947-1954-0060657-0.Google Scholar
[44] Neumann, D. A., Classification of continuous flows on 2-manifolds. Proc. Amer. Math. Soc. 48 (1975), 73–81. http://dx.doi.org/10.1090/S0002-9939-1975-0356138-6.Google Scholar
[45] Nikolaev, I. V. and Vulpe, N. I., Topological classification of quadratic Systems with a unique finite second order singularity with two zero eigenvalues. Izv. Akad. Nauk. Respub. Moldova Mat. 1993 (1993), 3–8, 107, 109.Google Scholar
[46] Pal, J. and Schlomiuk, D., Summing up the dynamics of quadratic Hamiltonian Systems with a center. Canad. J. Math. 49 (1997), 583–599. http://dx.doi.Org/10.4153/CJM-1997-027-0.Google Scholar
[47] Peixoto, M. M., ed., Dynamical Systems. Academic Press, New York, 1973, pp. 389–420.Google Scholar
[48] Schlomiuk, D., Algebraic particular integrals, integrability and the problem of the center. Trans. Amer. Math. Soc. 338 (1993), 799–841. http://dx.doi.org/10.1090/S0002-9947-1993-11061 93-6.Google Scholar
[49] Schlomiuk, D. and Vulpe, N., Planar quadratic vectorfields with invariant lines of total multiplicity at leastfive. Qual. Theory Dyn. Syst. 5 (2004), 135–194. http://dx.doi.Org/10.1007/BF02968134.Google Scholar
[50] Schlomiuk, D. and Vulpe, N., Integrals and phase portraits of planar quadratic differential Systems with invariant lines ofat leastfive total multiplicity. Rocky Mountain J. Math. 38 (2008), 2015–2075. http://dx.doi.Org/1 0.121 6/RMJ-2008-38-6-201 5.Google Scholar
[51] Schlomiuk, D. and Vulpe, N., Planar quadratic differential Systems with invariant straight lines of total multiplicity four. Nonlinear Anal. 68 (2008), 681–715. http://dx.doi.Org/10.1 01 6/j.na.2OO6.11.028.Google Scholar
[52] Schlomiuk, D. and Vulpe, N., Integrals and phase portraits of planar quadratic differential Systems with invariant lines of total multiplicity four, Bul. Acad. § tiin(e Repub. Mold. Mat. 2008 (2008), 27–83.Google Scholar
[53] Schlomiuk, D. and Vulpe, N., The füll study of planar quadratic Systems possessing a line of singularities at infinity. J. Dynam. Differential Equations 20 (2008), 737–775. http://dx.doi.Org/10.1007/s10884-008-911 7-2.Google Scholar
[54] Schlomiuk, D. and Vulpe, N., Bifurcation diagrams and moduli Spaces of planar quadratic vector fields with invariant lines of total multiplicity four and having exactly three real singularities at infinity. Qual. Theory Dyn. Syst. 9 (2010), 251–300. http://dx.doi.org/10.1007/s12346-010-0028-3.Google Scholar
[55] Schlomiuk, D. and Vulpe, N., Global classification ofthe planar Lotka-Volterra differential Systems according to their configurations of invariant straight lines. J. Fixed Point Theory Appl. 8 (2010), 177–245. http://dx.doi.Org/1 0.1007/s11 784-010-0031 -y.Google Scholar
[56] Schlomiuk, D. and Vulpe, N., Global topological classification of Lotka-Volterra quadratic differential Systems. Electron. J. Differential Equations (2012), no. 64.Google Scholar
[57] Voldman, M., Calin, L. T., and Vulpe, N. I., Affine invariant conditions for the topological distinction of quadratic Systems with a critical point of 4th multiplicity. Publ. Mat. 40 (1996), 431–441. http://dx.doi.Org/10.5565/PUBLMAT_40296_13.Google Scholar
[58] Voldman, M. and Vulpe, N. I., Affine invariant conditions for topologically distinguishing quadratic Systems without finite critical points. Izv. Akad. Nauk. Respub. Moldova Mat. 1995 (1995), 100–112, 114,117.Google Scholar
[59] Voldman, M. and Vulpe, N. I., Affine invariant conditions for topologically distinguishing quadratic Systems with nij = 1. Nonlinear Anal. 31 (1998), 171–179. http://dx.doi.org/10.101 6/S0362-546X(96)00302-1.Google Scholar
[60] Vulpe, N. I., Affine-invariant conditions for topological distinction of quadratic Systems in the presence of a center. DifferentsiaFnye Uravneniya 19 (1983), 371–379.Google Scholar
[61] Vulpe, N. I. and Likhovetskii, A. Y., Coefficient conditions for the topological discrimination of quadratic Systems ofDarboux type. Mat. Issled. 106 (1989), 34–49,178.Google Scholar
[62] Vulpe, N. I. and Nikolaev, I. V., Topological classification ofQS with a unique third order Singular point Izv. Akad. Nauk. Respub. Moldova Mat. (1992), 37-44.Google Scholar
[63] Vulpe, N. I. and Nikolaev, I. V., Topological classification of quadratic Systems with afour-fold Singular point. Differential Equations 29 (1993), 1449–1453.Google Scholar
[64] Vulpe, N. I. and Sibirskii, K. S., Affinely invariant coefficient conditions for the topological distinctness of quadratic Systems. Mat. Issled. 10 (1975), 15–28, 238.Google Scholar
[65] Vulpe, N. I. and Sibirskii, K. S., Geometrie classification of a quadratic differential System. DifferentsiaFnye Uravneniya 13 (1977), 803–814, 963.Google Scholar
[66] Yu, P. and Han, M., On limit cycles ofthe Lienard equation with Z2 symmetry. Chaos Solitons Fractals 31 (2007), 617–630. http://dx.doi.Org/10.1016/j.chaos.2005.10.013.Google Scholar
[67] Zoladek, H., Quadratic Systems with a center and their perturbations. J. Differential Equations 109 (1994), 223–273. http://dx.doi.Org/10.1006/jdeq.1994.1049Google Scholar