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On the birth and death of algebraic limit cycles in quadratic differential systems

Part of: Nonlinear dynamics Local and nonlocal bifurcation theory Qualitative theory

Published online by Cambridge University Press:  26 May 2020

JAUME LLIBRE ,
REGILENE OLIVEIRA Open the ORCID record for REGILENE OLIVEIRA [Opens in a new window]  and
YULIN ZHAO
JAUME LLIBRE
Affiliation:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain. email: [email protected]
REGILENE OLIVEIRA
Affiliation:
Departamento de Matemática, Instituto de Ciências Matemáticas e Computação, Universidade de São Paulo, Avenida Trabalhador São Carlense, 400, 13566-590São Carlos, SP, Brazil. email: [email protected]
YULIN ZHAO
Affiliation:
School of Mathematics (Zhuhai), Sun Yat-sen University, Zhuhai519082, Guangdong, People’s Republic of China. email: [email protected]
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Abstract

In 1958 started the study of the families of algebraic limit cycles in the class of planar quadratic polynomial differential systems. In the present we known one family of algebraic limit cycles of degree 2 and four families of algebraic limit cycles of degree 4, and that there are no limit cycles of degree 3. All the families of algebraic limit cycles of degree 2 and 4 are known, this is not the case for the families of degree higher than 4. We also know that there exist two families of algebraic limit cycles of degree 5 and one family of degree 6, but we do not know if these families are all the families of degree 5 and 6. Until today it is an open problem to know if there are algebraic limit cycles of degree higher than 6 inside the class of quadratic polynomial differential systems. Here we investigate the birth and death of all the known families of algebraic limit cycles of quadratic polynomial differential systems.

Keywords

Algebraic limit cyclesHopf bifurcationhomoclinic orbitsheteroclinic orbitsquadratic polynomial differential systems

MSC classification

Primary: 34C07: Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramifications)
Secondary: 34C05: Location of integral curves, singular points, limit cycles 37G15: Bifurcations of limit cycles and periodic orbits 70K05: Phase plane analysis, limit cycles
Type
Papers
Information
European Journal of Applied Mathematics , Volume 32 , Issue 2 , April 2021 , pp. 317 - 336
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

Alberich-Carramiñana, M., Ferragut, A. & Llibre, J. (2019) Quadratic planar differential systems having algebraic limit cycles, preprint.Google Scholar
Alvarez, M. J., Ferragut, A. & Jarque, X. (2011) A survey on the blow up technique. Int. J. Bifur. Chaos Appl. Sci. Eng. 21, 3103–3118.CrossRefGoogle Scholar
Chavarriga, J., Giacomini, H. & Llibre, J. (2001) Uniqueness of algebraic limit cycles for quadratic systems. J. Math. Anal. Appl. 261, 85–99.CrossRefGoogle Scholar
Chavarriga, J., Llibre, J. & Sorolla, J. (2003) Algebraic limit cycles of quadratic systems. J. Differ. Equations 200, 206–244.CrossRefGoogle Scholar
Chavarriga, J., Llibre, J. & Moulin Ollagnier, J. (2001) On a result of Darboux. LMS J. Comput. Math. 4, 197–210.CrossRefGoogle Scholar
Christopher, C., Llibre, J. & Świrszcz, G. (2005) Invariant algebraic curves of large degree for quadratic systems. J. Math. Anal.Appl. 303, 450–461.CrossRefGoogle Scholar
Dumortier, F., Llibre, J. & Artés, J. C. (2006) Qualitative Theory of Planar Differential Systems, Universitext, Springer, New York.Google Scholar
Dumortier, F., Roussarie, R. & Rousseau, C. (1994) Hilbert’s 16th problem for quadratic vector fields. J. Differ. Equations 110, 86–133.CrossRefGoogle Scholar
Evdokimenco, R. M. (1970) Construction of algebraic paths and the qualitative investigation in the large of the properties of integral curves of a system of differential equations. Differ. Equations 6, 1349–1358.Google Scholar
Evdokimenco, R. M. (1974) Behavior of integral curves of a dynamic system. Differ. Equations 9, 1095–1103.Google Scholar
Evdokimenco, R. M. (1979) Investigation in the large of a dynamic system. Differ. Equations 15, 215–221.Google Scholar
Filiptsov, V. F. (1973) Algebraic limit cycles. Differ. Equations 9, 983–986.Google Scholar
de Freitas, B. R., Llibre, J. & Medrado, J. C. (2018) Limit cycles of continuous and discontinuous piecewise-linear differential systems in $\mathbb{R}^3$ . J. Comput. Appl. Math. 338, 311–323.CrossRefGoogle Scholar
Hilbert, D. (1900) Mathematische Probleme. Lecture, Second Internat. Congr. Math. (Paris, 1900),Nachr. Ges. Wiss. G”ttingen Math. Phys. KL., 253–297; English translation, Bull. Amer. Math. Soc. 8 (1902), 437–479.Google Scholar
Ilyashenko, Yu. (2002) Centennial history of Hilbert’s 16-th problem. Bull. Amer. Math. Soc. 39, 301–354.CrossRefGoogle Scholar
Kuznetsov, Y. A. (2004) Elements of Applied Bifurcation Theory, 3rd ed. Applied Mathematical Sciences, Vol. 112, Springer-Verlag, New York.Google Scholar
Li, J. (2003) Hilbert’s 16th problem and bifurcations of planar polynomial vector fields. Int. J. Bifur. Chaos Appl. Sci. Eng. 13, 47–106.CrossRefGoogle Scholar
Llibre, J. & Itikawa, J. (2015) Limit cycles for continuous and discontinuous perturbations of uniform isochronous cubic centers. J. Comput. Appl. Math. 277, 171–191.CrossRefGoogle Scholar
Llibre, J. & Schlomiuk, D. (2004) The geometry of differential quadratic systems with a weak focus of third order. Can. J. Math. 56, 310–343.CrossRefGoogle Scholar
Llibre, J. & Świrszcz, G. (2007) Classification of quadratic systems admitting the existence of an algebraic limit cycle. Bull. Sci. Math. 131, 405–421.CrossRefGoogle Scholar
Makovínyiová, K. & Zimka, R. (2013) On the bifurcation of limit cycles in a dynamic model of a small open economy. Eur. J. Appl. Math. 24, 455–470.CrossRefGoogle Scholar
Markus, L. (1954) Global structure of ordinary differential equations in the plane. Trans. Amer. Math Soc. 76, 127–148.CrossRefGoogle Scholar
Neumann, D. A. (1975) Classification of continuous flows on 2–manifolds. Proc. Amer. Math. Soc. 48, 73–81.Google Scholar
Peixoto, M. M. (1973) On the classification of flows on 2–manifolds. In: Dynamical Systems (Proceedings of a Symposium Held at the University of Bahia, Salvador, 1971), Academic, New York, pp. 389–419.Google Scholar
Poincaré, H. (1951) Mémoire sur les courbes définies par les équations différentielles, Oeuvreus de Henri Poincaré, Vol. I, Gauthiers–Villars, Paris, pp. 95–114.Google Scholar
Rayleigh, L. (1945) The Theory of Sound, Dover, New York.Google Scholar
Van der Pol, B. (1927) On relaxation-oscillations. London Edinburgh Dublin Phil. Mag. J. Sci. 2(7), 978–992.CrossRefGoogle Scholar
Yablonskii, A. I. (1966) Limit cycles of a certain differential equations. Differ. Equations 2, 335–344 (In Russian).Google Scholar
Yuan-Xun, Q. (1958) On the algebraic limit cycles of second degree of the differential equation ${dy/dx\, = \,\sum\nolimits_{0 \le i + j \le 2} {{a_{ij}}{x^i}{y^j}} /\sum\nolimits_{0 \le i + j \le 2} {{b_{ij}}{x^i}{y^j}}}$ Acta Math. Sinica 8, 23–35.Google Scholar