The limit cycles of a class of quintic polynomial vector fields
Keywords
Limit cycle, periodic orbit, inverse integrating factor, polynomial vector fieldAbstract
Using the inverse integrating factor we study the limit cycles of a class of polynomial vector fields of degree $5$.References
A.A. Andronov, E.A. Leontovich, I.I. Gordon and A.L. Maier, Qualitative Theory of Second-Order Dynamical Systems, Wiley, New York, 1973.
R. Asheghi and H.R.Z. Zangeneh, Bifurcations of limit cycles for a quintic Hamiltonian system with a double cuspidal loop, Comput. Math. Appl. 59 (2010), 1409—1418.
J. Chavarriga, H. Giacomini and J. Giné, On a new type of bifurcation of limit cycles for a planar cubic system, Nonlinear Anal. 36 (1999), 139–149.
C. Du and Y. Liu, General center conditions and bifurcation of limit cycles for a quasisymmetric seventh degree system, Comput. Math. Appl. 56 (2008), 2957–2969.
F. Dumortier, J. Llibre and J.C. Artés, Qualitative Theory of Planar Differential Systems, UniversiText, Springer–Verlag, New York, 2006.
I.A. Garcı́a and M. Grau, A survey on the inverse integrating factor, Qual. Theory Dyn. Syst. 9 (2010), 115—166.
H. Giacomini, J. Llibre and M. Viano, On the nonexistence, existence, and uniquennes of limit cycles, Nonlinearity 9 (1996), 501–516.
H. Giacomini, J. Llibre and M. Viano, On the shape of limit cycles that bifurcate from Hamiltonian centers, Nonlinear Anal. 41 (2001), 523–537.
H. Giacomini, J. Llibre and M. Viano, The shape of limit cycles that bifurcate from non–Hamiltonian centers, Nonlinear Anal. 43 (2001), 837–859.
H. Giacomini, J. Llibre and M. Viano, Arbitrary order bifurcations for perturbed Hamiltonian planar systems via the reciprocal of an integrating factor, Nonlinear Anal. 48 (2002), 117–136.
H. Giacomini, J. Llibre and M. Viano, Semistable limit cycles that bifurcate from centers, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 13 (2003), 3489–3498.
E.A. González, Generic properties of polynomial vector fields at infinity, Trans. Amer. Math. Soc. 143 (1969), 201–222.
D. Hilbert, Mathematische Problem (lecture), Second Internat. Congress Math. Paris, 1900, Nachr. Ges. Wiss. Göttingen Math.–Phys. Kl. 1900, 253–297.
M.W. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra, Academic Press, New York, 1974.
Yu. S. Ilyashenko, Centennial history of Hilbert’s 16th problem, Bull. Amer. Math. Soc. 39 (2002), 301–354.
J. Llibre and G. Rodrı́guez, Configurations of limit cycles and planar polynomial vector fields, J. Differential Equations 198 (2004), 374–380.
Y. Yanqian et al., Theory of Limit Cycles, Translations of Math. Monographs, Vol. 66, Amer. Math. Soc., Providence, 1986.
Published
How to Cite
Issue
Section
Stats
Number of views and downloads: 0
Number of citations: 0
