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May  2015, 35(5): 1873-1890. doi: 10.3934/dcds.2015.35.1873

Stability of singular limit cycles for Abel equations

1. 

Departamento de Matemáticas, Universidad de Extremadura, Badajoz 06006, Spain, Spain

2. 

Dept. de Matemàtiques, Universitat Autònoma de Barcelona, Edifici C, 08193 Bellaterra, Barcelona

Received  March 2014 Revised  September 2014 Published  December 2014

We obtain a criterion for determining the stability of singular limit cycles of Abel equations $x'=A(t)x^3+B(t)x^2$. This stability controls the possible saddle-node bifurcations of limit cycles. Therefore, studying the Hopf-like bifurcations at $x=0$, together with the bifurcations at infinity of a suitable compactification of the equations, we obtain upper bounds of their number of limit cycles. As an illustration of this approach, we prove that the family $x'=a t(t-t_A)x^3+b (t-t_B)x^2$, with $a ,b>0$, has at most two positive limit cycles for any $t_B,t_A$.
Citation: José Luis Bravo, Manuel Fernández, Armengol Gasull. Stability of singular limit cycles for Abel equations. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 1873-1890. doi: 10.3934/dcds.2015.35.1873
References:
[1]

M. J. Álvarez, A. Gasull and H. Giacomini, A new uniqueness criterion for the number of periodic orbits of Abel equations, J. Differential Equations, 234 (2007), 161-176. doi: 10.1016/j.jde.2006.11.004.

[2]

M. A. M. Alwash and N. G. Lloyd, Nonautonomous equations related to polynomial two dimensional systems, Proc. Roy. Soc. Edinburgh Sect. A 105 (1987), 129-152. doi: 10.1017/S0308210500021971.

[3]

A. A. Andronov, E. A. Leontovich, I. I. Gordon and A. G. Maier, Qualitative Theory of Second-Order Dynamic Systems, Halsted Press (A division of John Wiley & Sons), Israel Program for Scientific Translations Jerusalem-London, 1973.

[4]

D. M. Benardete, V. W. Noonburg and B. Pollina, Qualitative tools for studying periodic solutions and bifurcations as applied to the periodically harvested logistic equation, Amer. Math. Monthly, 115 (2008), 202-219.

[5]

J. L. Bravo, M. Fernández and A. Gasull, Limit cycles for some Abel equations having coefficients without fixed signs, Int. J. Bif. Chaos, 19 (2009), 3869-3876. doi: 10.1142/S0218127409025195.

[6]

J. L. Bravo and J. Torregrosa, Abel-like equations with no periodic solutions, J. Math. Anal. Appl., 342 (2008), 931-942. doi: 10.1016/j.jmaa.2007.12.060.

[7]

L. A. Cherkas, Number of limit cycles of an autonomous second-order system, Diff. Eq., 12 (1976), 944-946, 960.

[8]

G. F. D. Duff, Limit-cycles and rotated vector fields, Ann. of Math., 57 (1953), 15-31. doi: 10.2307/1969724.

[9]

E. Fossas, J. M. Olm and H. Sira-Ramírez, Iterative approximation of limit cycles for a class of Abel equations, Phys. D, 237 (2008), 3159-3164. doi: 10.1016/j.physd.2008.05.011.

[10]

A. Gasull and A. Guillamon, Limit cycles for generalized Abel equations, Int. J. Bif. Chaos, 16 (2006), 3737-3745. doi: 10.1142/S0218127406017130.

[11]

A. Gasull and J. Llibre, Limit cycles for a class of Abel equations, SIAM J. Math. Anal., 21 (1990), 1235-1244. doi: 10.1137/0521068.

[12]

T. Harko and M. K. Mak, Relativistic dissipative cosmological models and Abel differential equation, Comput. Math. Appl., 46 (2003), 849-853. doi: 10.1016/S0898-1221(03)90147-7.

[13]

E. Kamke, Differentialgleichungen, Lösungsmethoden und Lösungen. I: Gewöhnliche Differentialgleichungen, Neunte Auflage, Mit einem Vorwort von Detlef Kamke, B. G. Teubner, Stuttgart, 1977.

[14]

A. Lins Neto, On the number of solutions of the equation $\frac{d x}{dt}=\sum_{j=0}^n a_j(t)x^j$, $0\leq t\leq 1$, for which $x(0)=x(1)$, Inv. Math., 59 (1980), 67-76. doi: 10.1007/BF01390315.

[15]

N. G. Lloyd, A note on the number of limit cycles in certain two-dimensional systems, J. London Math. Soc., 20 (1979), 277-286. doi: 10.1112/jlms/s2-20.2.277.

[16]

J. M. Olm and X. Ros-Oton, Existence of periodic solutions with nonconstant sign in a class of generalized Abel equations, Discrete Contin. Dyn. Syst., 33 (2013), 1603-1614. doi: 10.3934/dcds.2013.33.1603.

[17]

J. M. Olm, X. Ros-Oton and T. M. Seara, Periodic solutions with non-constant sign in Abel equations of the second kind, J. Math. Anal. Appl., 381 (2011), 582-589. doi: 10.1016/j.jmaa.2011.02.084.

[18]

D. E. Panayotounakos and T. I. Zarmpoutis, Construction of Exact Parametric or Closed Form Solutions of Some Unsolvable Classes of Nonlinear ODEs (Abel's Nonlinear ODEs of the First Kind and Relative Degenerate Equations), Int. J. Math. Math. Sci., 2011 (2011), Article ID 387429, 13 pp. doi: 10.1155/2011/387429.

[19]

A. A. Panov, The number of periodic solutions of polynomial differential equations, Math. Notes, 64 (1998), 622-628. doi: 10.1007/BF02316287.

[20]

L. M. Perko, Differential Equations and Dynamical Systems, Third edition, Texts in Applied Mathematics 7, Springer-Verlag, New York [etc.], 2001. doi: 10.1007/978-1-4613-0003-8.

[21]

V. A. Pliss, Non-Local Problems of the Theory of Oscillations, Academic Press, New York, 1966.

[22]

J. Sotomayor, Curvas Definidas Por Equações Diferenciais no Plano, IMPA, Rio de Janeiro, 1981.

show all references

References:
[1]

M. J. Álvarez, A. Gasull and H. Giacomini, A new uniqueness criterion for the number of periodic orbits of Abel equations, J. Differential Equations, 234 (2007), 161-176. doi: 10.1016/j.jde.2006.11.004.

[2]

M. A. M. Alwash and N. G. Lloyd, Nonautonomous equations related to polynomial two dimensional systems, Proc. Roy. Soc. Edinburgh Sect. A 105 (1987), 129-152. doi: 10.1017/S0308210500021971.

[3]

A. A. Andronov, E. A. Leontovich, I. I. Gordon and A. G. Maier, Qualitative Theory of Second-Order Dynamic Systems, Halsted Press (A division of John Wiley & Sons), Israel Program for Scientific Translations Jerusalem-London, 1973.

[4]

D. M. Benardete, V. W. Noonburg and B. Pollina, Qualitative tools for studying periodic solutions and bifurcations as applied to the periodically harvested logistic equation, Amer. Math. Monthly, 115 (2008), 202-219.

[5]

J. L. Bravo, M. Fernández and A. Gasull, Limit cycles for some Abel equations having coefficients without fixed signs, Int. J. Bif. Chaos, 19 (2009), 3869-3876. doi: 10.1142/S0218127409025195.

[6]

J. L. Bravo and J. Torregrosa, Abel-like equations with no periodic solutions, J. Math. Anal. Appl., 342 (2008), 931-942. doi: 10.1016/j.jmaa.2007.12.060.

[7]

L. A. Cherkas, Number of limit cycles of an autonomous second-order system, Diff. Eq., 12 (1976), 944-946, 960.

[8]

G. F. D. Duff, Limit-cycles and rotated vector fields, Ann. of Math., 57 (1953), 15-31. doi: 10.2307/1969724.

[9]

E. Fossas, J. M. Olm and H. Sira-Ramírez, Iterative approximation of limit cycles for a class of Abel equations, Phys. D, 237 (2008), 3159-3164. doi: 10.1016/j.physd.2008.05.011.

[10]

A. Gasull and A. Guillamon, Limit cycles for generalized Abel equations, Int. J. Bif. Chaos, 16 (2006), 3737-3745. doi: 10.1142/S0218127406017130.

[11]

A. Gasull and J. Llibre, Limit cycles for a class of Abel equations, SIAM J. Math. Anal., 21 (1990), 1235-1244. doi: 10.1137/0521068.

[12]

T. Harko and M. K. Mak, Relativistic dissipative cosmological models and Abel differential equation, Comput. Math. Appl., 46 (2003), 849-853. doi: 10.1016/S0898-1221(03)90147-7.

[13]

E. Kamke, Differentialgleichungen, Lösungsmethoden und Lösungen. I: Gewöhnliche Differentialgleichungen, Neunte Auflage, Mit einem Vorwort von Detlef Kamke, B. G. Teubner, Stuttgart, 1977.

[14]

A. Lins Neto, On the number of solutions of the equation $\frac{d x}{dt}=\sum_{j=0}^n a_j(t)x^j$, $0\leq t\leq 1$, for which $x(0)=x(1)$, Inv. Math., 59 (1980), 67-76. doi: 10.1007/BF01390315.

[15]

N. G. Lloyd, A note on the number of limit cycles in certain two-dimensional systems, J. London Math. Soc., 20 (1979), 277-286. doi: 10.1112/jlms/s2-20.2.277.

[16]

J. M. Olm and X. Ros-Oton, Existence of periodic solutions with nonconstant sign in a class of generalized Abel equations, Discrete Contin. Dyn. Syst., 33 (2013), 1603-1614. doi: 10.3934/dcds.2013.33.1603.

[17]

J. M. Olm, X. Ros-Oton and T. M. Seara, Periodic solutions with non-constant sign in Abel equations of the second kind, J. Math. Anal. Appl., 381 (2011), 582-589. doi: 10.1016/j.jmaa.2011.02.084.

[18]

D. E. Panayotounakos and T. I. Zarmpoutis, Construction of Exact Parametric or Closed Form Solutions of Some Unsolvable Classes of Nonlinear ODEs (Abel's Nonlinear ODEs of the First Kind and Relative Degenerate Equations), Int. J. Math. Math. Sci., 2011 (2011), Article ID 387429, 13 pp. doi: 10.1155/2011/387429.

[19]

A. A. Panov, The number of periodic solutions of polynomial differential equations, Math. Notes, 64 (1998), 622-628. doi: 10.1007/BF02316287.

[20]

L. M. Perko, Differential Equations and Dynamical Systems, Third edition, Texts in Applied Mathematics 7, Springer-Verlag, New York [etc.], 2001. doi: 10.1007/978-1-4613-0003-8.

[21]

V. A. Pliss, Non-Local Problems of the Theory of Oscillations, Academic Press, New York, 1966.

[22]

J. Sotomayor, Curvas Definidas Por Equações Diferenciais no Plano, IMPA, Rio de Janeiro, 1981.

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