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Qualitative analysis to the traveling wave solutions of Kakutani-Kawahara equation and its approximate damped oscillatory solution
Limit cycles of non-autonomous scalar ODEs with two summands
1. | Departamento de Matemáticas, Universidad de Extremadura, Badajoz, 06071 |
2. | Departamento de Matemáticas, Universidad de Extremadura, Facultad de Ciencias, 06071 Badajoz |
References:
[1] |
N. Alkoumi and P. J. Torres, On the number of limit cycles of a generalized Abel equation, Czech. Math. J., 61 (2011), 73-83.
doi: 10.1007/s10587-011-0018-x. |
[2] |
N. Alkoumi and P. J. Torres, Estimates on the number of limit cycles of a generalized Abel equation, Discrete Contin. Dyn. Syst., 31 (2011), 25-34.
doi: 10.3934/dcds.2011.31.25. |
[3] |
A. Álvarez, J. L. Bravo and M. Fernández, The number of limit cycles for generalized Abel equations with periodic coefficients of definite sign, Commun. Pure Appl. Anal., 8 (2009), 1493-1501.
doi: 10.3934/cpaa.2009.8.1493. |
[4] |
A. Álvarez, J. L. Bravo and M. Fernández, Abel-like differential equations with a unique limit cycle, Nonlinear Anal. T.M.A., 74 (2011), 3694-3702.
doi: 10.1016/j.na.2011.02.049. |
[5] |
A. Álvarez, J. L. Bravo and M. Fernández, Uniqueness of limit cycles for polynomial first-order differential equations, J. Math. Anal. Appl., 360 (2009), 168-189.
doi: 10.1016/j.jmaa.2009.06.031. |
[6] |
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. |
[7] |
M. A. M. Alwash, Periodic solutions of Abel differential equations, J. Math. Anal. Appl., 329 (2007), 1161-1169.
doi: 10.1016/j.jmaa.2006.07.039. |
[8] |
M. A. M. Alwash and N. G. Lloyd, Nonautonomous equations related to polynomial two dimensional systems, Proc. Roy. Soc.Edinburgh, 105A (1987), 129-152.
doi: 10.1017/S0308210500021971. |
[9] |
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. |
[10] |
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. |
[11] |
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. |
[12] |
M. Chamberland and A. Gasull, Chini equations and isochronous centers in three-dimensional differential systems. Qual. Theory Dyn. Syst., 9 (2010), 29-38.
doi: 10.1007/s12346-010-0019-4. |
[13] |
J. Devlin, N. G. Lloyd and J. M. Pearson, Cubic systems and Abel equations, J. Differential Equations, 147 (1998), 435-454.
doi: 10.1006/jdeq.1998.3420. |
[14] |
A. Gasull and A. Guillamon, Limit cycles for generalized Abel equations, Int. J. Bif. Chaos, 16 (2006), 3737-3745.
doi: 10.1142/S0218127406017130. |
[15] |
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. |
[16] |
A. Gasull, R. Prohens and J. Torregrosa, Limit cycles for rigid cubic systems, J. Math. Anal. Appl., 303 (2005), 391-404.
doi: 10.1016/j.jmaa.2004.07.030. |
[17] |
A. Gasull and J. Torregrosa, Some results on rigid systems, In International Conference on Differential Equations (Equadiff-2003), World Sci. Publ., Hackensack, NJ. (2005), 340-345. |
[18] |
Yu. Ilyashenko, Centennial history of Hilbert's 16th problem, Bull. Amer. Math. Soc., 39 (2002), 301-354.
doi: 10.1090/S0273-0979-02-00946-1. |
[19] |
A. Lins Neto, On the number of solutions of the equation $\frac{d x}{d t}=\sum_{j=0} ^n a_j(t)x^j$, $0 \leq t \leq 1$, for which $x(0)=x(1)$}, Inv. Math., 59 (1980), 67-76. |
[20] |
N. G. Lloyd, The number of periodic solutions of the equation $\dot z = z^N+ p_1(t) z^{N-1} +\cdots +p_N(t)$, Proc. London Math. Soc., 27 (1973), 667-700.
doi: 10.1112/plms/s3-27.4.667. |
[21] |
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. |
[22] |
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. |
[23] |
V. A. Pliss, "Non-Local Problems of the Theory of Oscillations," Academic Press, New York, 1966. |
[24] |
Wolfram Research, Inc., "Mathematica, Version 8.0," Champaign, IL (2010). |
show all references
References:
[1] |
N. Alkoumi and P. J. Torres, On the number of limit cycles of a generalized Abel equation, Czech. Math. J., 61 (2011), 73-83.
doi: 10.1007/s10587-011-0018-x. |
[2] |
N. Alkoumi and P. J. Torres, Estimates on the number of limit cycles of a generalized Abel equation, Discrete Contin. Dyn. Syst., 31 (2011), 25-34.
doi: 10.3934/dcds.2011.31.25. |
[3] |
A. Álvarez, J. L. Bravo and M. Fernández, The number of limit cycles for generalized Abel equations with periodic coefficients of definite sign, Commun. Pure Appl. Anal., 8 (2009), 1493-1501.
doi: 10.3934/cpaa.2009.8.1493. |
[4] |
A. Álvarez, J. L. Bravo and M. Fernández, Abel-like differential equations with a unique limit cycle, Nonlinear Anal. T.M.A., 74 (2011), 3694-3702.
doi: 10.1016/j.na.2011.02.049. |
[5] |
A. Álvarez, J. L. Bravo and M. Fernández, Uniqueness of limit cycles for polynomial first-order differential equations, J. Math. Anal. Appl., 360 (2009), 168-189.
doi: 10.1016/j.jmaa.2009.06.031. |
[6] |
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. |
[7] |
M. A. M. Alwash, Periodic solutions of Abel differential equations, J. Math. Anal. Appl., 329 (2007), 1161-1169.
doi: 10.1016/j.jmaa.2006.07.039. |
[8] |
M. A. M. Alwash and N. G. Lloyd, Nonautonomous equations related to polynomial two dimensional systems, Proc. Roy. Soc.Edinburgh, 105A (1987), 129-152.
doi: 10.1017/S0308210500021971. |
[9] |
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. |
[10] |
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. |
[11] |
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. |
[12] |
M. Chamberland and A. Gasull, Chini equations and isochronous centers in three-dimensional differential systems. Qual. Theory Dyn. Syst., 9 (2010), 29-38.
doi: 10.1007/s12346-010-0019-4. |
[13] |
J. Devlin, N. G. Lloyd and J. M. Pearson, Cubic systems and Abel equations, J. Differential Equations, 147 (1998), 435-454.
doi: 10.1006/jdeq.1998.3420. |
[14] |
A. Gasull and A. Guillamon, Limit cycles for generalized Abel equations, Int. J. Bif. Chaos, 16 (2006), 3737-3745.
doi: 10.1142/S0218127406017130. |
[15] |
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. |
[16] |
A. Gasull, R. Prohens and J. Torregrosa, Limit cycles for rigid cubic systems, J. Math. Anal. Appl., 303 (2005), 391-404.
doi: 10.1016/j.jmaa.2004.07.030. |
[17] |
A. Gasull and J. Torregrosa, Some results on rigid systems, In International Conference on Differential Equations (Equadiff-2003), World Sci. Publ., Hackensack, NJ. (2005), 340-345. |
[18] |
Yu. Ilyashenko, Centennial history of Hilbert's 16th problem, Bull. Amer. Math. Soc., 39 (2002), 301-354.
doi: 10.1090/S0273-0979-02-00946-1. |
[19] |
A. Lins Neto, On the number of solutions of the equation $\frac{d x}{d t}=\sum_{j=0} ^n a_j(t)x^j$, $0 \leq t \leq 1$, for which $x(0)=x(1)$}, Inv. Math., 59 (1980), 67-76. |
[20] |
N. G. Lloyd, The number of periodic solutions of the equation $\dot z = z^N+ p_1(t) z^{N-1} +\cdots +p_N(t)$, Proc. London Math. Soc., 27 (1973), 667-700.
doi: 10.1112/plms/s3-27.4.667. |
[21] |
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. |
[22] |
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. |
[23] |
V. A. Pliss, "Non-Local Problems of the Theory of Oscillations," Academic Press, New York, 1966. |
[24] |
Wolfram Research, Inc., "Mathematica, Version 8.0," Champaign, IL (2010). |
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