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Evaluating cyclicity of cubic systems with algorithms of computational algebra
1. | Lehrstuhl D für Mathematik, RWTH Aachen University, Templergraben 64, D-52062 Aachen, Germany |
2. | Technische University of Kaiserslautern Fachbereich Mathematik Erwin-Schrödinger Str. 48,D-67653 Kaiserslautern, Germany |
3. | CAMTP - Center for Applied Mathematics and Theoretical Physics,University of Maribor, Krekova 2 , SI-2000 Maribor, Slovenia |
References:
[1] |
N. N. Bautin, On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type, Mat. Sbornik N. S., 30 (1952), 181-196; Translations Amer. Math. Soc., 100 (1954), 181-196. |
[2] |
B. Buchberger, An algorithm for finding the basis elements of the residue class ring of a zero dimensional polynomial ideal, J. Symbolic Comput., 41 (2006), 475-511.
doi: 10.1016/j.jsc.2005.09.007. |
[3] |
C. Christopher, Estimating limit cycles bifurcations, in "Trends in Mathematics, Differential Equations with Symbolic Computations" (Eds. D. Wang and Z. Zheng), Birkhäuser-Verlag, (2005), 23-36.
doi: 10.1007/3-7643-7429-2_2. |
[4] |
C. J. Christopher and C. Rousseau, Nondegenerate linearizable centres of complex planar quadratic and symmetric cubic systems in $\mathbbC^2$, Publ. Mat., 45 (2001), 95-123.
doi: 10.5565/PUBLMAT_45101_04. |
[5] |
D. Cox, J. Little and D. O'Shea, "Ideals, Varieties and Algorithms," Springer-Verlag, New York, 1992.
doi: 10.1216/rmjm/1181071923. |
[6] |
A. Cima, A. Gasull, V. Mañosa and F. Mañosas, Algebraic properties of the Liapunov and periodic constants, Rocky Mountain J. Math., 27 (1997), 471-501.
doi: 10.1216/rmjm/1181071923. |
[7] |
W. Decker, S. Laplagne, G. Pfister and H. A. Schönemann, Singular 3-1 library for computing the primary decomposition and radical of ideals, primdec.lib, 2010. |
[8] |
J.-P. Françoise and Y. Yomdin, Bernstein inequalities and applications to analytic geometry and differential equations, J. Functional Analysis, 146 (1997), 185-205.
doi: 10.1006/jfan.1996.3029. |
[9] |
P. Gianni, B. Trager and G. Zacharias, Gröbner bases and primary decomposition of polynomials, J. Symbolic Comput., 6 (1988), 146-167.
doi: 10.1016/S0747-7171(88)80040-3. |
[10] |
J. Giné, On some open problems in planar differential systems and Hilbert's 16th problem, Chaos Solitons Fractals, 31 (2007), 1118-1134.
doi: 10.1016/j.chaos.2005.10.057. |
[11] |
G.-M. Greuel and G. Pfister, "A Singular Introduction to Commutative Algebra," Springer-Verlag, New York, 2002. |
[12] |
W. Decker, G.-M. Greuel, G. Pfister and H. Schönemann, Singular 3-1-2. A computer algebra system for polynomial computations, Centre for Computer Algebra, University of Kaiserslautern (2010). Available from: http://www.singular.uni-kl.de. |
[13] |
M. Han, H. Zang and T. Zhang, A new proof to Bautin's theorem, Chaos Solitons Fractals, 31 (2007), 218-223.
doi: 10.1016/j.chaos.2005.09.051. |
[14] |
Yu. Ilyashenko and S. Yakovenko, "Lectures on Analytic Differential Equations," Graduate Studies in Mathematics, 86, (American Mathematical Society, Providence), 2008. |
[15] |
A. Jarrah, R. Laubenbacher and V. G. Romanovski, The Sibirsky component of the center variety of polynomial differential systems, J. Symb. Comput., 35 (2003), 577-589.
doi: 10.1016/S0747-7171(03)00016-6. |
[16] |
V. Levandovskyy, A. Logar and V. G. Romanovski, The cyclicity of a cubic system, Open Syst. Inf. Dyn., 16 (2009), 429-439.
doi: 10.1142/S1230161209000323. |
[17] |
V. Levandovskyy, V. G. Romanovski and D. S. Shafer, The cyclicity of a cubic system with nonradical Bautin ideal, J. Differential Equations, 246 (2009), 1274-1287.
doi: 10.1016/j.jde.2008.07.026. |
[18] |
Y.-R. Liu and J.-B. Li, Theory of values of singular point in complex autonomous differential systems, Sci. China Ser. A, 33 (1989), 10-23. |
[19] |
J. Li, Hilbert's 16th problem and bifurcations of planar polynomial vector fields, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 47-106.
doi: 10.1142/S0218127403006352. |
[20] |
N. G. Lloyd, J. M. Pearson and V. G. Romanovsky, Computing integrability conditions for a cubic differential system, Comput. Math. Appl., 32 (1996), 99-107.
doi: 10.1016/S0898-1221(96)00188-5. |
[21] |
V. G. Romanovski, Time-reversibility in 2-Dim systems, Open Syst. Inf. Dyn., 15 (2008), 359-370.
doi: 10.1142/S1230161208000249. |
[22] |
V. G. Romanovski and D. S. Shafer, Time-reversibility in two-dimensional polynomial systems, in "Trends in Mathematics, Differential Equations with Symbolic Computations" (Eds. D. Wang and Z. Zheng), Birkhäuser-Verlag, (2005), 67-83.
doi: 10.1007/3-7643-7429-2_5. |
[23] |
V. G. Romanovski and D. S. Shafer, "The Center and Cyclicity Problems: A Computational Algebra Approach," Birkhäuser Boston, Inc., Boston, MA, 2009. |
[24] |
R. Roussarie, "Bifurcations of Planar Vector Fields and Hilbert's Sixteenth Problem," Progress in Mathematics, 164, Birkhäuser, Basel, 1998. |
[25] |
K. S. Sibirskii, On the number of limit cycles in the neighborhood of a singular point, Differ. Uravn. (Russian), 1 (1965), 53-66; Differ. Equ. (English translation), 1 (1965), 36-47. |
[26] |
S. Yakovenko, A geometric proof of the Bautin theorem, Concerning the Hilbert Sixteenth Problem. Advances in Mathematical Sciences, Vol. 23; Amer. Math. Soc. Transl., 165 (1995), 203-219. |
[27] |
H. Żołądek, Quadratic systems with center and their perturbations, J. Differential Equations, 109 (1994), 223-273.
doi: 10.1006/jdeq.1994.1049. |
[28] |
H. Żołądek, On a certain generalization of Bautin's theorem, Nonlinearity, 7 (1994), 273-279.
doi: 10.1088/0951-7715/7/1/013. |
show all references
References:
[1] |
N. N. Bautin, On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type, Mat. Sbornik N. S., 30 (1952), 181-196; Translations Amer. Math. Soc., 100 (1954), 181-196. |
[2] |
B. Buchberger, An algorithm for finding the basis elements of the residue class ring of a zero dimensional polynomial ideal, J. Symbolic Comput., 41 (2006), 475-511.
doi: 10.1016/j.jsc.2005.09.007. |
[3] |
C. Christopher, Estimating limit cycles bifurcations, in "Trends in Mathematics, Differential Equations with Symbolic Computations" (Eds. D. Wang and Z. Zheng), Birkhäuser-Verlag, (2005), 23-36.
doi: 10.1007/3-7643-7429-2_2. |
[4] |
C. J. Christopher and C. Rousseau, Nondegenerate linearizable centres of complex planar quadratic and symmetric cubic systems in $\mathbbC^2$, Publ. Mat., 45 (2001), 95-123.
doi: 10.5565/PUBLMAT_45101_04. |
[5] |
D. Cox, J. Little and D. O'Shea, "Ideals, Varieties and Algorithms," Springer-Verlag, New York, 1992.
doi: 10.1216/rmjm/1181071923. |
[6] |
A. Cima, A. Gasull, V. Mañosa and F. Mañosas, Algebraic properties of the Liapunov and periodic constants, Rocky Mountain J. Math., 27 (1997), 471-501.
doi: 10.1216/rmjm/1181071923. |
[7] |
W. Decker, S. Laplagne, G. Pfister and H. A. Schönemann, Singular 3-1 library for computing the primary decomposition and radical of ideals, primdec.lib, 2010. |
[8] |
J.-P. Françoise and Y. Yomdin, Bernstein inequalities and applications to analytic geometry and differential equations, J. Functional Analysis, 146 (1997), 185-205.
doi: 10.1006/jfan.1996.3029. |
[9] |
P. Gianni, B. Trager and G. Zacharias, Gröbner bases and primary decomposition of polynomials, J. Symbolic Comput., 6 (1988), 146-167.
doi: 10.1016/S0747-7171(88)80040-3. |
[10] |
J. Giné, On some open problems in planar differential systems and Hilbert's 16th problem, Chaos Solitons Fractals, 31 (2007), 1118-1134.
doi: 10.1016/j.chaos.2005.10.057. |
[11] |
G.-M. Greuel and G. Pfister, "A Singular Introduction to Commutative Algebra," Springer-Verlag, New York, 2002. |
[12] |
W. Decker, G.-M. Greuel, G. Pfister and H. Schönemann, Singular 3-1-2. A computer algebra system for polynomial computations, Centre for Computer Algebra, University of Kaiserslautern (2010). Available from: http://www.singular.uni-kl.de. |
[13] |
M. Han, H. Zang and T. Zhang, A new proof to Bautin's theorem, Chaos Solitons Fractals, 31 (2007), 218-223.
doi: 10.1016/j.chaos.2005.09.051. |
[14] |
Yu. Ilyashenko and S. Yakovenko, "Lectures on Analytic Differential Equations," Graduate Studies in Mathematics, 86, (American Mathematical Society, Providence), 2008. |
[15] |
A. Jarrah, R. Laubenbacher and V. G. Romanovski, The Sibirsky component of the center variety of polynomial differential systems, J. Symb. Comput., 35 (2003), 577-589.
doi: 10.1016/S0747-7171(03)00016-6. |
[16] |
V. Levandovskyy, A. Logar and V. G. Romanovski, The cyclicity of a cubic system, Open Syst. Inf. Dyn., 16 (2009), 429-439.
doi: 10.1142/S1230161209000323. |
[17] |
V. Levandovskyy, V. G. Romanovski and D. S. Shafer, The cyclicity of a cubic system with nonradical Bautin ideal, J. Differential Equations, 246 (2009), 1274-1287.
doi: 10.1016/j.jde.2008.07.026. |
[18] |
Y.-R. Liu and J.-B. Li, Theory of values of singular point in complex autonomous differential systems, Sci. China Ser. A, 33 (1989), 10-23. |
[19] |
J. Li, Hilbert's 16th problem and bifurcations of planar polynomial vector fields, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 47-106.
doi: 10.1142/S0218127403006352. |
[20] |
N. G. Lloyd, J. M. Pearson and V. G. Romanovsky, Computing integrability conditions for a cubic differential system, Comput. Math. Appl., 32 (1996), 99-107.
doi: 10.1016/S0898-1221(96)00188-5. |
[21] |
V. G. Romanovski, Time-reversibility in 2-Dim systems, Open Syst. Inf. Dyn., 15 (2008), 359-370.
doi: 10.1142/S1230161208000249. |
[22] |
V. G. Romanovski and D. S. Shafer, Time-reversibility in two-dimensional polynomial systems, in "Trends in Mathematics, Differential Equations with Symbolic Computations" (Eds. D. Wang and Z. Zheng), Birkhäuser-Verlag, (2005), 67-83.
doi: 10.1007/3-7643-7429-2_5. |
[23] |
V. G. Romanovski and D. S. Shafer, "The Center and Cyclicity Problems: A Computational Algebra Approach," Birkhäuser Boston, Inc., Boston, MA, 2009. |
[24] |
R. Roussarie, "Bifurcations of Planar Vector Fields and Hilbert's Sixteenth Problem," Progress in Mathematics, 164, Birkhäuser, Basel, 1998. |
[25] |
K. S. Sibirskii, On the number of limit cycles in the neighborhood of a singular point, Differ. Uravn. (Russian), 1 (1965), 53-66; Differ. Equ. (English translation), 1 (1965), 36-47. |
[26] |
S. Yakovenko, A geometric proof of the Bautin theorem, Concerning the Hilbert Sixteenth Problem. Advances in Mathematical Sciences, Vol. 23; Amer. Math. Soc. Transl., 165 (1995), 203-219. |
[27] |
H. Żołądek, Quadratic systems with center and their perturbations, J. Differential Equations, 109 (1994), 223-273.
doi: 10.1006/jdeq.1994.1049. |
[28] |
H. Żołądek, On a certain generalization of Bautin's theorem, Nonlinearity, 7 (1994), 273-279.
doi: 10.1088/0951-7715/7/1/013. |
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