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September  2012, 11(5): 2023-2035. doi: 10.3934/cpaa.2012.11.2023

Evaluating cyclicity of cubic systems with algorithms of computational algebra

Viktor Levandovskyy 1, , Gerhard Pfister 2, and  Valery G. Romanovski 3,

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

Received  June 2011 Revised  December 2011 Published  March 2012

We describe an algorithmic approach to studying limit cycle bifurcations in a neighborhood of an elementary center or focus of a polynomial system. Using it we obtain an upper bound for cyclicity of a family of cubic systems. Then using a theorem by Christopher [3] we study bifurcation of limit cycles from each component of the center variety. We obtain also the sharp bound for the cyclicity of a generic time-reversible cubic system.
Keywords: polynomial systems of ODEs., bifurcation, Limit cycles, cyclicity.
Mathematics Subject Classification: Primary: 34C07; Secondary: 34C2.
Citation: Viktor Levandovskyy, Gerhard Pfister, Valery G. Romanovski. Evaluating cyclicity of cubic systems with algorithms of computational algebra. Communications on Pure & Applied Analysis, 2012, 11 (5) : 2023-2035. doi: 10.3934/cpaa.2012.11.2023
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

D. Cox, J. Little and D. O'Shea, "Ideals, Varieties and Algorithms," Springer-Verlag, New York, 1992. doi: 10.1216/rmjm/1181071923.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

G.-M. Greuel and G. Pfister, "A Singular Introduction to Commutative Algebra," Springer-Verlag, New York, 2002.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[14]

Yu. Ilyashenko and S. Yakovenko, "Lectures on Analytic Differential Equations," Graduate Studies in Mathematics, 86, (American Mathematical Society, Providence), 2008.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

V. G. Romanovski, Time-reversibility in 2-Dim systems, Open Syst. Inf. Dyn., 15 (2008), 359-370. doi: 10.1142/S1230161208000249.  Google Scholar

[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.  Google Scholar

[23]

V. G. Romanovski and D. S. Shafer, "The Center and Cyclicity Problems: A Computational Algebra Approach," Birkhäuser Boston, Inc., Boston, MA, 2009.  Google Scholar

[24]

R. Roussarie, "Bifurcations of Planar Vector Fields and Hilbert's Sixteenth Problem," Progress in Mathematics, 164, Birkhäuser, Basel, 1998.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

H. Żołądek, Quadratic systems with center and their perturbations, J. Differential Equations, 109 (1994), 223-273. doi: 10.1006/jdeq.1994.1049.  Google Scholar

[28]

H. Żołądek, On a certain generalization of Bautin's theorem, Nonlinearity, 7 (1994), 273-279. doi: 10.1088/0951-7715/7/1/013.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

D. Cox, J. Little and D. O'Shea, "Ideals, Varieties and Algorithms," Springer-Verlag, New York, 1992. doi: 10.1216/rmjm/1181071923.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

G.-M. Greuel and G. Pfister, "A Singular Introduction to Commutative Algebra," Springer-Verlag, New York, 2002.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[14]

Yu. Ilyashenko and S. Yakovenko, "Lectures on Analytic Differential Equations," Graduate Studies in Mathematics, 86, (American Mathematical Society, Providence), 2008.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

V. G. Romanovski, Time-reversibility in 2-Dim systems, Open Syst. Inf. Dyn., 15 (2008), 359-370. doi: 10.1142/S1230161208000249.  Google Scholar

[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.  Google Scholar

[23]

V. G. Romanovski and D. S. Shafer, "The Center and Cyclicity Problems: A Computational Algebra Approach," Birkhäuser Boston, Inc., Boston, MA, 2009.  Google Scholar

[24]

R. Roussarie, "Bifurcations of Planar Vector Fields and Hilbert's Sixteenth Problem," Progress in Mathematics, 164, Birkhäuser, Basel, 1998.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

H. Żołądek, Quadratic systems with center and their perturbations, J. Differential Equations, 109 (1994), 223-273. doi: 10.1006/jdeq.1994.1049.  Google Scholar

[28]

H. Żołądek, On a certain generalization of Bautin's theorem, Nonlinearity, 7 (1994), 273-279. doi: 10.1088/0951-7715/7/1/013.  Google Scholar

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