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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.

[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. doi: 10.1016/j.jsc.2005.09.007.

[3]

C. Christopher, Estimating limit cycles bifurcations,, in, (2005), 23. 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. doi: 10.5565/PUBLMAT_45101_04.

[5]

D. Cox, J. Little and D. O'Shea, "Ideals, Varieties and Algorithms,", Springer-Verlag, (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. 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. 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. 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. doi: 10.1016/j.chaos.2005.10.057.

[11]

G.-M. Greuel and G. Pfister, "A Singular Introduction to Commutative Algebra,", Springer-Verlag, (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, (2010).

[13]

M. Han, H. Zang and T. Zhang, A new proof to Bautin's theorem,, Chaos Solitons Fractals, 31 (2007), 218. doi: 10.1016/j.chaos.2005.09.051.

[14]

Yu. Ilyashenko and S. Yakovenko, "Lectures on Analytic Differential Equations,", Graduate Studies in Mathematics, 86 (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. 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. 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. 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.

[19]

J. Li, Hilbert's 16th problem and bifurcations of planar polynomial vector fields,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 47. 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. 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. doi: 10.1142/S1230161208000249.

[22]

V. G. Romanovski and D. S. Shafer, Time-reversibility in two-dimensional polynomial systems,, in, (2005), 67. 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,", Birkh\, (2009).

[24]

R. Roussarie, "Bifurcations of Planar Vector Fields and Hilbert's Sixteenth Problem,", Progress in Mathematics, 164 (1998).

[25]

K. S. Sibirskii, On the number of limit cycles in the neighborhood of a singular point,, Differ. Uravn. (Russian), 1 (1965), 53.

[26]

S. Yakovenko, A geometric proof of the Bautin theorem,, Concerning the Hilbert Sixteenth Problem. Advances in Mathematical Sciences, 165 (1995), 203.

[27]

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

[28]

H. Żołądek, On a certain generalization of Bautin's theorem,, Nonlinearity, 7 (1994), 273. 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.

[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. doi: 10.1016/j.jsc.2005.09.007.

[3]

C. Christopher, Estimating limit cycles bifurcations,, in, (2005), 23. 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. doi: 10.5565/PUBLMAT_45101_04.

[5]

D. Cox, J. Little and D. O'Shea, "Ideals, Varieties and Algorithms,", Springer-Verlag, (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. 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. 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. 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. doi: 10.1016/j.chaos.2005.10.057.

[11]

G.-M. Greuel and G. Pfister, "A Singular Introduction to Commutative Algebra,", Springer-Verlag, (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, (2010).

[13]

M. Han, H. Zang and T. Zhang, A new proof to Bautin's theorem,, Chaos Solitons Fractals, 31 (2007), 218. doi: 10.1016/j.chaos.2005.09.051.

[14]

Yu. Ilyashenko and S. Yakovenko, "Lectures on Analytic Differential Equations,", Graduate Studies in Mathematics, 86 (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. 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. 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. 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.

[19]

J. Li, Hilbert's 16th problem and bifurcations of planar polynomial vector fields,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 47. 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. 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. doi: 10.1142/S1230161208000249.

[22]

V. G. Romanovski and D. S. Shafer, Time-reversibility in two-dimensional polynomial systems,, in, (2005), 67. 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,", Birkh\, (2009).

[24]

R. Roussarie, "Bifurcations of Planar Vector Fields and Hilbert's Sixteenth Problem,", Progress in Mathematics, 164 (1998).

[25]

K. S. Sibirskii, On the number of limit cycles in the neighborhood of a singular point,, Differ. Uravn. (Russian), 1 (1965), 53.

[26]

S. Yakovenko, A geometric proof of the Bautin theorem,, Concerning the Hilbert Sixteenth Problem. Advances in Mathematical Sciences, 165 (1995), 203.

[27]

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

[28]

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

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