American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Smooth attractors for the Brinkman-Forchheimer equations with fast growing nonlinearities
  • CPAA Home
  • This Issue
  • Next Article
    Double resonance for Dirichlet problems with unbounded indefinite potential and combined nonlinearities
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. 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. doi: 10.1016/j.jsc.2005.09.007. Google Scholar

[3]

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

[5]

D. Cox, J. Little and D. O'Shea, "Ideals, Varieties and Algorithms,", Springer-Verlag, (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. 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. 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. 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. 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, (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, (2010). Google Scholar

[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. Google Scholar

[14]

Yu. Ilyashenko and S. Yakovenko, "Lectures on Analytic Differential Equations,", Graduate Studies in Mathematics, 86 (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. 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. 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. 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. 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. 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. 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. doi: 10.1142/S1230161208000249. Google Scholar

[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. Google Scholar

[23]

V. G. Romanovski and D. S. Shafer, "The Center and Cyclicity Problems: A Computational Algebra Approach,", Birkh\, (2009). Google Scholar

[24]

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

[26]

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

[27]

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

[28]

H. Żołądek, On a certain generalization of Bautin's theorem,, Nonlinearity, 7 (1994), 273. 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. 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. doi: 10.1016/j.jsc.2005.09.007. Google Scholar

[3]

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

[5]

D. Cox, J. Little and D. O'Shea, "Ideals, Varieties and Algorithms,", Springer-Verlag, (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. 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. 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. 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. 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, (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, (2010). Google Scholar

[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. Google Scholar

[14]

Yu. Ilyashenko and S. Yakovenko, "Lectures on Analytic Differential Equations,", Graduate Studies in Mathematics, 86 (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. 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. 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. 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. 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. 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. 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. doi: 10.1142/S1230161208000249. Google Scholar

[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. Google Scholar

[23]

V. G. Romanovski and D. S. Shafer, "The Center and Cyclicity Problems: A Computational Algebra Approach,", Birkh\, (2009). Google Scholar

[24]

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

[26]

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

[27]

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

[28]

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

[1]

Jaume Llibre, Claudia Valls. Algebraic limit cycles for quadratic polynomial differential systems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2475-2485. doi: 10.3934/dcdsb.2018070
[2]

Magdalena Caubergh, Freddy Dumortier, Stijn Luca. Cyclicity of unbounded semi-hyperbolic 2-saddle cycles in polynomial Lienard systems. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 963-980. doi: 10.3934/dcds.2010.27.963
[3]

Freddy Dumortier. Sharp upperbounds for the number of large amplitude limit cycles in polynomial Lienard systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1465-1479. doi: 10.3934/dcds.2012.32.1465
[4]

Armengol Gasull, Hector Giacomini. Upper bounds for the number of limit cycles of some planar polynomial differential systems. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 217-229. doi: 10.3934/dcds.2010.27.217
[5]

Xingwu Chen, Jaume Llibre, Weinian Zhang. Averaging approach to cyclicity of hopf bifurcation in planar linear-quadratic polynomial discontinuous differential systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3953-3965. doi: 10.3934/dcdsb.2017203
[6]

Salomón Rebollo-Perdomo, Claudio Vidal. Bifurcation of limit cycles for a family of perturbed Kukles differential systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4189-4202. doi: 10.3934/dcds.2018182
[7]

Jianhe Shen, Maoan Han. Bifurcations of canard limit cycles in several singularly perturbed generalized polynomial Liénard systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 3085-3108. doi: 10.3934/dcds.2013.33.3085
[8]

José-Luis Bravo, Manuel Fernández. Limit cycles of non-autonomous scalar ODEs with two summands. Communications on Pure & Applied Analysis, 2013, 12 (2) : 1091-1102. doi: 10.3934/cpaa.2013.12.1091
[9]

Maoan Han, Tonghua Zhang. Some bifurcation methods of finding limit cycles. Mathematical Biosciences & Engineering, 2006, 3 (1) : 67-77. doi: 10.3934/mbe.2006.3.67
[10]

Jaume Llibre, Yilei Tang. Limit cycles of discontinuous piecewise quadratic and cubic polynomial perturbations of a linear center. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1769-1784. doi: 10.3934/dcdsb.2018236
[11]

Maoan Han. On some properties and limit cycles of Lienard systems. Conference Publications, 2001, 2001 (Special) : 426-434. doi: 10.3934/proc.2001.2001.426
[12]

Zhanyuan Hou, Stephen Baigent. Heteroclinic limit cycles in competitive Kolmogorov systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4071-4093. doi: 10.3934/dcds.2013.33.4071
[13]

Valery A. Gaiko. The geometry of limit cycle bifurcations in polynomial dynamical systems. Conference Publications, 2011, 2011 (Special) : 447-456. doi: 10.3934/proc.2011.2011.447
[14]

Ricardo M. Martins, Otávio M. L. Gomide. Limit cycles for quadratic and cubic planar differential equations under polynomial perturbations of small degree. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3353-3386. doi: 10.3934/dcds.2017142
[15]

Fabio Scalco Dias, Luis Fernando Mello. The center--focus problem and small amplitude limit cycles in rigid systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1627-1637. doi: 10.3934/dcds.2012.32.1627
[16]

Song-Mei Huan, Xiao-Song Yang. On the number of limit cycles in general planar piecewise linear systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2147-2164. doi: 10.3934/dcds.2012.32.2147
[17]

Luca Dieci, Cinzia Elia, Dingheng Pi. Limit cycles for regularized discontinuous dynamical systems with a hyperplane of discontinuity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3091-3112. doi: 10.3934/dcdsb.2017165
[18]

Jackson Itikawa, Jaume Llibre, Ana Cristina Mereu, Regilene Oliveira. Limit cycles in uniform isochronous centers of discontinuous differential systems with four zones. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3259-3272. doi: 10.3934/dcdsb.2017136
[19]

Mats Gyllenberg, Ping Yan. On the number of limit cycles for three dimensional Lotka-Volterra systems. Discrete & Continuous Dynamical Systems - B, 2009, 11 (2) : 347-352. doi: 10.3934/dcdsb.2009.11.347
[20]

Dingheng Pi. Limit cycles for regularized piecewise smooth systems with a switching manifold of codimension two. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 881-905. doi: 10.3934/dcdsb.2018211

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences