Cyclicity of a class of polynomial nilpotent center singularities

Previous Article
Recent progresses in boundary layer theory
DCDS Home
This Issue
Next Article
Multidimensional stability of planar traveling waves for the scalar nonlocal Allen-Cahn equation

May 2016, 36(5): 2497-2520. doi: 10.3934/dcds.2016.36.2497

Cyclicity of a class of polynomial nilpotent center singularities

Isaac A. García ^1, and Douglas S. Shafer ^2,

1.	Departament de Matemàtica, Universitat de Lleida, Avda. Jaume II, 69. 25001. Lleida.
2.	Mathematics Department, University of North Carolina at Charlotte, Charlotte, North Carolina 28223, United States

Received July 2014 Revised July 2015 Published October 2015

Abstract
Related Papers

In this work we extend techniques based on computational algebra for bounding the cyclicity of nondegenerate centers to nilpotent centers in a natural class of polynomial systems, those of the form $\dot x = y + P_{2m + 1}(x,y)$, $\dot y = Q_{2m + 1}(x,y)$, where $P_{2m+1}$ and $Q_{2m+1}$ are homogeneous polynomials of degree $2m + 1$ in $x$ and $y$. We use the method to obtain an upper bound (which is sharp in this case) on the cyclicity of all centers in the cubic family and all centers in a broad subclass in the quintic family.

Keywords: nilpotent center., limit cycle, Cyclicity.

Mathematics Subject Classification: 37G15, 37G10, 34C0.

Citation: Isaac A. García, Douglas S. Shafer. Cyclicity of a class of polynomial nilpotent center singularities. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2497-2520. doi: 10.3934/dcds.2016.36.2497

References:

[1]	A. Algaba, C. García and M. Reyes, The center problem for a family of systems of differential equations having a nilpotent singular point, J. Math. Anal. Appl., 340 (2008), 32-43. doi: 10.1016/j.jmaa.2007.07.043.
[2]	A. Algaba, C. García and M. Reyes, Local bifurcation of limit cycles and integrability of a class of nilpotent systems of differential equations, Appl. Math. Comput., 215 (2009), 314-323. doi: 10.1016/j.amc.2009.04.077.
[3]	V. V. Amel'kin, N. A. Lukashevich and A. P. Sadovskii, Nonlinear Oscillations in Second-Order Systems, Minsk, 1982.
[4]	A. Andreev, Solution of the problem of the center and the focus in one case (Russian), Akad. Nauk SSSR. Prikl. Mat. Meh., 17 (1953), 333-338.
[5]	A. Andreev, Investigation on the behaviour of the integral curves of a system of two differential equations in the neighborhood of a singular point, Translations Amer. Math. Soc., 8 (1958), 187-207.
[6]	A. F. Andreev, A. P. Sadovskiĭ and V. A. Tsikalyuk, The center-focus problem for a system with homogeneous nonlinearities in the case of zero eigenvalues of the linear part, Differ. Equ., 39 (2003), 155-164. doi: 10.1023/A:1025192613518.
[7]	J. Chavarriga, H. Giacomini, J. Giné and J. Llibre, Local analytic integrability for nilpotent centers, Ergodic Theory Dynam. Systems, 23 (2003), 417-428. doi: 10.1017/S014338570200127X.
[8]	C. Christopher, Estimating limit cycles bifurcations from centers, in Trends in Mathematics, Differential Equations with Symbolic Computations, Birkhäuser-Verlag, Basel, 2005, 23-35. doi: 10.1007/3-7643-7429-2_2.
[9]	D. Eisenbud, C. Huneke and W. Vasconcelos, Direct methods for primary decomposition, Invent. Math., 110 (1992), 207-235. doi: 10.1007/BF01231331.
[10]	B. Ferčec, V. Levandovskyy, V. G. Romanovski and D. S. Shafer, Bifurcation of critical periods of polynomial systems, J. Differential Equations, 259 (2015), 3825-3853. doi: 10.1016/j.jde.2015.05.004.
[11]	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.
[12]	A. M. Lyapunov, Stability of Motion, Mathematics in Science and Engineering, Vol 30, Academic Press, New York-London, 1966.
[13]	J. F. Mattei and R. Moussu, Holonomie et intégrales premières, Annales Scientifiques de l'École Normale Supérieure, 13 (1980), 469-523.
[14]	V. G. Romanovski, Cyclicity of the equilibrium state of the center or focus type of a system (Russian), Vestnik Leningrad. Univ. Mat. Mekh. Astronom. vyp., 4 (1986), 82-87, 125.
[15]	V. G. Romanovski and D. S. Shafer, The Center and Cyclicity Problems: A Computational Algebra Approach, Birkhäuser Boston, Inc., Boston, MA, 2009. doi: 10.1007/978-0-8176-4727-8.
[16]	A. P. Sadovskii, The problem of center and focus (Russian), Differents. Uravn., 4 (1968), 2002-2009.

show all references

References:

[1]	A. Algaba, C. García and M. Reyes, The center problem for a family of systems of differential equations having a nilpotent singular point, J. Math. Anal. Appl., 340 (2008), 32-43. doi: 10.1016/j.jmaa.2007.07.043.
[2]	A. Algaba, C. García and M. Reyes, Local bifurcation of limit cycles and integrability of a class of nilpotent systems of differential equations, Appl. Math. Comput., 215 (2009), 314-323. doi: 10.1016/j.amc.2009.04.077.
[3]	V. V. Amel'kin, N. A. Lukashevich and A. P. Sadovskii, Nonlinear Oscillations in Second-Order Systems, Minsk, 1982.
[4]	A. Andreev, Solution of the problem of the center and the focus in one case (Russian), Akad. Nauk SSSR. Prikl. Mat. Meh., 17 (1953), 333-338.
[5]	A. Andreev, Investigation on the behaviour of the integral curves of a system of two differential equations in the neighborhood of a singular point, Translations Amer. Math. Soc., 8 (1958), 187-207.
[6]	A. F. Andreev, A. P. Sadovskiĭ and V. A. Tsikalyuk, The center-focus problem for a system with homogeneous nonlinearities in the case of zero eigenvalues of the linear part, Differ. Equ., 39 (2003), 155-164. doi: 10.1023/A:1025192613518.
[7]	J. Chavarriga, H. Giacomini, J. Giné and J. Llibre, Local analytic integrability for nilpotent centers, Ergodic Theory Dynam. Systems, 23 (2003), 417-428. doi: 10.1017/S014338570200127X.
[8]	C. Christopher, Estimating limit cycles bifurcations from centers, in Trends in Mathematics, Differential Equations with Symbolic Computations, Birkhäuser-Verlag, Basel, 2005, 23-35. doi: 10.1007/3-7643-7429-2_2.
[9]	D. Eisenbud, C. Huneke and W. Vasconcelos, Direct methods for primary decomposition, Invent. Math., 110 (1992), 207-235. doi: 10.1007/BF01231331.
[10]	B. Ferčec, V. Levandovskyy, V. G. Romanovski and D. S. Shafer, Bifurcation of critical periods of polynomial systems, J. Differential Equations, 259 (2015), 3825-3853. doi: 10.1016/j.jde.2015.05.004.
[11]	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.
[12]	A. M. Lyapunov, Stability of Motion, Mathematics in Science and Engineering, Vol 30, Academic Press, New York-London, 1966.
[13]	J. F. Mattei and R. Moussu, Holonomie et intégrales premières, Annales Scientifiques de l'École Normale Supérieure, 13 (1980), 469-523.
[14]	V. G. Romanovski, Cyclicity of the equilibrium state of the center or focus type of a system (Russian), Vestnik Leningrad. Univ. Mat. Mekh. Astronom. vyp., 4 (1986), 82-87, 125.
[15]	V. G. Romanovski and D. S. Shafer, The Center and Cyclicity Problems: A Computational Algebra Approach, Birkhäuser Boston, Inc., Boston, MA, 2009. doi: 10.1007/978-0-8176-4727-8.
[16]	A. P. Sadovskii, The problem of center and focus (Russian), Differents. Uravn., 4 (1968), 2002-2009.

[1]	Min Hu, Tao Li, Xingwu Chen. Bi-center problem and Hopf cyclicity of a Cubic Liénard system. Discrete and Continuous Dynamical Systems - B, 2020, 25 (1) : 401-414. doi: 10.3934/dcdsb.2019187
[2]	Ben Niu, Weihua Jiang. Dynamics of a limit cycle oscillator with extended delay feedback. Discrete and Continuous Dynamical Systems - B, 2013, 18 (5) : 1439-1458. doi: 10.3934/dcdsb.2013.18.1439
[3]	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
[4]	Jaume Llibre, Dana Schlomiuk. On the limit cycles bifurcating from an ellipse of a quadratic center. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 1091-1102. doi: 10.3934/dcds.2015.35.1091
[5]	Magdalena Caubergh, Freddy Dumortier, Robert Roussarie. Alien limit cycles in rigid unfoldings of a Hamiltonian 2-saddle cycle. Communications on Pure and Applied Analysis, 2007, 6 (1) : 1-21. doi: 10.3934/cpaa.2007.6.1
[6]	Jihua Yang, Liqin Zhao. Limit cycle bifurcations for piecewise smooth integrable differential systems. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2417-2425. doi: 10.3934/dcdsb.2017123
[7]	Stijn Luca, Freddy Dumortier, Magdalena Caubergh, Robert Roussarie. Detecting alien limit cycles near a Hamiltonian 2-saddle cycle. Discrete and Continuous Dynamical Systems, 2009, 25 (4) : 1081-1108. doi: 10.3934/dcds.2009.25.1081
[8]	Fangfang Jiang, Junping Shi, Qing-guo Wang, Jitao Sun. On the existence and uniqueness of a limit cycle for a Liénard system with a discontinuity line. Communications on Pure and Applied Analysis, 2016, 15 (6) : 2509-2526. doi: 10.3934/cpaa.2016047
[9]	Meilan Cai, Maoan Han. Limit cycle bifurcations in a class of piecewise smooth cubic systems with multiple parameters. Communications on Pure and Applied Analysis, 2021, 20 (1) : 55-75. doi: 10.3934/cpaa.2020257
[10]	Sze-Bi Hsu, Junping Shi. Relaxation oscillation profile of limit cycle in predator-prey system. Discrete and Continuous Dynamical Systems - B, 2009, 11 (4) : 893-911. doi: 10.3934/dcdsb.2009.11.893
[11]	Jaume Llibre. Limit cycles of continuous piecewise differential systems separated by a parabola and formed by a linear center and a quadratic center. Discrete and Continuous Dynamical Systems - S, 2022 doi: 10.3934/dcdss.2022034
[12]	Fabio Scalco Dias, Luis Fernando Mello. The center--focus problem and small amplitude limit cycles in rigid systems. Discrete and Continuous Dynamical Systems, 2012, 32 (5) : 1627-1637. doi: 10.3934/dcds.2012.32.1627
[13]	Jaume Llibre, Yilei Tang. Limit cycles of discontinuous piecewise quadratic and cubic polynomial perturbations of a linear center. Discrete and Continuous Dynamical Systems - B, 2019, 24 (4) : 1769-1784. doi: 10.3934/dcdsb.2018236
[14]	Iliya D. Iliev, Chengzhi Li, Jiang Yu. Bifurcations of limit cycles in a reversible quadratic system with a center, a saddle and two nodes. Communications on Pure and Applied Analysis, 2010, 9 (3) : 583-610. doi: 10.3934/cpaa.2010.9.583
[15]	Lijun Wei, Xiang Zhang. Limit cycle bifurcations near generalized homoclinic loop in piecewise smooth differential systems. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2803-2825. doi: 10.3934/dcds.2016.36.2803
[16]	Bourama Toni. Upper bounds for limit cycle bifurcation from an isochronous period annulus via a birational linearization. Conference Publications, 2005, 2005 (Special) : 846-853. doi: 10.3934/proc.2005.2005.846
[17]	Jihua Yang, Erli Zhang, Mei Liu. Limit cycle bifurcations of a piecewise smooth Hamiltonian system with a generalized heteroclinic loop through a cusp. Communications on Pure and Applied Analysis, 2017, 16 (6) : 2321-2336. doi: 10.3934/cpaa.2017114
[18]	Qiongwei Huang, Jiashi Tang. Bifurcation of a limit cycle in the ac-driven complex Ginzburg-Landau equation. Discrete and Continuous Dynamical Systems - B, 2010, 14 (1) : 129-141. doi: 10.3934/dcdsb.2010.14.129
[19]	Huanhuan Tian, Maoan Han. Limit cycle bifurcations of piecewise smooth near-Hamiltonian systems with a switching curve. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5581-5599. doi: 10.3934/dcdsb.2020368
[20]	Wenye Liu, Maoan Han. Limit cycle bifurcations of near-Hamiltonian systems with multiple switching curves and applications. Discrete and Continuous Dynamical Systems - S, 2022 doi: 10.3934/dcdss.2022053

2021 Impact Factor: 1.588

Tools

Metrics

Other articles
by authors

on AIMS
Isaac A. García Douglas S. Shafer
on Google Scholar
Isaac A. García Douglas S. Shafer

[Back to Top]