# American Institute of Mathematical Sciences

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

## Cyclicity of a class of polynomial nilpotent center singularities

 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

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.
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.

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##### 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.
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