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Multidimensional stability of planar traveling waves for the scalar nonlocal Allen-Cahn equation
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 |
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. |
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