American Institute of Mathematical Sciences

Advanced
December  2009, 25(4): 1129-1141. doi: 10.3934/dcds.2009.25.1129

Polynomial differential equations with small coefficients

M. A. M. Alwash 1,

1. 

Department of Mathematics, West Los Angeles College, Los Angeles, CA 90230-3519, United States

Received  January 2009 Revised  May 2009 Published  September 2009

Classes of polynomial non-autonomous differential equations of degree $n$ are considered. An explicit bound on the size of the coefficients is given which implies that each equation in the class has exactly $n$ complex periodic solutions. In most of the classes the upper bound can be improved when we consider real periodic solutions. We present a proof to a recent conjecture about the number of periodic solutions. The results are used to give upper bounds for the number of limit cycles of polynomial two-dimensional systems.
Keywords: Periodic solutions, Abel differential equation, limit cycles, polynomial differential equations, Hilbert's sixteenth problem..
Mathematics Subject Classification: Primary: 34C25, 34C07, 34C05; Secondary: 37G1.
Citation: M. A. M. Alwash. Polynomial differential equations with small coefficients. Discrete & Continuous Dynamical Systems, 2009, 25 (4) : 1129-1141. doi: 10.3934/dcds.2009.25.1129
[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]

Amelia Álvarez, José-Luis Bravo, Manuel Fernández. The number of limit cycles for generalized Abel equations with periodic coefficients of definite sign. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1493-1501. doi: 10.3934/cpaa.2009.8.1493
[3]

Jaume Llibre, Claudia Valls. Rational limit cycles of Abel equations. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1077-1089. doi: 10.3934/cpaa.2021007
[4]

Jaume Llibre, Ana Rodrigues. On the limit cycles of the Floquet differential equation. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1129-1136. doi: 10.3934/dcdsb.2014.19.1129
[5]

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, 2017, 37 (6) : 3353-3386. doi: 10.3934/dcds.2017142
[6]

Naeem M. H. Alkoumi, Pedro J. Torres. Estimates on the number of limit cycles of a generalized Abel equation. Discrete & Continuous Dynamical Systems, 2011, 31 (1) : 25-34. doi: 10.3934/dcds.2011.31.25
[7]

José Luis Bravo, Manuel Fernández, Armengol Gasull. Stability of singular limit cycles for Abel equations. Discrete & Continuous Dynamical Systems, 2015, 35 (5) : 1873-1890. doi: 10.3934/dcds.2015.35.1873
[8]

Armengol Gasull, Hector Giacomini. Upper bounds for the number of limit cycles of some planar polynomial differential systems. Discrete & Continuous Dynamical Systems, 2010, 27 (1) : 217-229. doi: 10.3934/dcds.2010.27.217
[9]

Tao Li, Jaume Llibre. Limit cycles of piecewise polynomial differential systems with the discontinuity line xy = 0. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021136
[10]

Regilene Oliveira, Cláudia Valls. On the Abel differential equations of third kind. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1821-1834. doi: 10.3934/dcdsb.2020004
[11]

Regilene Oliveira, Cláudia Valls. Corrigendum: On the Abel differential equations of third kind. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021157
[12]

Eugenia N. Petropoulou, Panayiotis D. Siafarikas. Polynomial solutions of linear partial differential equations. Communications on Pure & Applied Analysis, 2009, 8 (3) : 1053-1065. doi: 10.3934/cpaa.2009.8.1053
[13]

Kun-Peng Jin, Jin Liang, Ti-Jun Xiao. Uniform polynomial stability of second order integro-differential equations in Hilbert spaces with positive definite kernels. Discrete & Continuous Dynamical Systems - S, 2021, 14 (9) : 3141-3166. doi: 10.3934/dcdss.2021077
[14]

Anatoli F. Ivanov, Sergei Trofimchuk. Periodic solutions and their stability of a differential-difference equation. Conference Publications, 2009, 2009 (Special) : 385-393. doi: 10.3934/proc.2009.2009.385
[15]

P. Dormayer, A. F. Ivanov. Symmetric periodic solutions of a delay differential equation. Conference Publications, 1998, 1998 (Special) : 220-230. doi: 10.3934/proc.1998.1998.220
[16]

Jaume Llibre, Luci Any Roberto. On the periodic solutions of a class of Duffing differential equations. Discrete & Continuous Dynamical Systems, 2013, 33 (1) : 277-282. doi: 10.3934/dcds.2013.33.277
[17]

Yingjie Bi, Siyu Liu, Yong Li. Periodic solutions of differential-algebraic equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1383-1395. doi: 10.3934/dcdsb.2019232
[18]

Nguyen Thieu Huy, Ngo Quy Dang. Dichotomy and periodic solutions to partial functional differential equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3127-3144. doi: 10.3934/dcdsb.2017167
[19]

Mahmoud M. El-Borai. On some fractional differential equations in the Hilbert space. Conference Publications, 2005, 2005 (Special) : 233-240. doi: 10.3934/proc.2005.2005.233
[20]

Josep M. Olm, Xavier Ros-Oton. Existence of periodic solutions with nonconstant sign in a class of generalized Abel equations. Discrete & Continuous Dynamical Systems, 2013, 33 (4) : 1603-1614. doi: 10.3934/dcds.2013.33.1603

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (77)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences