# American Institute of Mathematical Sciences

October  2015, 20(8): 2657-2661. doi: 10.3934/dcdsb.2015.20.2657

## Analytic integrability of a class of planar polynomial differential systems

 1 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia 2 Departamento de Matemática, Instituto Superior Técnico , Universidade Técnica de Lisboa, Av. Rovisco Pais 1049-001, Lisboa, Portugal

Received  October 2014 Revised  January 2015 Published  August 2015

In this paper we find necessary and sufficient conditions in order that the differential systems of the form $\dot x = x f(y)$, $\dot y =g(y)$, with $f$ and $g$ polynomials, have a first integral which is analytic in the variable $x$ and meromorphic in the variable $y$. We also characterize their analytic first integrals in both variables $x$ and $y$.
These polynomial differential systems are important because after a convenient change of variables they contain all quasi--homogeneous polynomial differential systems in $\mathbb{R}^2$.
Citation: Jaume Llibre, Claudia Valls. Analytic integrability of a class of planar polynomial differential systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2657-2661. doi: 10.3934/dcdsb.2015.20.2657
##### References:
 [1] J. Giné, M. Grau and J. Llibre, Polynomial and rational first integrals for planar quasi-homogeneous polynomial differential systems,, Discrete and Continuous Dynamical Systems, 33 (2013), 4531. doi: 10.3934/dcds.2013.33.4531. Google Scholar [2] E. Isaacson and H. B. Keller, Analysis of Numerical Methods,, Dover Publications, (1994). Google Scholar [3] J. Llibre and X. Zhang, Polynomial first integrals for quasi-homogeneous polynomial differential systems,, Nonlinearity, 15 (2002), 1269. doi: 10.1088/0951-7715/15/4/313. Google Scholar

show all references

##### References:
 [1] J. Giné, M. Grau and J. Llibre, Polynomial and rational first integrals for planar quasi-homogeneous polynomial differential systems,, Discrete and Continuous Dynamical Systems, 33 (2013), 4531. doi: 10.3934/dcds.2013.33.4531. Google Scholar [2] E. Isaacson and H. B. Keller, Analysis of Numerical Methods,, Dover Publications, (1994). Google Scholar [3] J. Llibre and X. Zhang, Polynomial first integrals for quasi-homogeneous polynomial differential systems,, Nonlinearity, 15 (2002), 1269. doi: 10.1088/0951-7715/15/4/313. Google Scholar
 [1] Jaume Giné, Maite Grau, Jaume Llibre. Polynomial and rational first integrals for planar quasi--homogeneous polynomial differential systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4531-4547. doi: 10.3934/dcds.2013.33.4531 [2] Yilei Tang, Long Wang, Xiang Zhang. Center of planar quintic quasi--homogeneous polynomial differential systems. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2177-2191. doi: 10.3934/dcds.2015.35.2177 [3] Michal Fečkan, Michal Pospíšil. Discretization of dynamical systems with first integrals. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3543-3554. doi: 10.3934/dcds.2013.33.3543 [4] Yanqin Xiong, Maoan Han. Planar quasi-homogeneous polynomial systems with a given weight degree. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 4015-4025. doi: 10.3934/dcds.2016.36.4015 [5] 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 [6] Jaume Llibre, Roland Rabanal. Center conditions for a class of planar rigid polynomial differential systems. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1075-1090. doi: 10.3934/dcds.2015.35.1075 [7] Rehana Naz, Fazal M. Mahomed. Characterization of partial Hamiltonian operators and related first integrals. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 723-734. doi: 10.3934/dcdss.2018045 [8] Elena Celledoni, Brynjulf Owren. Preserving first integrals with symmetric Lie group methods. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 977-990. doi: 10.3934/dcds.2014.34.977 [9] Dirk Aeyels, Filip De Smet, Bavo Langerock. Area contraction in the presence of first integrals and almost global convergence. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 135-157. doi: 10.3934/dcds.2007.18.135 [10] Richard A. Norton, David I. McLaren, G. R. W. Quispel, Ari Stern, Antonella Zanna. Projection methods and discrete gradient methods for preserving first integrals of ODEs. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2079-2098. doi: 10.3934/dcds.2015.35.2079 [11] 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 [12] Yilei Tang. Global dynamics and bifurcation of planar piecewise smooth quadratic quasi-homogeneous differential systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2029-2046. doi: 10.3934/dcds.2018082 [13] 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 [14] Claude Froeschlé, Massimiliano Guzzo, Elena Lega. First numerical evidence of global Arnold diffusion in quasi-integrable systems. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 687-698. doi: 10.3934/dcdsb.2005.5.687 [15] Francisco Braun, Jaume Llibre, Ana Cristina Mereu. Isochronicity for trivial quintic and septic planar polynomial Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5245-5255. doi: 10.3934/dcds.2016029 [16] Bin Wang, Arieh Iserles. Dirichlet series for dynamical systems of first-order ordinary differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 281-298. doi: 10.3934/dcdsb.2014.19.281 [17] Zhaosheng Feng, Guangyue Gao, Jing Cui. Duffing--van der Pol--type oscillator system and its first integrals. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1377-1391. doi: 10.3934/cpaa.2011.10.1377 [18] J. R. Ward. Periodic solutions of first order systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 381-389. doi: 10.3934/dcds.2013.33.381 [19] Weigu Li, Jaume Llibre, Hao Wu. Polynomial and linearized normal forms for almost periodic differential systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 345-360. doi: 10.3934/dcds.2016.36.345 [20] Alain Jacquemard, Weber Flávio Pereira. On periodic orbits of polynomial relay systems. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 331-347. doi: 10.3934/dcds.2007.17.331

2018 Impact Factor: 1.008