May  2020, 25(5): 1821-1834. doi: 10.3934/dcdsb.2020004

On the Abel differential equations of third kind

1. 

Departamento de Matemática, ICMC-Universidade de São Paulo, Avenida Trabalhador São-carlense, 400, 13566-590, São Carlos, SP, Brazil

2. 

Departamento de Matemática, Instituto Superior Técnico, Universidade Técnica de Lisboa, Av. Rovisco Pais, 1049–001, Lisboa, Portugal

* Corresponding author

Received  December 2018 Revised  May 2019 Published  May 2020 Early access  December 2019

Abel equations of the first and second kind have been widely studied, but one question that never has been addressed for the Abel polynomial differential systems is to understand the behavior of its solutions (without knowing explicitly them), or in other words, to obtain its qualitative behavior. This is a very hard task that grows exponentially as the number of parameters in the equation increases. In this paper, using Poincaré compactification we classify the topological phase portraits of a special kind of quadratic differential system, the Abel quadratic equations of third kind. We also describe the maximal number of polynomial solutions that Abel polynomial differential equations can have.

Citation: Regilene Oliveira, Cláudia Valls. On the Abel differential equations of third kind. Discrete and Continuous Dynamical Systems - B, 2020, 25 (5) : 1821-1834. doi: 10.3934/dcdsb.2020004
References:
[1]

M. Bhargava and H. Kaufman, Degrees of polynomial solutions of a class of Riccati-type differential equations, Collect. Math., 16 (1964), 211-223. 

[2]

M. Bhargava and H. Kaufman, Existence of polynomial solutions of a class of Riccati-type differential equations, Collect. Math., 17 (1965), 135-143. 

[3]

M. Bhargava and H. Kaufman, Some properties of polynomial solutions of a class of Riccati-type differential equations, Collect. Math., 18 (1966/1967), 3-6. 

[4]

J. G. Campbell, A criterion for the polynomial solutions of a certain Riccati equation, Amer. Math. Monthly, 59 (1952), 388-389.  doi: 10.2307/2306810.

[5]

J. G. Campbell and M. Golomb, On the polynomial solutions of a Riccati equation, Amer. Math. Monthly, 61 (1954), 402-404.  doi: 10.2307/2307902.

[6]

F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, UniversiText, Springer-verlag, Berlin, 2006.

[7]

A. GasullJ. Torregrosa and X. Zhang, The number of polynomial solutions of polynomial Riccati equations, J. Differential Equations, 261 (2016), 5071-5093.  doi: 10.1016/j.jde.2016.07.019.

[8]

A. GasullL.-R. Sheng and J. Llibre, Chordal quadratic systems, Rocky Mountain J. Math., 16 (1986), 751-782.  doi: 10.1216/RMJ-1986-16-4-751.

[9]

J. GinéM. Grau and J. Llibre, On the polynomial limit cycles of polynomial differential equations, Israel J. Math., 181 (2011), 461-475.  doi: 10.1007/s11856-011-0019-3.

[10]

E. Hille, Ordinary Differential Equations in the Complex Domain, Dover Publications, Inc., NY, 1997.

[11]

P. de Jager, Phase portraits for quadratic systems with a higher order singularity with two zero eigenvalues, J. Differential Equations, 87 (1990), 169-204.  doi: 10.1016/0022-0396(90)90021-G.

[12]

D. A. Neumann, Classification of continuous flows on $2$-manifolds, Proc. Amer. Math. Soc., 48 (1975), 73-81.  doi: 10.1090/S0002-9939-1975-0356138-6.

[13]

M. PollicottH. Wang and H. Weiss, Extracting the time-dependent transmission rate from infection data via solution of an inverse ODE problem, J. Biol. Dyn., 6 (2012), 509-523.  doi: 10.1080/17513758.2011.645510.

[14]

E. D. Rainville, Necessary conditions for polynomial solutions of certain Riccati equations, Amer. Math. Monthly, 43 (1936), 473-476.  doi: 10.1080/00029890.1936.11987882.

show all references

References:
[1]

M. Bhargava and H. Kaufman, Degrees of polynomial solutions of a class of Riccati-type differential equations, Collect. Math., 16 (1964), 211-223. 

[2]

M. Bhargava and H. Kaufman, Existence of polynomial solutions of a class of Riccati-type differential equations, Collect. Math., 17 (1965), 135-143. 

[3]

M. Bhargava and H. Kaufman, Some properties of polynomial solutions of a class of Riccati-type differential equations, Collect. Math., 18 (1966/1967), 3-6. 

[4]

J. G. Campbell, A criterion for the polynomial solutions of a certain Riccati equation, Amer. Math. Monthly, 59 (1952), 388-389.  doi: 10.2307/2306810.

[5]

J. G. Campbell and M. Golomb, On the polynomial solutions of a Riccati equation, Amer. Math. Monthly, 61 (1954), 402-404.  doi: 10.2307/2307902.

[6]

F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, UniversiText, Springer-verlag, Berlin, 2006.

[7]

A. GasullJ. Torregrosa and X. Zhang, The number of polynomial solutions of polynomial Riccati equations, J. Differential Equations, 261 (2016), 5071-5093.  doi: 10.1016/j.jde.2016.07.019.

[8]

A. GasullL.-R. Sheng and J. Llibre, Chordal quadratic systems, Rocky Mountain J. Math., 16 (1986), 751-782.  doi: 10.1216/RMJ-1986-16-4-751.

[9]

J. GinéM. Grau and J. Llibre, On the polynomial limit cycles of polynomial differential equations, Israel J. Math., 181 (2011), 461-475.  doi: 10.1007/s11856-011-0019-3.

[10]

E. Hille, Ordinary Differential Equations in the Complex Domain, Dover Publications, Inc., NY, 1997.

[11]

P. de Jager, Phase portraits for quadratic systems with a higher order singularity with two zero eigenvalues, J. Differential Equations, 87 (1990), 169-204.  doi: 10.1016/0022-0396(90)90021-G.

[12]

D. A. Neumann, Classification of continuous flows on $2$-manifolds, Proc. Amer. Math. Soc., 48 (1975), 73-81.  doi: 10.1090/S0002-9939-1975-0356138-6.

[13]

M. PollicottH. Wang and H. Weiss, Extracting the time-dependent transmission rate from infection data via solution of an inverse ODE problem, J. Biol. Dyn., 6 (2012), 509-523.  doi: 10.1080/17513758.2011.645510.

[14]

E. D. Rainville, Necessary conditions for polynomial solutions of certain Riccati equations, Amer. Math. Monthly, 43 (1936), 473-476.  doi: 10.1080/00029890.1936.11987882.

Figure 1.  Global phase portraits in the Poincaré disk of systems (ⅰ)–(ⅳ): from (1)–(20). Here $ S $ denotes the number of separatrices and $ R $ the number of canonical region of each phase portrait
Figure 2.  Global phase portraits in the Poincaré disk of systems (ⅰ)–(ⅱ): from (21)–(40). Here $ S $ denotes the number of separatrices and $ R $ the number of canonical region of each phase portrait
Figure 3.  Global phase portraits in the Poincaré disk of systems (ⅱ): from (41)–(44). Here $ S $ denotes the number of separatrices and $ R $ the number of canonical region of each phase portrait
Figure 4.  Blowing down process for system (ⅰ) at origin when $ k_0 = k_1 = 0 $ and $ k_2< 3/4^{1/3} $ (a), $ k_2 = 3/4^{1/3} $ (b), and $ k_2> 3/4^{1/3} $ (c)
Figure 5.  The position of the separatrices of the two saddle-nodes $ (\pm \sqrt{-k_0},0) $ for system (ⅰ) with $ k_0<0 $, $ k_2<3/4^{1/3} $ and $ (k_1+\sqrt{-k_0}k_2)(k_1- \sqrt{-k_0}k_2)\neq0 $
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