doi: 10.3934/dcdsb.2021157

Corrigendum: 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: Regilene Oliveira

Received  February 2021 Revised  April 2021 Early access  June 2021

In this paper, using the Poincaré compactification technique we classify the topological phase portraits of a special kind of quadratic differential system, the Abel quadratic equations of third kind. In [1] where such investigation was presented for the first time some phase portraits were not correct and some were missing. Here we provide the complete list of non equivalent phase portraits that the Abel quadratic equations of third kind can exhibit and the bifurcation diagram of a $ 3 $-parametric subfamily of it.

Citation: Regilene Oliveira, Cláudia Valls. Corrigendum: On the Abel differential equations of third kind. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021157
References:
[1]

R. Oliveira and C. Valls, On the Abel differential equations of third kind, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 1821-1834.  doi: 10.3934/dcdsb.2020004.  Google Scholar

show all references

References:
[1]

R. Oliveira and C. Valls, On the Abel differential equations of third kind, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 1821-1834.  doi: 10.3934/dcdsb.2020004.  Google Scholar

Figure 1.  Global phase portraits in the Poincaré disk of of system $(ⅰ)$
Figure 2.  Global phase portraits in the Poincaré disk of system $(ⅰ)$
Figure 3.  Bifurcation diagram of system $(ⅰ)$ for $ k_1 = 1 $ and $ k_1 = 0 $. The phase portrait $ P_2 $ is topologically equivalent to $ L_{8} $. In this diagram, the phase portraits in the surfaces $ S_i $, for $ i = 4, 5, 6, 9, 10, 18 $ and in the curves $ L_j $, for $ j = 2, 4, 5, 6 $ admit a separatrix connection and although such surfaces and curves do not need to be algebraic the surfaces $ S_9 $ and $ S_{10} $ are contained in one component of the algebraic surface given by $ -64 k_0^3 - k_1^6 + 16 k_0^2 k_1^2 k_2 - 3 k_0 k_1^4 k_2^2 + 16 k_0^3 k_2^3 - 3 k_0^2 k_1^2 k_2^4 - k_0^3 k_2^6 = 0 $
Figure 4.  Global phase portraits in the Poincaré disk of systems $(ⅱ)$. The phase portraits $ V_{13} $ and $ V_{15} $ of system $(ⅱ)$ are topologically equivalent to the phase portraits $ S_3 $ and $ S_2 $ of system $(ⅰ)$, respectively. Here we remark that in the phase portraits $ S_3 $ and $ S_2 $ of system $(ⅰ)$, such system has two complex points and a saddle–node as finite singular points but in system $(ⅱ)$ the phase portraits $ V_{13} $ and $ V_{15} $ have a saddle–node as the unique finite singular point. The phase portrait $ S_{21} $ has a separatrix connection between the finite saddle-node and the infinite saddle
[1]

Antoni Ferragut, Jaume Llibre, Adam Mahdi. Polynomial inverse integrating factors for polynomial vector fields. Discrete & Continuous Dynamical Systems, 2007, 17 (2) : 387-395. doi: 10.3934/dcds.2007.17.387

[2]

Jackson Itikawa, Jaume Llibre. Global phase portraits of uniform isochronous centers with quartic homogeneous polynomial nonlinearities. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 121-131. doi: 10.3934/dcdsb.2016.21.121

[3]

Jaume Llibre, Claudia Valls. Centers for polynomial vector fields of arbitrary degree. Communications on Pure & Applied Analysis, 2009, 8 (2) : 725-742. doi: 10.3934/cpaa.2009.8.725

[4]

Adriana Buică, Jaume Giné, Maite Grau. Essential perturbations of polynomial vector fields with a period annulus. Communications on Pure & Applied Analysis, 2015, 14 (3) : 1073-1095. doi: 10.3934/cpaa.2015.14.1073

[5]

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

[6]

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

[7]

Primitivo B. Acosta-Humánez, J. Tomás Lázaro, Juan J. Morales-Ruiz, Chara Pantazi. On the integrability of polynomial vector fields in the plane by means of Picard-Vessiot theory. Discrete & Continuous Dynamical Systems, 2015, 35 (5) : 1767-1800. doi: 10.3934/dcds.2015.35.1767

[8]

Isaac A. García, Jaume Giné. Non-algebraic invariant curves for polynomial planar vector fields. Discrete & Continuous Dynamical Systems, 2004, 10 (3) : 755-768. doi: 10.3934/dcds.2004.10.755

[9]

Olivier Hénot. On polynomial forms of nonlinear functional differential equations. Journal of Computational Dynamics, 2021, 8 (3) : 307-323. doi: 10.3934/jcd.2021013

[10]

Jaume Llibre, Jesús S. Pérez del Río, J. Angel Rodríguez. Structural stability of planar semi-homogeneous polynomial vector fields applications to critical points and to infinity. Discrete & Continuous Dynamical Systems, 2000, 6 (4) : 809-828. doi: 10.3934/dcds.2000.6.809

[11]

Jaume Llibre, Y. Paulina Martínez, Claudio Vidal. Phase portraits of linear type centers of polynomial Hamiltonian systems with Hamiltonian function of degree 5 of the form $ H = H_1(x)+H_2(y)$. Discrete & Continuous Dynamical Systems, 2019, 39 (1) : 75-113. doi: 10.3934/dcds.2019004

[12]

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

[13]

Qi Zhang, Huaizhong Zhao. Backward doubly stochastic differential equations with polynomial growth coefficients. Discrete & Continuous Dynamical Systems, 2015, 35 (11) : 5285-5315. doi: 10.3934/dcds.2015.35.5285

[14]

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

[15]

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

[16]

László Mérai, Igor E. Shparlinski. Unlikely intersections over finite fields: Polynomial orbits in small subgroups. Discrete & Continuous Dynamical Systems, 2020, 40 (2) : 1065-1073. doi: 10.3934/dcds.2020070

[17]

Luis Silvestre. Hölder continuity for integro-differential parabolic equations with polynomial growth respect to the gradient. Discrete & Continuous Dynamical Systems, 2010, 28 (3) : 1069-1081. doi: 10.3934/dcds.2010.28.1069

[18]

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

[19]

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

[20]

Elimhan N. Mahmudov. Optimal control of evolution differential inclusions with polynomial linear differential operators. Evolution Equations & Control Theory, 2019, 8 (3) : 603-619. doi: 10.3934/eect.2019028

2020 Impact Factor: 1.327

Article outline

Figures and Tables

[Back to Top]