Figure 1.
Global phase portraits in the Poincaré disk of of system $(ⅰ)$
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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 |
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 [
Keywords:
Abel polynomial differential equations,
vector fields,
phase portraits,
polynomial solutions.
Mathematics Subject Classification:
Primary: 37C15, 37C10.
Citation:
Regilene Oliveira, Cláudia Valls.
Corrigendum: On the Abel differential equations of third kind. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021157
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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. |
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. |
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
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