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Article Contents

Corrigendum: On the Abel differential equations of third kind

• * Corresponding author: Regilene Oliveira
• 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.

Mathematics Subject Classification: Primary: 37C15, 37C10.

 Citation:

• 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] 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.

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