Abstract
We study a family of quadratic vector fields in Class I \(\dot{x}=y, \dot{y}=-x-\alpha y+\mu x^2-y^2\), where \((\alpha , \mu ) \in \mathbb {R}^2\). To study the equilibria at infinity on the Poincaré disk of this system completely, we follow the method of generalized normal sectors of Tang and Zhang (Nonlinearity 17:1407–1426, 2004) and give further two new criterions, which allows us to obtain not only the qualitative properties of the equilibria but also asymptotic expressions of these orbits connecting the equilibria at infinity of this system. Further, the complete bifurcation diagram including saddle connection bifurcation curves of this system is given. Moreover, by qualitative properties of the equilibria, the nonexistence of limit cycle and rotated properties about \(\alpha \) and \(\mu \), all global phase portraits on the Poincaré disk of this system are also obtained and the number is 19.
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Acknowledgements
The paper is supported partially by the National Natural Science Foundation of China (Nos. 11801079, 11671403). The authors are grateful to the reviewers for their helpful and valuable suggestions and comments.
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Jia, M., Chen, H. & Chen, H. Bifurcation Diagram and Global Phase Portraits of a Family of Quadratic Vector Fields in Class I. Qual. Theory Dyn. Syst. 19, 64 (2020). https://doi.org/10.1007/s12346-020-00402-4
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DOI: https://doi.org/10.1007/s12346-020-00402-4