Generalized Analytic Integrability of a Class of Polynomial Differential Systems in $\mathbb{C}^{2}$

2021 ◽  
Vol 173 (1) ◽  
Author(s):  
Jaume Llibre ◽  
Yuzhou Tian
Keyword(s):  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Analytic Integrability

scholarly journals Analytic integrability of quadratic–linear polynomial differential systems

2009 ◽  
Vol 31 (1) ◽  
pp. 245-258 ◽  
Author(s):  
JAUME LLIBRE ◽  
CLÀUDIA VALLS
Keyword(s):  
Singular Point ◽  
Explicit Expression ◽  
First Integral ◽  
First Integrals ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Analytic Integrability ◽  
Linear Polynomial ◽  
Analytic First Integral

AbstractFor the quadratic–linear polynomial differential systems with a finite singular point, we classify the ones which have a global analytic first integral, and provide the explicit expression of their first integrals.

Three-Dimensional Hopf Bifurcation for a Class of Cubic Kolmogorov Model

2014 ◽  
Vol 24 (03) ◽  
pp. 1450036 ◽  
Author(s):  
Chaoxiong Du ◽  
Qinlong Wang ◽  
Wentao Huang
Keyword(s):  
Hopf Bifurcation ◽  
Singular Point ◽  
Three Dimensional ◽  
Singular Values ◽  
Bifurcation Problem ◽  
Disturbed Condition ◽  
Differential Systems ◽  
Kolmogorov Model ◽  
Polynomial Differential Systems ◽  
Fine Focus

We study the Hopf bifurcation for a class of three-dimensional cubic Kolmogorov model by making use of our method (i.e. singular values method). We show that the positive singular point (1, 1, 1) of an investigated model can become a fine focus of 5 order, and moreover, it can bifurcate five small limit cycles under certain coefficients with disturbed condition. In terms of three-dimensional cubic Kolmogorov model, published references can hardly be seen, and our results are new. At the same time, it is worth pointing out that our method is valid to study the Hopf bifurcation problem for other three-dimensional polynomial differential systems.

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