scholarly journals Algebraic invariant curves of plane polynomial differential systems

2001 ◽  
Vol 34 (3) ◽  
pp. 663-672 ◽  
Author(s):  
Alexei Tsygvintsev
Keyword(s):  
Invariant Curves ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Algebraic Invariant

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.

scholarly journals The Cubic Polynomial Differential Systems with two Circles as Algebraic Limit Cycles

2018 ◽  
Vol 18 (1) ◽  
pp. 183-193 ◽  
Author(s):  
Jaume Giné ◽  
Jaume Llibre ◽  
Claudia Valls
Keyword(s):  
Limit Cycles ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Cubic Polynomial ◽  
Algebraic Limit Cycles ◽  
Algebraic Limit

AbstractIn this paper we characterize all cubic polynomial differential systems in the plane having two circles as invariant algebraic limit cycles.

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