scholarly journals Linearizability Problem of Resonant Degenerate Singular Point for Polynomial Differential Systems

2012 ◽  
Vol 2012 ◽  
pp. 1-19
Author(s):  
Yusen Wu ◽  
Cui Zhang ◽  
Luju Liu
Keyword(s):  
Singular Point ◽  
Necessary Conditions ◽  
Recursive Algorithm ◽  
Septic System ◽  
Open Problems ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Degenerate Singular Point

The linearizability (or isochronicity) problem is one of the open problems for polynomial differential systems which is far to be solved in general. A progressive way to find necessary conditions for linearizability is to compute period constants. In this paper, we are interested in the linearizability problem ofp : −qresonant degenerate singular point for polynomial differential systems. Firstly, we transform degenerate singular point into the origin via a homeomorphism. Moreover, we establish a new recursive algorithm to compute the so-called generalized period constants for the origin of the transformed system. Finally, to illustrate the effectiveness of our algorithm, we discuss the linearizability problems of 1 : −1 resonant degenerate singular point for a septic system. We stress that similar results are hardly seen in published literatures up till now. Our work is completely new and extends existing ones.

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 Formal Weierstrass Nonintegrability Criterion for Some Classes of Polynomial Differential Systems in ℂ2

2020 ◽  
Vol 30 (04) ◽  
pp. 2050064
Author(s):  
Jaume Giné ◽  
Jaume Llibre
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Differential System ◽  
Integrable Systems ◽  
Necessary Conditions ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Formal Power

In this paper, we present a criterion for determining the formal Weierstrass nonintegrability of some polynomial differential systems in the plane [Formula: see text]. The criterion uses solutions of the form [Formula: see text] of the differential system in the plane and their associated cofactors, where [Formula: see text] is a formal power series. In particular, the criterion provides the necessary conditions in order that some polynomial differential systems in [Formula: see text] would be formal Weierstrass integrable. Inside this class there exist non-Liouvillian integrable systems. Finally we extend the theory of formal Weierstrass integrability to Puiseux Weierstrass 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.

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