Polynomial first integrals for quasi-homogeneous polynomial differential systems

Nonlinearity ◽  
2002 ◽  
Vol 15 (4) ◽  
pp. 1269-1280 ◽  
Author(s):  
Jaume Llibre ◽  
Xiang Zhang
Keyword(s):  
Homogeneous Polynomial ◽  
First Integrals ◽  
Differential Systems ◽  
Polynomial Differential Systems

Projective dynamics of homogeneous systems: local invariants, syzygies and the Global Residue Theorem

2012 ◽  
Vol 55 (3) ◽  
pp. 577-589 ◽  
Author(s):  
Z. Balanov ◽  
A. Kononovich ◽  
Y. Krasnov
Keyword(s):  
Explicit Formula ◽  
Homogeneous Polynomial ◽  
First Integrals ◽  
Global Dynamics ◽  
Bounded Solutions ◽  
Residue Theorem ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Homogeneous Systems ◽  
Local Invariants

AbstractWe give an explicit formula for the projective dynamics of planar homogeneous polynomial differential systems in terms of natural local invariants and we establish explicit algebraic connections (syzygies) between these invariants (leading to restrictions on possible global dynamics). We discuss multidimensional generalizations together with applications to the existence of first integrals and bounded solutions.

scholarly journals Center of planar quintic quasi--homogeneous polynomial differential systems

2015 ◽  
Vol 35 (5) ◽  
pp. 2177-2191 ◽  
Author(s):  
Yilei Tang ◽  
◽  
Long Wang ◽  
Xiang Zhang ◽  
◽  
...  
Keyword(s):  
Homogeneous Polynomial ◽  
Differential Systems ◽  
Polynomial Differential Systems

scholarly journals Planar quasi-homogeneous polynomial differential systems and their integrability

2013 ◽  
Vol 255 (10) ◽  
pp. 3185-3204 ◽  
Author(s):  
Belén García ◽  
Jaume Llibre ◽  
Jesús S. Pérez del Río
Keyword(s):  
Homogeneous Polynomial ◽  
Differential Systems ◽  
Polynomial Differential Systems

Center Problems and Limit Cycle Bifurcations in a Class of Quasi-Homogeneous Systems

2015 ◽  
Vol 25 (10) ◽  
pp. 1550135 ◽  
Author(s):  
Yanqin Xiong ◽  
Maoan Han ◽  
Yong Wang
Keyword(s):  
Limit Cycle ◽  
Differential System ◽  
Homogeneous Polynomial ◽  
Sufficient Conditions ◽  
Differential Systems ◽  
First Order ◽  
Polynomial Differential Systems ◽  
Perturbed System ◽  
Generalized Polynomial ◽  
Necessary And Sufficient

In this paper, we first classify all centers of a class of quasi-homogeneous polynomial differential systems of degree 5. Then we extend this kind of systems to a generalized polynomial differential system and provide the necessary and sufficient conditions to have a center at the origin. Furthermore, we study the Poincaré bifurcation for its perturbed system as it has a center at the origin, find the Poincaré cyclicity up to first order of ε.

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