ALGEBRAIC ASPECTS OF INTEGRABILITY FOR POLYNOMIAL DIFFERENTIAL SYSTEMS

Yangyou Pan; Xiang Zhang

doi:10.11948/2013005

2013 Volume 3 Issue 1

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Yangyou Pan, Xiang Zhang. ALGEBRAIC ASPECTS OF INTEGRABILITY FOR POLYNOMIAL DIFFERENTIAL SYSTEMS[J]. Journal of Applied Analysis & Computation, 2013, 3(1): 51-69. doi: 10.11948/2013005

Citation:

Yangyou Pan, Xiang Zhang. ALGEBRAIC ASPECTS OF INTEGRABILITY FOR POLYNOMIAL DIFFERENTIAL SYSTEMS[J]. Journal of Applied Analysis & Computation, 2013, 3(1): 51-69. doi: 10.11948/2013005

ALGEBRAIC ASPECTS OF INTEGRABILITY FOR POLYNOMIAL DIFFERENTIAL SYSTEMS

Yangyou Pan¹,
Xiang Zhang²

1 Department of Mathematics and Computer Science, Chizhou University, Chizhou 274000, China;
2 Department of Mathematics, and MOE-LSC, Shanghai Jiao Tong University, Shanghai 200240, China

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Abstract

Abstract

In this article we summarize the results on algebraic aspects of integrability for polynomial differential systems and its application, which include the Darboux, elementary and Liouvelle integrability. Darboux theory of integrability was found by Darboux in 1878, and it becomes extremely useful in study of the center focus problem, of bifurcation, of limit cycle problem and of global dynamics. The importance of Darboux theory of integrability is also presented by the Singer's theorem for planar polynomial differential system. That is, if a polynomial system is Liouville integrable, then it is Darboux integrable, i.e. the system has a Darboux first integral or a Darboux integrating factor.
- Darboux integrability /
- elementary integrability /
- Liouville integrability /
- invariant algebraic curves
MSC: 34A34;34C20;34C41;37G05

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References

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