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International Journal of Mathematical Research

February 2017, Volume 6, 2, pp 46-52

Period Monotonicity for Weight-Homogeneous Differential Systems

Khalil I.T. Al-Dosary

,

Hishyar Kh. Abdullah

Khalil I.T. Al-Dosary 1 ,

Hishyar Kh. Abdullah 1 
  1. College of Sciences, University of Sharjah, Sharjah, United Arab Emirates 1

Pages: 46-52

DOI: 10.18488/journal.24.2017.62.46.52

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Article History:

Received: 20 March, 2017
Revised: 30 May, 2017
Accepted: 23 June, 2017
Published: 13 July, 2017


Abstract:

In this article, integrability, center, and monotonicity of associated period function for  -quasi-homogeneous vector fields are investigated. We are concerned with family of vector field given by sum, finite or infinite number of quasi-homogeneous polynomials not necessarily to be sharing the same weights. The investigation is done by utilizing method of computing focal values. As an application of the result, a particular family of (p, q)-quasi-homogeneous vector field is studied to find conditions for center, monotonicity and consequently an explicit form for the associated period function.

Contribution/ Originality

Keywords:

Integrability, Centre, Period function, Quasi-homogeneous polynomial, Center-focus problem, Vector field.

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Reference:

 [1]        A. Algaba, E. Freire, E. Gamero, and C. Garca, "Monodromy, centre-focus and integrability problems for quasi-homogeneous polynomial systems," Nonlinear Analysis, vol. 72, pp. 1726-1736, 2010. View at Google Scholar | View at Publisher

[2]         A. Algaba, N. Fuentes, and C. Garca, "Centre of quasi-homogeneous polynomial planar systems," Nonlinear Analysis: Real World Applications, vol. 13, pp. 419-431, 2012. View at Google Scholar | View at Publisher

[3]         M. J. Alvarez, A. Gasull, and R. Prohens, "Global behavior of the period function of the sum of two quasi-homogeneous vecto fields," Journal of Mathematical Analysis and Applications, vol. 449, pp. 1553-1569, 2017. View at Google Scholar | View at Publisher

[4]         L. Tao, L. Feng, Y. Liu, and L. Shimin, "Center-focus determination and limit cycles bifurcation for $ p: q $ homogeneous weight singular point," arXiv preprint arXiv:1608.08890, 2016.

[5]         J. Chavarriga and M. Sabatini, "A survey of isochronous centres," Qualitative Theory of Dynamical Systems, vol. 1, pp. 1-70, 1999. View at Google Scholar 

[6]         J. Gine, "On the degenerate centre problem," International Journal of Bifurcation and Chaos Applied Sciences and Engineering, vol. 21, pp. 1382-1392, 2011.

[7]         J. Gine and J. Llibre, "On the centre conditions for analytic monodromic degenerate singularities," International Journal of Bifurcation and Chaos Applied Sciences and Engineering, vol. 22, p. 1250303, 2012. View at Google Scholar | View at Publisher

[8]         Y. Xiong, M. Han, and Y. Wang, "Centre problems and limit cycle bifurcations in a class of quasi-homogeneous system," International Journal of Bifurcation and Chaos, vol. 25, p. 1550135, 2015. View at Google Scholar | View at Publisher

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Funding:

The authors would like to express their gratitude to the University of Sharjah for support

Competing Interests:

The authors declare that they have no competing interests.

Acknowledgement:

Both authors contributed equally to the conception and design of the study.

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