The Necessary Condition for the Existence of Periodic Solutions of Certain Three Dimensional Nonlinear Dynamical Systems

Ting Ting Quan; Jing Li; Min Sun

doi:10.4028/www.scientific.net/AMM.252.40

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HomeApplied Mechanics and MaterialsApplied Mechanics and Materials Vol. 252The Necessary Condition for the Existence of...

The Necessary Condition for the Existence of Periodic Solutions of Certain Three Dimensional Nonlinear Dynamical Systems

Abstract:

In this paper, we investigate a class of three dimensional nonlinear dynamical systems whose unperturbed systems have a family of periodic orbits. Firstly, we establish the moving Frenet Frame on these closed orbits. Secondly, the successor functions are defined by the orbits which go through the normal plane. Finally, by judging the existence of solutions of the equations obtained from the Successor functions, we obtain the necessary condition for the existence of periodic solutions of these three dimensional nonlinear dynamical systems. The result has important significance for the basic research of applied mechanics.

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Applied Mechanics and Materials (Volume 252)

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40-43

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https://doi.org/10.4028/www.scientific.net/AMM.252.40

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Online since:

December 2012

Authors:

Ting Ting Quan, Jing Li, Min Sun

Keywords:

Frenet Frame, Periodic Solution, Successor Function, Three Dimensional Nonlinear Dynamical Systems

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References

[1] J. Li. Hilbert's 16th Problem and Bifurcations of Planar Polynomial Vector Fields. International Journal of Bifurcation and Chaos. 13 (2002), pp.47-106.

DOI: 10.1142/s0218127403006352

Google Scholar

[2] J. Li, Y. Tian, W. Zhang and S. F. Miao. Bifurcation of Multiple Limit Cycles for a Rotor-Active Magnetic Bearings System with Time-Varying Stiffness. International Journal of Bifurcation and Chaos. 18 (2008), pp.755-778.

DOI: 10.1142/s021812740802063x

Google Scholar

[3] B. Mehri, N. Mahdavi-Amiri. Periodic Solutions of Certain Three Dimensional Autonomous Systems. J. Sci. I. R. Iran, 10(1999), pp.117-120.

Google Scholar

[4] M. Bayat, B. Mehri. A necessary Condition for the Existence of Periodic Solutions of Certain Three Dimensional Autonomous Systems. Applied Mathematics Letters. 22 (2009), pp.1292-1296.

DOI: 10.1016/j.aml.2009.01.045

Google Scholar

[5] K.L. Wu and Y.L. Zhao. On the Number of Zeros of Abelian Integral for a Cubic Isochronous Center. International Journal of Bifurcation and Chaos. 1 (2012), pp.1-9.

DOI: 10.1142/s0218127412500162

Google Scholar

[6] J. Li, Y. Tian and W. Zhang. Investigation of the Relation between Singular Points and the Number of Limit Cycles for a Rotor-AMB System. Chaos, Solitons and Fractals. 39 (2009), pp.1627-1640.

DOI: 10.1016/j.chaos.2007.06.044

Google Scholar

[7] M. Han, J. Li. Lower Bounds for the Hilbert Number of Polynomial Systems. Journal of Differential Equations. 252 (2012), pp.3278-3304.

DOI: 10.1016/j.jde.2011.11.024

Google Scholar

[8] E. Perdios, C. G. Zagouras, O. Ragos. Three-dimensional Bifurcations of Periodic Solutions Around the Triangular Equilibrium Points of the Restricted Three-body Problem. Celestial Mechanics & Dynamical Astronony. 51 (1991), pp.349-362.

DOI: 10.1007/bf00052927

Google Scholar

[9] Kenneth and R. Meyer. Continuation of Periodic Solutions in Three Dimensions. Physica D. 112 (1998), pp.310-318.

DOI: 10.1016/s0167-2789(97)00219-4

Google Scholar