Transition curves and bifurcations of a class of fractional mathieu-type equations

Research output: Journal Publications and Reviews (RGC: 21, 22, 62)21_Publication in refereed journal

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  • A. Y T Leung
  • Zhongjin Guo
  • H. X. Yang


Original languageEnglish
Article number1250275
Journal / PublicationInternational Journal of Bifurcation and Chaos
Issue number11
Publication statusPublished - Nov 2012


A general version of the fractional Mathieu equation and the corresponding fractional Mathieu-Duffing equation are established for the first time and investigated via the harmonic balance method. The approximate expressions for the transition curves separating the regions of stability are derived. It is shown that a change in the fractional derivative order remarkably affects the shape and location of the transition curves in the n = 1 tongue. However, the shape of the transition curve does not change very much for different fractional orders for the n = 0 tongue. The steady state approximate responses of the corresponding fractional Mathieu-Duffing equation are obtained by means of harmonic balance, polynomial homotopy continuation and technique of linearization. The curves with respect to fractional order versus response amplitude, driving amplitude versus response amplitude with different fractional orders are shown. It can be found that the bifurcation point and stability of branch solutions is different under different fractional orders of system. When the fractional order increases to some value, the symmetric breaking, saddle-node bifurcation as well as period-doubling bifurcation phenomena are found and exhibited analytically by taking the driving amplitude as the bifurcation parameter. © 2012 World Scientific Publishing Company.

Research Area(s)

  • bifurcation, Fractional Mathieu-type equation, harmonic balance method, polynomial homotopy continuation, transition curves