TY - JOUR
T1 - Transition curves and bifurcations of a class of fractional mathieu-type equations
AU - Leung, A. Y T
AU - Guo, Zhongjin
AU - Yang, H. X.
PY - 2012/11
Y1 - 2012/11
N2 - 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.
AB - 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.
KW - bifurcation
KW - Fractional Mathieu-type equation
KW - harmonic balance method
KW - polynomial homotopy continuation
KW - transition curves
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U2 - 10.1142/S0218127412502756
DO - 10.1142/S0218127412502756
M3 - 21_Publication in refereed journal
VL - 22
JO - International Journal of Bifurcation and Chaos in Applied Sciences and Engineering
JF - International Journal of Bifurcation and Chaos in Applied Sciences and Engineering
SN - 0218-1274
IS - 11
M1 - 1250275
ER -