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**Computation of the simplest normal form of a resonant double Hopf bifurcation system with the complex normal form method.**
*(English)*
Zbl 1176.70026

Summary: The complex normal form method is presented to find the simplest normal form (SNF) of an odd ratio double Hopf bifurcation system. New nonlinear transformations and rules of getting the form of the SNF coefficients are presented to find an efficient approach to get the final explicit expressions of the SNF. During the course of obtaining the key equations, based on the algorithm of complex operations, all the matrix deduction is substituted by complex operation and no former matrix operation is involved compared with the matrix representation method. Explicit iterative formulas have been derived to determine the coefficients of the SNF and associated nonlinear transformations up to an arbitrary order. Meanwhile, a rule of analyzing the general expression of the SNF is also summarized, which enables to predict the components of the final explicit equations before any substantive analyses. As the result of these innovations is deduced, the computation process is basically simplified. An ordinary differential equation (ODE) system shows the feasibility and convenience of the new method.

### MSC:

70K45 | Normal forms for nonlinear problems in mechanics |

70K50 | Bifurcations and instability for nonlinear problems in mechanics |

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\textit{W. Wang} and \textit{Q. Zhang}, Nonlinear Dyn. 57, No. 1--2, 219--229 (2009; Zbl 1176.70026)

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### References:

[1] | Cheng, Z., Cao, J.D.: Bifurcation and stability analysis of a neural network model with distributed delays. Nonlinear Dyn. 46(4), 363–373 (2006) · Zbl 1169.92001 |

[2] | Nayfeh, A.H.: Method of Normal Forms. Wiley, New York (1993) · Zbl 1220.37002 |

[3] | Yu, P., Bi, Q.: Symbolic computation of normal forms for semi-simple cases. Nonlinear Dyn. 27(1), 19–53 (2002) · Zbl 0994.65140 |

[4] | Leung, A.Y.T., Zhang, Q.C.: Normal form computation without central manifold reduction. J. Sound Vib. 266(2), 261–279 (2003) |

[5] | Ushiki, S.: Normal form for singularities of vector fields. Jpn. J. Ind. Appl. Math. 1, 1–34 (1984) · Zbl 0578.58029 |

[6] | Wang, D.: An introduction to the normal form theory of ordinary differential equations. Adv. Math. 19(1), 38–71 (1990) · Zbl 0709.34012 |

[7] | Yu, P.: Simplest normal forms of Hopf and generalized Hopf bifurcations. Int. J. Bifurc. Chaos 9(10), 1917–1939 (1999) · Zbl 1089.37528 |

[8] | Yu, P.: Computation of the simplest normal forms with perturbation parameters based on Lie transform and rescaling. J. Comput. Appl. Math. 144(1–2), 359–373 (2002) · Zbl 1019.65043 |

[9] | Zhang, Q., He, X., Hu, L.: Computation of the simplest normal form from a bifurcation system with parameters. J. Vib. Eng. 18(4), 495–499 (2005) |

[10] | Zhang, Q., Hu, S., Wang, W.: Computation of the simplest normal form of semi-simple system without central manifold reduction. Tianjin Daxue Xuebao 39(7), 773–776 (2006) |

[11] | Xu, J.: Co-dimension 2 bifurcations and chaos in cantilevered pipe conveying time-varying fluid with three-to-one internal resonances. Acta Mech. Sinica 16(3), 245–255 (2003) |

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