# 显式计算Z2对称系统的Hopf和Bautin分岔Explicit Computations of Hopf and Bautin Bifurcations in Z2-Symmetric Systems

DOI: 10.12677/AAM.2014.32009, PDF, HTML, XML, 下载: 2,578  浏览: 9,860  科研立项经费支持

Abstract: By using a homogical method, we drive out computational formulae for normal forms of the Hopf and Bautin bifurcations in Z2-symmetric systems. For practical bifurcation analysis of Hopf and Bautin in a Z2-symmetric system, we can use these formulae to compute the first and the second Lyapunov coefficients, and check whether the bifurcation is degenerate. Furthermore, we can use the formulae of unfolding parameters to decide the topological structures when parameters perturb in a neighborhood of the critical values. So, we construct the relation between the parameters and the structures for Hopf and Bautin bifurcations in any Z2-symmetric systems.

 [1] Guckenheimer, J. and Holmes, P.J. (1983) Nonlinear oscillations, dynamical systems and bifurcations of vector fields. Springer, New York. [2] Chow, S.N., Li, C.Z. and Wang, D. (1994) Normal forms and bifurcation of planar vector fields. Cambridge University Press, Cambridge. [3] Kuznetsov, Yu.A. (2004) Elements of applied bifurcation theory. 3rd Edition, Springer, New York. [4] Hassard, B., Kazarinoff, N. and Wan, Y.H. (1981) Theory and applications of Hopf bifurcation. Cambridge University Press, London. [5] Kuznetsov, Yu.A. (1999) Numerical normalization techniques for all codim 2 bifurcations of equilibria in ODEs. SIAM Journal on Numerical Analysis, 36, 1104-1124. [6] Shilnikov, L.P., Shilnikov, A.L., Turaev, D.V. and Chua, L. (2001) Methods of qualitative theory in nonlinear dynam- ics. World Scientific, Singapore. [7] Peng, G.J. and Jiang, Y.L. (2013) Computation of universal unfolding of the double-zero bifurcation in -symmetric systems by a homological method. Journal of Difference Equations and Applications, 19, 1501-1512. [8] Freire, E., Rodríguez-Luis, A.J., Gamero, E. and Ponce, E. (1993) A case study for homoclinic chaos in an autonomous electronic circuit. A trip from Takens-Bogdanov to Hopf-Šil’nikov. Physica D, 62, 230-253.