zbMATH — the first resource for mathematics

Numerical analysis of the flip bifurcation of maps. (English) Zbl 0729.65050
Discrete dynamical systems depending on a paramter \(\alpha\) are considered: \(x(t+1)=f_{\alpha}(t).\) It is assumed that an \(n\times n\) matrix \(A_{\alpha}\) and a smooth map \(g_{\alpha}\) with \(f_{\alpha}(x)=A_{\alpha}x+g_{\alpha}(x)\) and \(g_{\alpha}(0)=0\), \(\partial g/\partial x|_{\alpha =0}=0\) exists. Problems of this type are of interest in connection with limit cycles in autonomous systems and period doubling of periodic solutions of time periodic systems.
One eigenvalue of \(A_{\alpha}\) is supposed to cross the unit circle for \(\alpha =0\) with nonzero velocity. Under these assumptions ”flip bifurcation” takes place. The stability properties can be analyzed by investigating the “center manifold” described by a series expansion. A procedure for computing the relevant coefficient is presented.
Reviewer: R.Tracht (Essen)

65K10 Numerical optimization and variational techniques
93C55 Discrete-time control/observation systems
Full Text: DOI
[1] Arnold, V.I., Geometrical methods in the theory of ordinary differential equations, (1982), Springer-Verlag New York
[2] Arnold, V.I.; Afraimovich, V.S.; Il’yashenko, Yu.S.; Shil’nikov, L.P., Bifurcation theory (in Russian), () · Zbl 1038.37500
[3] Carr, J., Applications of the center manifold theory, (1981), Springer-Verlag New York
[4] Guckenheimer, J.; Holmes, Ph., Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, (1986), Springer-Verlag New York
[5] Hassard, B.D.; Kazarinoff, N.D.; Wan, Y.-H., Theory and applications of the Hopf bifurcation, (1980), Cambridge U.P Cambridge · Zbl 0474.34002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.