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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)

MSC:
65K10 Numerical optimization and variational techniques
93C55 Discrete-time control/observation systems
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References:
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