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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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