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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: $x(t+1)=f\sb{\alpha}(t).$ It is assumed that an $n\times n$ matrix $A\sb{\alpha}$ and a smooth map $g\sb{\alpha}$ with $f\sb{\alpha}(x)=A\sb{\alpha}x+g\sb{\alpha}(x)$ and $g\sb{\alpha}(0)=0$, $\partial g/\partial x\vert\sb{\alpha =0}=0$ 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 $A\sb{\alpha}$ 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)

65K10Optimization techniques (numerical methods)
93C55Discrete-time control systems
Full Text: DOI
[1] Arnold, V. I.: Geometrical methods in the theory of ordinary differential equations. (1982)
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[3] Carr, J.: Applications of the center manifold theory. (1981) · Zbl 0464.58001
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