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Zbl 0729.65050
Kuznetsov, Yu.A.; Rinaldi, S.
Numerical analysis of the flip bifurcation of maps. (English)
[J] Appl. Math. Comput. 43, No.3, 231-236 (1991). ISSN 0096-3003

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. \par 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.
[R.Tracht (Essen)]

MSC 2000:: *65K10 Optimization techniques (numerical methods)
93C55 Discrete-time control systems

Keywords: Discrete dynamical systems; limit cycles; autonomous systems; period doubling; periodic solutions; time periodic systems; eigenvalue; flip bifurcation; stability; center manifold

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