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Numerical analysis of the flip bifurcation of maps. (English) Zbl 0729.65050

Discrete dynamical systems depending on a paramter α are considered: x(t+1)=f α (t)· It is assumed that an n×n matrix A α and a smooth map g α with f α (x)=A α x+g α (x) and g α (0)=0, g/x| α=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 α is supposed to cross the unit circle for α=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:
65K10Optimization techniques (numerical methods)
93C55Discrete-time control systems