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Multiplier of a periodic orbit of an iteration

For a period-

k

orbit

x_{0}, \dots, x_{k - 1}

of a differentiable map, the multiplier is

(F^{k})^{'} (x_{0}) = \prod_{j = 0}^{k - 1} F^{'} (x_{j}) .

The orbit is locally asymptotically stable when the modulus of this product is less than one.

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Solution