Pathwise Properties of Random Quadratic Mapping

Abstract

In this thesis, we consider the random dynamical system from a sequence of random quadratic mapping f k (x) = k x(1 − x), where k can choose µ or λ randomly, where 1 < µ < λ 1 + √ 5. That means we consider, where { k : k 1} is a sequence with k = µ or λ and X 0 ∈ [0, 1]. As to this random dynamical system, we prove the existence of the stationary solution when 1 < µ < λ 3 and the existence of random periodic solution of period 2 for 2i = 2i+1 (i ∈ Z) when 3.00547 µ < λ 1 + √ 5.