Sojourn times in laminar states of deterministic dynamical systems with intermittency
The author considers a kind of chaotic deterministic dynamical system, which is relevant from the stochastic viewpoint. The dynamical systems considered in this note are the iterates \(T^k\) of a mapping \(T\) from \([0,1]\) onto itself, that is \[ T(x)= \begin{cases} x+4x^2\quad & \text{for }x\in [0, \textstyle {1\over 2}],\\ x-4(1-x^2)\quad & \text{for }x\in [\textstyle {1\over 2},1]. \end{cases} \] The author investigates the behaviour of the trajectory and shows that the trajectory \(\{T^k(x)\}^n_{k=0}\) belongs to laminar states \(A\) and \(B\) for an extremely long time. The author also studies the limit of the ratio of the sojourn time of the trajectory in \(A\) to that in \(B\). The peculiarity of the dynamical systems under consideration is that they admit ergodic invariant measures.
scientific article; zbMATH DE number 1289185