Effects of phase space discretization on the long-time behavior of dynamical systems
The effects of the phase space discretizations and of the roundoff errors on the period, L, of the loops of the iterates \(f^ n(x_ 0)=f(f^{n- 1}(x_ 0))\), \(n\geq 1\), of a given function f(.): I\(=[-1,1]\to R\), \(x_ 0\in I\), are investigated through a large number of computer experiments with functions of the form: \(f_{\mu}(x)=1-\mu | x|^ z\). Dividing the interval \(I=[-1,1]\) into N cells of size \(\Delta\) and plotting log L against log N for \(\Delta\to 0\) the authors conclude that the period L of the loops of the iterates \(f^ n(x_ 0)\), \(n\geq 1\), \(x_ 0\in I\), satisfies a relation of the form: \(L\leq N^{\epsilon}\) where \(\epsilon\) is empirically guessed. For ergodic discrete dynamical systems represented by the system defined by the function \(f(x)=1-2x^ 2\) as well as for ''critical systems'' defined by functions of the form: \(f(x)=1-\mu_ cx^ z\) where \(\mu_ c=\mu_ c(z)\) is a critical value, some theoretical arguments to explain the numerical results are given.
scientific article; zbMATH DE number 4001353