A piecewise-linear invertible noodle map
The smooth noodle map is given by \[ (1)\quad x_{n+1}=\alpha x_ n(x^ 2_ n-1)-y_ n-z_ n,\quad y_{n+1}=\beta x_ n,\quad z_{n+1}=\gamma (x^ 2_ n-0.33)+\delta z_ n. \] The above map is of interest for its occurrence in realistic systems. But such map is not easily amenable to analysis. So the map which is amenable to analysis is the following piecewise-linear analogue \[ (2)\quad x_{n+1}=3.3(x_ n- 1)-2-y_ n-z_ n,\quad if\quad x_ n>1, \] \[ x_{n+1}=-2x_ n-y_ n-z_ n,\quad if\quad 1\geq x_ n\geq -1, \] \[ x_{n+1}=3.3(x_ n+1)+2-y_ n-z_ n,\quad if\quad x_ n<-1, \] \[ y_{n+1}=\beta x_ n,\quad z_{n+1}=\gamma (| x_ n| -1)+\delta z_ n. \] The maps (1) and (2) are invertible. The qualitative behaviour of the map (2) is discussed.
scientific article; zbMATH DE number 3956055
scientific article