This a subsequent question to the one I asked regarding linear flows: How to show that a finite dimensional linear system of ODEs cannot be chaotic?
Consider the system $$x_{n+1}=F_x(x_n,y_n,z_n),$$ $$y_{n+1}=F_y(x_n,y_n,z_n),$$ $$z_{n+1}=F_z(x_n,y_n,z_n)$$ where $F_x,F_y,F_z$ are linear in $x_n,y_n,z_n$. For example, the system $$x_{n+1}=5x_n+y_n+z_n,$$ $$y_{n+1}=2x_n-y_n,$$ $$z_{n+1}=-3z_n+x_n-y_n.$$
Can such a system be chaotic? That is, is the case for such a system of linear maps analogous to the case of linear flows?