We consider a dynamical system to have memory if it remembers the current state as well as the state before that. The dynamics is defined as follows: $x_{ n+1}=T_{ \alpha}(x_{ n-1},x_{ n})=\tau (\alpha \cdot x_{ n}+(1-\alpha)\cdot x_{ n-1}),$ where $\tau$ is a one-dimensional map on $I=[0,1]$ and $0<\alpha <1$ determines how much memory is being used. $T_{ \alpha}$ does not define a dynamical system since it maps $U=I\times I$ into $I$. In this note we let $\tau $ to be the symmetric tent map. We shall prove that for $0<\alpha <0.46,$ the orbits of $\{ x_{ n}\}$ are described statistically by an absolutely continuous invariant measure (acim) in two dimensions. As $\alpha $ approaches $0.5 $ from below, that is, as we approach a balance between the memory state and the present state, the support of the acims become thinner until at $\alpha =0.5$, all points have period 3 or eventually possess period 3. For $0.5<\alpha <0.75$, we have a global attractor: for all starting points in $U$ except $(0,0)$, the orbits are attracted to the fixed point $(2/3,2/3).$ At $\alpha=0.75,$ we have slightly more complicated periodic behavior.
Categories: math.DS