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Statistical and Deterministic Dynamics of Maps with Memory

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: $x_{n+1}=T_α(x_{n-1},x_{n})=τ(α\cdot x_{n}+(1-α)\cdot x_{n-1}),$ where $τ$ is a one-dimensional map on $I=[0,1]$ and $0<α<1$ determines how much memory is being used. $T_α$ does not define a dynamical system since it maps $U=I\times I$ into $I$. In this note we let $τ$ to be the symmetric tent map. We shall prove that for $0<α<0.46,$ the orbits of $\{x_{n}\}$ are described statistically by an absolutely continuous invariant measure (acim) in two dimensions. As $α$ 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 $α=0.5$, all points have period 3 or eventually possess period 3. For $0.5<α<0.75$, 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 $α=0.75,$ we have slightly more complicated periodic behavior.