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: $x_{n+1}=T_{\alpha}(x_{n-1},x_{n})=\tau (\alpha \cdot x_{n}+(1-\alpha)\cdot x_{n-1}),$where$\tau$isaone-dimensionalmapon$I=[0,1]$and$0<\alpha <1$determineshowmuchmemoryisbeingused\mathrm{.}$T_{\alpha}$ 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 $\alpha =0.5$,allpointshaveperiod3oreventuallypossessperiod3.For$0.5<\alpha <0.75$,wehaveaglobalattractor:forallstartingpointsin$U$except$(0,0)$,theorbitsareattractedtothefixedpoint$(2/3,2/3).$At$\alpha=0.75,$ we have slightly more complicated periodic behavior.