Statistical and Deterministic Dynamics of Maps with Memory

TL;DR

This paper investigates the dynamics of a memory-influenced map combining statistical and deterministic analysis, revealing how the system's behavior transitions from statistical measures to periodic and fixed points as the memory parameter varies.

Contribution

It introduces a novel analysis of maps with memory, specifically examining the transition from statistical to periodic behavior as the memory parameter changes.

Findings

For small memory parameter, orbits are described by an absolutely continuous invariant measure.

At a critical value, the support of the measure becomes thinner, leading to period-3 behavior.

Beyond a certain point, the system converges to a fixed point, with complex behavior at the boundary.

Abstract

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.