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Statistical and Deterministic Dynamics of Maps with Memory

Title:

Statistical and Deterministic Dynamics of Maps with Memory

Gora, Pawel, Boyarsky, Abraham, Li, Zhenyang and Proppe, Harald (2017) Statistical and Deterministic Dynamics of Maps with Memory. Discrete and Continuous Dynamical System - A, 37 (8). pp. 4347-4378. ISSN ISSN: 1078-0947

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Official URL: http://dx.doi.org/10.3934/dcds.2017186

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.

Divisions:Concordia University > Faculty of Arts and Science > Mathematics and Statistics
Item Type:Article
Refereed:Yes
Authors:Gora, Pawel and Boyarsky, Abraham and Li, Zhenyang and Proppe, Harald
Journal or Publication:Discrete and Continuous Dynamical System - A
Date:August 2017
Digital Object Identifier (DOI):10.3934/dcds.2017186
ID Code:985780
Deposited By: PAWEL GORA
Deposited On:03 Sep 2019 17:00
Last Modified:04 Sep 2019 00:00
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