# American Institute of Mathematical Sciences

August  2017, 37(8): 4347-4378. doi: 10.3934/dcds.2017186

## Statistical and deterministic dynamics of maps with memory

 1 Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Blvd. West, Montreal, Quebec H3G 1M8, Canada 2 Department of Mathematics, Honghe University, Mengzi, Yunnan 661100, China 3 Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Blvd. West, Montreal, Quebec H3G 1M8, Canada

* Corresponding author: Paweł Góra

Received  April 2016 Revised  May 2017 Published  April 2017

Fund Project: The research of the authors was supported by NSERC grants. The research of Z. Li was also supported by NNSF of China (No. 11601136) and Doctor/Master grant at Honghe University (No. XJ16B07)

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 $\tau$ 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 $x_{n-1}$ and the present state $x_{n}$, 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.

Citation: Paweł Góra, Abraham Boyarsky, Zhenyang LI, Harald Proppe. Statistical and deterministic dynamics of maps with memory. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4347-4378. doi: 10.3934/dcds.2017186
##### References:
 [1] P. Góra, A. Boyarsky and Z. Li, Singular SRB measures for a non 1-1 map of the unit square, Journal of Stat. Physics, 165 (2016), 409-433, available at http://arxiv.org/abs/1607. 01658, full-text view-only version: http://rdcu.be/kod0 doi: 10.1007/s10955-016-1620-y. Google Scholar [2] F. Dyson, Birds and Frogs, Notices of Amer. Math. Soc., 56 (2009), 212-223. Google Scholar [3] J. Maynard Smith, Mathematical Ideas in Biology, Cambridge University Press, 1968. doi: 10.1017/CBO9780511565144. Google Scholar [4] B. Saussol, Absolutely continuous invariant measures for multidimensional expanding maps, Israel J. Math., 116 (2000), 223-248. doi: 10.1007/BF02773219. Google Scholar [5] M. Tsujii, Absolutely continuous invariant measures for piecewise real-analytic expanding maps on the plane, Commun. Math Phys., 208 (2000), 605-622. doi: 10.1007/s002200050003. Google Scholar [6] G. -C. Wu and D. Baleanu, Discrete chaos in fractional delayed logistic maps, Nonlinear Dynam., 80 (2015), 1697-1703. doi: 10.1007/s11071-014-1250-3. Google Scholar [7] L. Zou, A lower bound for the smallest singular value, J. Math. Inequal., 6 (2012), 625-629. doi: 10.7153/jmi-06-60. Google Scholar

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##### References:
 [1] P. Góra, A. Boyarsky and Z. Li, Singular SRB measures for a non 1-1 map of the unit square, Journal of Stat. Physics, 165 (2016), 409-433, available at http://arxiv.org/abs/1607. 01658, full-text view-only version: http://rdcu.be/kod0 doi: 10.1007/s10955-016-1620-y. Google Scholar [2] F. Dyson, Birds and Frogs, Notices of Amer. Math. Soc., 56 (2009), 212-223. Google Scholar [3] J. Maynard Smith, Mathematical Ideas in Biology, Cambridge University Press, 1968. doi: 10.1017/CBO9780511565144. Google Scholar [4] B. Saussol, Absolutely continuous invariant measures for multidimensional expanding maps, Israel J. Math., 116 (2000), 223-248. doi: 10.1007/BF02773219. Google Scholar [5] M. Tsujii, Absolutely continuous invariant measures for piecewise real-analytic expanding maps on the plane, Commun. Math Phys., 208 (2000), 605-622. doi: 10.1007/s002200050003. Google Scholar [6] G. -C. Wu and D. Baleanu, Discrete chaos in fractional delayed logistic maps, Nonlinear Dynam., 80 (2015), 1697-1703. doi: 10.1007/s11071-014-1250-3. Google Scholar [7] L. Zou, A lower bound for the smallest singular value, J. Math. Inequal., 6 (2012), 625-629. doi: 10.7153/jmi-06-60. Google Scholar
Absolute values of the eigenvalues of the derivatives of $G_1$ (red) and $G_2$ (green) as functions of $\alpha$
Examples of partitions for map $G$
Partition into $A_1$ and $A_2$ for a) $\alpha=0.34$ and b) $\alpha=0.74$
a) Singular values for matrices $D_2D_1$ and $D_1D_1$. The lower curve intersects level 1 at $\alpha_1\sim 0.24760367$. b) Singular values for matrices $D_2D_2$ and $D_1D_2$. The lower curve intersects level 1 at $\sim 0.3709557543$
Singular values of $D_1D_2D_2$ or $D_2D_2D_2$
First two images of $A_1$ for a) $\alpha=0.25290169942$ and b) $\alpha=0.320169942$
a) Functions $cx,cy,cc$ in Proposition 9. b)Functions $cx+cc$ and $cx+cy+cc$ in Proposition 9
Functions $cx, cy, cc$ and their sums in Proposition 10
Region $G(A_2)\cap A_1$ and its image for a) $\alpha=0.29$ and b) $\alpha=0.34$
Four first images of $G(A_2)\cap A_1$, $\alpha> 0.39$
Further images of $G(G^3(B)\cap A_2)\cap A_1$ for a) $\alpha=0.391$ and b) $\alpha=0.394$
Further images of $C_1=G(G^3(B)\cap A_2)\cap A_2$ (thick brown), for a) $\alpha=0.343$ and b) $\alpha=0.355$
The image of $G^3(B)\cap A_2$ for a) $\alpha=0.415$ and b) $\alpha=0.432$
Images of points which stayed for 6 steps in $A_2$
When the sequence $D_1D_2^5D_1D_2^6$ becomes inadmissible
Sequence $D_1D_2^6$ becomes inadmissible
Images of $O_6$: a) 6 images for $\alpha=0.446$, b) 9 images for $\alpha=0.451$
Support of acim for $\alpha=0.3$ and $\alpha=0.4$
Support of acim for $\alpha=0.43$ and $\alpha=0.46$
Support of conjectured acim for $\alpha=0.49$ and $\alpha=0.495$
a: Support of conjectured acim for $\alpha=0.493$. b: Close-up of one of the clusters in part a
Regions for $\alpha=3/4$
Images $G(B_2)$ and $G(G(B_2))$, $\alpha=3/4$
Images of a) the upper part and b) the lower part of $G(G(B_2))$
Trapping region $T$ for $1/2 < \alpha\le \sim 0.593$. Case $\alpha=0.533$ is shown
a)The graph of $z-t$ and b) of $y(z_i)-y_w$ for the proof of Proposition 23
a) $T_3$ and its images, b) enlargement of $T_3$ and $G^3(T_3)$
$\alpha =0.63$ (case ii)) a) Trapping region $T$ (red) and its image $G(T)$ (dashed black). b) Region $W$ and its images, $G^4(W)\subset T$
$\alpha =0.594$ (case i)) a) Region $W$ and its images in green except for $G^3(W)$ in magenta, $G^5(W)\subset T$. b) Enlargement of the intersection of $W$ and $G^3(W)$ which causes $G^4(W)\not\subset T$
$\alpha =0.69$ (case ⅲ)) a) Trapping region $T$ (red) and its image $G(T)$ (dashed black). b) Region $W$ and its images, $G^4(W)\subset T$
$\alpha=0.734$ a) the trapping region $T$ (red) and its image $G(T)$ (dashed black). b) shows $W$ and its images with $G^4(W)\subset T$
$\alpha=0.734$ a)the old trapping region of Proposition 24 and the points $G(p_4)$, $G^2(p_4)$, $G^3(p_4)$. b) enlarged $T$, $G^3(W)$ and $G^4(W)$
$\alpha=0743$ a) Trapping region $T$ (red) and its image $G(T)$ (dashed black). The dashed red line is an eigenline going through $X_0$. b) Region $W$ and its images (green), $G^4(W)\subset T$
$\alpha=0743$ a) Lower part of $G^3(W)$ and b) upper part of $G^4(W)$
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