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

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B. Saussol, Absolutely continuous invariant measures for multidimensional expanding maps, Israel J. Math., 116 (2000), 223-248.  doi: 10.1007/BF02773219.  Google Scholar

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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

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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

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show all references

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

Figure 1.  Absolute values of the eigenvalues of the derivatives of $G_1$ (red) and $G_2$ (green) as functions of $\alpha$
Figure 2.  Examples of partitions for map $G$
Figure 3.  Partition into $A_1$ and $A_2$ for a) $\alpha=0.34$ and b) $\alpha=0.74$
Figure 4.  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 $\sim 0.3709557543$
Figure 5.  Singular values of $D_1D_2D_2$ or $D_2D_2D_2$
Figure 6.  First two images of $A_1$ for a) $\alpha=0.25290169942$ and b) $\alpha=0.320169942$
Figure 7.  a) Functions $cx,cy,cc$ in Proposition 9. b)Functions $cx+cc$ and $cx+cy+cc$ in Proposition 9
Figure 8.  Functions $cx, cy, cc$ and their sums in Proposition 10
Figure 9.  Region $G(A_2)\cap A_1$ and its image for a) $\alpha=0.29$ and b) $\alpha=0.34$
Figure 10.  Four first images of $G(A_2)\cap A_1$, $\alpha> 0.39$
Figure 11.  Further images of $G(G^3(B)\cap A_2)\cap A_1$ for a) $\alpha=0.391$ and b) $\alpha=0.394$
Figure 12.  Further images of $C_1=G(G^3(B)\cap A_2)\cap A_2$ (thick brown), for a) $\alpha=0.343$ and b) $\alpha=0.355$
Figure 13.  The image of $G^3(B)\cap A_2$ for a) $\alpha=0.415$ and b) $\alpha=0.432$
Figure 14.  Images of points which stayed for 6 steps in $A_2$
Figure 15.  When the sequence $D_1D_2^5D_1D_2^6$ becomes inadmissible
Figure 16.  Sequence $D_1D_2^6$ becomes inadmissible
Figure 17.  Images of $O_6$: a) 6 images for $\alpha=0.446$, b) 9 images for $\alpha=0.451$
Figure 18.  Support of acim for $\alpha=0.3$ and $\alpha=0.4$
Figure 19.  Support of acim for $\alpha=0.43$ and $\alpha=0.46$
Figure 20.  Support of conjectured acim for $\alpha=0.49$ and $\alpha=0.495$
Figure 21.  a: Support of conjectured acim for $\alpha=0.493$. b: Close-up of one of the clusters in part a
Figure 22.  Regions for $\alpha=3/4$
Figure 23.  Images $G(B_2)$ and $G(G(B_2))$, $\alpha=3/4$
Figure 24.  Images of a) the upper part and b) the lower part of $G(G(B_2))$
Figure 25.  Trapping region $T$ for $1/2 < \alpha\le \sim 0.593$. Case $\alpha=0.533$ is shown
Figure 26.  a)The graph of $z-t$ and b) of $y(z_i)-y_w$ for the proof of Proposition 23
Figure 27.  a) $T_3$ and its images, b) enlargement of $T_3$ and $G^3(T_3)$
Figure 28.  $\alpha =0.63$ (case ii)) a) Trapping region $T$ (red) and its image $G(T)$ (dashed black). b) Region $W$ and its images, $G^4(W)\subset T$
Figure 29.  $\alpha =0.594$ (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 $G^4(W)\not\subset T$
Figure 30.  $\alpha =0.69$ (case ⅲ)) a) Trapping region $T$ (red) and its image $G(T)$ (dashed black). b) Region $W$ and its images, $G^4(W)\subset T$
Figure 31.  $\alpha=0.734$ a) the trapping region $T$ (red) and its image $G(T)$ (dashed black). b) shows $W$ and its images with $G^4(W)\subset T$
Figure 32.  $\alpha=0.734$ 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 $G^4(W)$
Figure 33.  $\alpha=0743$ 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), $G^4(W)\subset T$
Figure 34.  $\alpha=0743$ a) Lower part of $G^3(W)$ and b) upper part of $G^4(W)$
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