Article Contents
Article Contents

# Statistical and deterministic dynamics of maps with memory

• * Corresponding author: Paweł Góra
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.

Mathematics Subject Classification: Primary: 37A05; Secondary: 37A10, 37E05, 37E30.

 Citation:

• 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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