A Proof for the Existence of Chaos in Diffusively Coupled Map Lattices with Open Boundary Conditions

<table>The control result of the DCML (<a href="https://static.hindawi.com/articles/ddns/volume-2011/174376/figures/#EEq3.4">3.18</a>), with the parameter values <svg height="10.9125" id="M369" style="vertical-align:-0.17555pt" version="1.1" viewbox="0 0 46.900002 10.9125" width="46.900002" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,10.9125)">
<g transform="translate(72,-63.27)">
<text transform="matrix(1,0,0,-1,-71.95,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑎</tspan>
<tspan style="font-size: 12.50px; " x="9.7398367" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="21.780226" y="0">1</tspan>
<tspan style="font-size: 12.50px; " x="28.031727" y="0">.</tspan>
<tspan style="font-size: 12.50px; " x="31.157476" y="0">8</tspan>
</text>
</g>
</g>
</svg>, <svg height="10.9125" id="M370" style="vertical-align:-0.17555pt" version="1.1" viewbox="0 0 54.212502 10.9125" width="54.212502" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,10.9125)">
<g transform="translate(72,-63.27)">
<text transform="matrix(1,0,0,-1,-71.95,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝜖</tspan>
<tspan style="font-size: 12.50px; " x="9.3397408" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="21.380131" y="0">0</tspan>
<tspan style="font-size: 12.50px; " x="27.63163" y="0">.</tspan>
<tspan style="font-size: 12.50px; " x="30.75738" y="0">0</tspan>
<tspan style="font-size: 12.50px; " x="37.008881" y="0">1</tspan>
</text>
</g>
</g>
</svg>, and <svg height="14.7875" id="M371" style="vertical-align:-3.13504pt" version="1.1" viewbox="0 0 61.775002 14.7875" width="61.775002" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,14.7875)">
<g transform="translate(72,-60.17)">
<text transform="matrix(1,0,0,-1,-71.95,63.35)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝛼</tspan>
</text>
<text transform="matrix(1,0,0,-1,-64.9,60.22)">
<tspan style="font-size: 8.75px; " x="0" y="0">1</tspan>
</text>
<text transform="matrix(1,0,0,-1,-56.55,63.35)">
<tspan style="font-size: 12.50px; " x="0" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="12.027886" y="0">0</tspan>
<tspan style="font-size: 12.50px; " x="18.279387" y="0">.</tspan>
<tspan style="font-size: 12.50px; " x="21.405136" y="0">8</tspan>
<tspan style="font-size: 12.50px; " x="27.656635" y="0">5</tspan>
</text>
</g>
</g>
</svg>. The feedback control starts at the 71st iteration.</table>

Discrete Dynamics in Nature and Society

fig6

Figure 6

Figure 6: A Proof for the Existence of Chaos in Diffusively Coupled Map Lattices with Open Boundary Conditions 