Examining the Chaotic Behavior in Dynamical Systems by Means of the 0-1 Test

<table>Poincaré surface of section at <svg height="10.9125" id="M104" style="vertical-align:-0.17555pt" version="1.1" viewbox="0 0 58.025002 10.9125" width="58.025002" 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">𝐸</tspan>
<tspan style="font-size: 12.50px; " x="12.390473" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="24.430862" y="0">0</tspan>
<tspan style="font-size: 12.50px; " x="30.682362" y="0">.</tspan>
<tspan style="font-size: 12.50px; " x="33.808113" y="0">2</tspan>
<tspan style="font-size: 12.50px; " x="40.059612" y="0">1</tspan>
</text>
</g>
</g>
</svg> (in fifth order Hamiltonian). We can distinguish the four families (I, II, III, and IV) plus an unstable invariant point <svg height="13.45" id="M105" style="vertical-align:-2.21957pt" version="1.1" viewbox="0 0 21.549999 13.45" width="21.549999" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,13.45)">
<g transform="translate(72,-61.24)">
<text transform="matrix(1,0,0,-1,-71.95,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">(</tspan>
<tspan style="font-size: 12.50px; " x="4.1634989" y="0">𝑈</tspan>
<tspan style="font-size: 12.50px; " x="12.978114" y="0">)</tspan>
</text>
</g>
</g>
</svg>, which characterizes the case as weakly chaotic.</table>

Journal of Applied Mathematics

fig3

Figure 3

Figure 3: Examining the Chaotic Behavior in Dynamical Systems by Means of the 0-1 Test 