Compute and graph the Poincare map with \(t{\bf{ = }}2\pi n,n{\bf{ = }}0,1, \ldots ,20\) for equation\(\left( 4 \right)\), taking \(A{\bf{ = F = }}1,\phi {\bf{ = }}0,\omega {\bf{ = }}\frac{1}{3}\) and\({\bf{b}} = {\bf{0}}.22\). Describe the attractor for this system.

For \(A{\bf{ = F = }}1,\phi {\bf{ = }}0,\omega {\bf{ = }}\frac{1}{3},{\bf{b}} = {\bf{0}}.22\) the Poincare maps are given by:

\(\begin{aligned}{c}{x_n}{\bf{ = }}{{\bf{e}}^{{\bf{ - }}\frac{{11n\pi }}{{50}}}}\sin \left( {\frac{{\sqrt {8911} }}{{150}}n\pi } \right){\bf{ + }}\frac{{450}}{{\sqrt {169801} }}\sin \left( {2n\pi - \frac{{400}}{{99}}} \right)\\{v_n}{\bf{ = }}\frac{{{e^{{\bf{ - }}\frac{{11n\pi }}{{50}}}}}}{{300}}\left( {\sqrt {8911} \cos \left( {\frac{{\sqrt {8911} }}{{150}}n\pi } \right){\bf{ - }}33\sin \left( {\frac{{\sqrt {8911} }}{{150}}n\pi } \right)} \right){\bf{ + }}\frac{{450}}{{\sqrt {169801} }}\cos \left( {2n\pi {\bf{ - }}\frac{{400}}{{99}}} \right)\end{aligned}\)

One will compute the points \(\left( {{x_n},{v_n}} \right)\) for\({\bf{n = }}\overline {{\bf{0,20}}} \). The values for\({x_n},{v_n}\) are given in the table below.

\(\begin{aligned}{*{20}{l}}{{{\bf{x}}_{\bf{n}}}}&{{{\bf{v}}_{\bf{n}}}}\\{{\bf{ - 1}}{\bf{.06006}}}&{{\bf{0}}{\bf{.577026}}}\\{{\bf{ - 0}}{\bf{.599846}}}&{{\bf{0}}{\bf{.149443}}}\\{{\bf{ - 1}}{\bf{.2423}}}&{{\bf{0}}{\bf{.228101}}}\\{{\bf{ - 1}}{\bf{.10342}}}&{{\bf{0}}{\bf{.304278}}}\\{{\bf{ - 0}}{\bf{.997156}}}&{{\bf{0}}{\bf{.25437}}}\\{{\bf{ - 1}}{\bf{.07409}}}&{{\bf{0}}{\bf{.255012}}}\\{{\bf{ - 1}}{\bf{.0703}}}&{{\bf{0}}{\bf{.267285}}}\\{{\bf{ - 1}}{\bf{.05249}}}&{{\bf{0}}{\bf{.262264}}}\\{{\bf{ - 1}}{\bf{.06049}}}&{{\bf{0}}{\bf{.261172}}}\\{{\bf{ - 1}}{\bf{.0618}}}&{{\bf{0}}{\bf{.262864}}}\\{{\bf{ - 1}}{\bf{.05927}}}&{{\bf{0}}{\bf{.262468}}}\\{{\bf{ - 1}}{\bf{.05994}}}&{{\bf{0}}{\bf{.2622}}}\\{{\bf{ - 1}}{\bf{.06031}}}&{{\bf{0}}{\bf{.262406}}}\\{{\bf{ - 1}}{\bf{.06}}}&{{\bf{0}}{\bf{.262392}}}\\{{\bf{ - 1}}{\bf{.06003}}}&{{\bf{0}}{\bf{.262346}}}\\{{\bf{ - 1}}{\bf{.0601}}}&{{\bf{0}}{\bf{.262367}}}\\{{\bf{ - 1}}{\bf{.06006}}}&{{\bf{0}}{\bf{.26237}}}\\{{\bf{ - 1}}{\bf{.06006}}}&{{\bf{0}}{\bf{.262364}}}\\{{\bf{ - 1}}{\bf{.06007}}}&{{\bf{0}}{\bf{.262366}}}\\{{\bf{ - 1}}{\bf{.06006}}}&{{\bf{0}}{\bf{.262367}}}\\{{\bf{ - 1}}{\bf{.06006}}}&{{\bf{0}}{\bf{.262366}}}\end{aligned}\)

The Poincare section is given in the figure below.

\(\begin{aligned}{c}\mathop {\lim }\limits_{n \to \infty } {x_n}{\bf{ = - 1}}{\bf{.060}}\\\mathop {\lim }\limits_{n \to \infty } {v_n}{\bf{ = 0}}{\bf{.2624}}\end{aligned}\)

So, the Poincare section converges to the point \(\left( {{\bf{ - 1}}.0604,0.2624} \right){\bf{ - }}\) red point, and that point is the attractor for this system.