Q3E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

Compute and graph the points of the Poincare map with\({\bf{t = 2\pi n,n = 0,1, \ldots ,20}}\)for equation (4), taking \({\bf{A = F = 1,f = 0,\omega  = 1,}}\) and\({\bf{b =  - 0}}{\bf{.1}}\). What is happening to these points as\({\bf{n}} \to \infty \)?

Step-by-Step Solution

Answer

The points are unbounded when\({\bf{n}} \to \infty \).

Step 1: Defining the Poincare maps

For \({\bf{A = F = 1,f = 0,\omega  = 1,b =   - 0}}{\bf{.1}}\) the Poincare maps are given by:

 

\(\begin{aligned}{c}{{\bf{x}}_{\bf{n}}}{\bf{ = }}{{\bf{e}}^{{\bf{n\pi /10}}}}{\bf{sin}}\left( {\frac{{\sqrt {{\bf{399}}} }}{{{\bf{10}}}}{\bf{n\pi }}} \right){\bf{ + 10sin(2n\pi ), }}\\{{\bf{v}}_{\bf{n}}}{\bf{ = }}\frac{{\bf{1}}}{{{\bf{20}}}}{{\bf{e}}^{{\bf{n\pi /10}}}}\left( {\sqrt {{\bf{399}}} {\bf{cos}}\left( {\frac{{\sqrt {{\bf{399}}} }}{{{\bf{10}}}}{\bf{n\pi }}} \right){\bf{ + sin}}\left( {\frac{{\sqrt {{\bf{399}}} }}{{{\bf{10}}}}{\bf{n\pi }}} \right)} \right){\bf{ + 10cos(2n\pi )}}\end{aligned}\)

 

One will compute the points \(\left( {{{\bf{x}}_{\bf{n}}}{\bf{,}}{{\bf{v}}_{\bf{n}}}} \right)\) for \({\bf{n = }}\overline {{\bf{0,20}}} {\bf{.}}\)

Step 2: Computing \({{\bf{x}}_{\bf{n}}}\) and \({{\bf{v}}_{\bf{n}}}\)

The values for \({{\bf{x}}_{\bf{n}}}{\bf{, }}{{\bf{v}}_{\bf{n}}}\) are given in the table below.

\(\begin{aligned}{*{20}{l}}{{{\bf{x}}_{\bf{n}}}}&{{{\bf{v}}_{\bf{n}}}}\\{\bf{0}}&{{\bf{10}}{\bf{.9987}}}\\{{\bf{ - 0}}{\bf{.0107596}}}&{{\bf{11}}{\bf{.3668}}}\\{{\bf{ - 0}}{\bf{.0294611}}}&{{\bf{11}}{\bf{.8704}}}\\{{\bf{ - 0}}{\bf{.0605}}}&{{\bf{12}}{\bf{.5594}}}\\{{\bf{ - 0}}{\bf{.110433}}}&{{\bf{13}}{\bf{.5019}}}\\{{\bf{ - 0}}{\bf{.188977}}}&{{\bf{14}}{\bf{.7913}}}\\{{\bf{ - 0}}{\bf{.31044}}}&{{\bf{16}}{\bf{.555}}}\\{{\bf{ - 0}}{\bf{.495797}}}&{{\bf{18}}{\bf{.9673}}}\\{{\bf{ - 0}}{\bf{.775651}}}&{{\bf{22}}{\bf{.2667}}}\\{{\bf{ - 1}}{\bf{.19448}}}&{{\bf{26}}{\bf{.779}}}\\{{\bf{ - 1}}{\bf{.81673}}}&{{\bf{32}}{\bf{.9496}}}\\{{\bf{ - 2}}{\bf{.73544}}}&{{\bf{41}}{\bf{.3875}}}\\{{\bf{ - 4}}{\bf{.08461}}}&{{\bf{52}}{\bf{.9252}}}\\{{\bf{ - 6}}{\bf{.05673}}}&{{\bf{68}}{\bf{.7003}}}\\{{\bf{ - 8}}{\bf{.92771}}}&{{\bf{90}}{\bf{.2677}}}\\{{\bf{ - 13}}{\bf{.0922}}}&{{\bf{119}}{\bf{.752}}}\\{{\bf{ - 19}}{\bf{.1134}}}&{{\bf{160}}{\bf{.058}}}\\{{\bf{ - 27}}{\bf{.7944}}}&{{\bf{215}}{\bf{.152}}}\\{{\bf{ - 40}}{\bf{.2775}}}&{{\bf{290}}{\bf{.457}}}\\{{\bf{ - 58}}{\bf{.1856}}}&{{\bf{393}}{\bf{.379}}}\\{{\bf{ - 83}}{\bf{.8213}}}&{{\bf{534}}{\bf{.038}}}\end{aligned}\)

Step 3: Check whether the points are bounded or not

The Poincare section is given in the figure below. Now,

 

\(\mathop {{\bf{lim}}}\limits_{{\bf{n}} \to \infty } {{\bf{x}}_{\bf{n}}}{\bf{ = }}\mathop {{\bf{lim}}}\limits_{{\bf{n}} \to \infty } \left( {{{\bf{e}}^{{\bf{n\pi /10}}}}{\bf{sin}}\left( {\frac{{\sqrt {{\bf{399}}} }}{{{\bf{10}}}}{\bf{n\pi }}} \right){\bf{ + 10sin(2n\pi )}}} \right){\bf{ = }}\infty \)

 

So, the points are unbounded when\({\bf{n}} \to \infty \).