Q12E Using a numerical scheme such as... [FREE SOLUTION]

91影视

Fundamentals Of Differential Equations And Boundary Value Problems

R. Kent Nagle, Edward B. Saff, Arthur David Snider

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 616 pages

ISBN: 9780321977069

Chapter 5: Q12E (page 304)

Using a numerical scheme such as Runge鈥揔utta or a software package, calculate the Poincare map for equation (7) when b = 0.3, ${\bf{\gamma }}$= 1.2, and F = 0.2. (Notice that the closer you start to the limiting point, the sooner the transient part will die out.) Compare your map with Figure 5.46(a) on page 300. Redo for F = 0.28.

Short Answer

Expert verified

By using the software, the Poincare maps are plotted and the value in the first part is that a solution is $\frac{{2\pi }}{\gamma }$ periodic, and in the second part a solution is sub harmonic of the period$\frac{{4\pi }}{\gamma }$.

Step by step solution

Sketch the Poincare map for${\bf{F = 0}}{\bf{.2}}$.

Here the equation of (7) is given as:

${\bf{x''(t) + bx' - x(t) + }}{{\bf{x}}^{\bf{3}}}{\bf{ = Fsin\gamma t}}$

Put the given values when$b{\rm{ }} = {\rm{ }}0.3,\,\gamma = {\rm{ }}1.2$, and $F = 0.2$

In this case, the limit is set at a single point and it doesn't depend on the initial value of the equation. So we can conclude that a solution is $\frac{{2\pi }}{\gamma }$ periodic.

Sketch the graph when ${\bf{F = 0}}{\bf{.28}}$

Here a solution is the sub harmonic of the period$\frac{{4\pi }}{\gamma }$.

Therefore, the value in the first part is that a solution is $\frac{{2\pi }}{\gamma }$ periodic, and in the second part a solution is sub harmonic of the period$\frac{{4\pi }}{\gamma }$.