Chapter 8.4, Problem 11E

Interpretation Introduction

Interpretation:

• By fixing parameters $\text{k =1, b =}\frac{\text{4}}{\text{3}}\text{, F = 2}$ and by using the numerical integration plot, the phase portrait for the averaged system, by varying parameter $\text{'a'}$ from negative to positive.

• To show for $\text{a = 2}\text{.8}$ there are two stable points.

• Numerically integrate the original forced Duffing equation and plot $\text{x(t)}$ for $\text{'a'}$ increasing slowly from $\text{a = -1 to a = 5}$ and decreases slowly back to $\text{a = -1}$

Concept Introduction:

• The system equation for the linear oscillator is $\ddot{\text{x}}\text{ + x = 0}$, if the system is perturbed by small perturbation constant the system equation becomes $\ddot{\text{x}}\text{ + x + εh}\left(\text{x,}\dot{\text{x}}\right)\text{= 0}$

  Here, $0<\epsilon \ll 1$ and $\text{h}\left(\text{x,}\dot{\text{x}}\right)$ is smooth function.

  This system is known as weakly nonlinear oscillator.

• The polar transformation of Cartesian co-ordinates is

  $\begin{array}{l}\text{x = rcosθ}\\ \text{y = rsinθ}\end{array}$

• The averaged equations for ${\text{r}}^{\prime }\text{ and r}{\varphi }^{\prime }$ are

  ${\text{r}}^{\prime }\text{=}\frac{\text{1}}{\text{2π}}{\displaystyle {\int }_{\text{0}}^{\text{2π}}\text{h}\left(\text{θ}\right)}\text{sinθdθ}\equiv \langle \text{h sinθ}\rangle$

  $\text{r}{\varphi }^{\prime }\text{=}\frac{\text{1}}{\text{2π}}{\displaystyle {\int }_{\text{0}}^{\text{2π}}\text{h}\left(\text{θ}\right)}\text{cosθdθ}\equiv \langle \text{h cosθ}\rangle$

  Where, $\text{h}\left(\text{θ}\right)\text{=h}\left(\text{x,}\dot{\text{x}}\right)\text{=h}\left(\text{r cosθ},-\text{r sinθ}\right)$

• The value of ${\text{r}}_{\text{0}}$ is calculated as

  ${\text{r}}_0\text{=}\sqrt{{\left(\text{x}\left(\text{0}\right)\right)}^{\text{2}}\text{+}{\left(\dot{\text{x}}\left(\text{0}\right)\right)}^{\text{2}}}$

  The value of ${\varphi }_0$ is calculated by

  ${\varphi }_0={\tan }^{-}\left(\frac{\dot{\text{x}}\left(\text{0}\right)}{\text{x}\left(\text{0}\right)}\right)$