Chapter 7.6, Problem 19E

Interpretation Introduction

Interpretation:

For a time $\text{τ = ωt}$, show that the equation transformsinto the ${\text{ω}}^{\text{2}}\text{x"+x+ε}{\left(\text{x}\right)}^{\text{3}}\text{=0}$. Also show that $\text{O}\left(\text{1}\right){\text{:x}}_{\text{0}}^{\text{"}}{\text{+x}}_{\text{0}}\text{= 0}$, $\text{O}\left(\text{ε}\right){\text{:x}}_{\text{1}}^{\text{"}}{\text{+ x}}_{\text{1}}{\text{= - 2ω}}_{\text{1}}{\text{x}}_{\text{0}}^{\text{"}}{\text{- x}}_{\text{0}}^{\text{3}}$, and the initial condition becomes ${\text{x}}_{\text{k}}\left(\text{0}\right){\text{= a, x}}_{\text{k}}^{\text{'}}\left(\text{0}\right)\text{=0 for all k > 0}$. Solve $\text{O}\left(\text{1}\right)$ for ${\text{x}}_{\text{0}}$. Show that after substitution of ${\text{x}}_{\text{0}}$, the $\text{O}\left(\text{ε}\right)$ equation becomes ${\text{x}}_{\text{1}}^{\text{"}}{\text{+ x}}_{\text{1}}\text{=}\left({\text{2ω}}_{\text{1}}\text{a- }\frac{\text{3}}{\text{4}}{\text{a}}^{\text{3}}\right)\text{cos τ- }\frac{\text{1}}{\text{4}}{\text{a}}^{\text{3}}\text{cos}\left(\text{3τ}\right)$. Also solve it for ${\text{x}}_{\text{1}}$.

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 a smooth function.

This system is known as a weakly nonlinear oscillator.

The expression of the amplitude of any limit cycle for the original system is ${\text{r = r}}_{\text{0}}\text{+ O}\left(\text{ε}\right)$

The expression of the frequency of any limit cycle for the original system is $\text{ω = 1+ε }\varphi \text{'}$

Taylor series expansion of $\text{x}\left(\text{t,ε}\right)$ is $\text{x}\left(\text{t,ε}\right)\text{ - x}\left(\text{t,T}\right)\text{ + O}\left(\text{ε}\right)$. Here, $\text{O}\left(\text{ε}\right)$ are the higher-order terms of the Taylor series expansion.

Taylor’s series expansion ${\left(1-x\right)}^{-1}$ is ${\left(\text{1- x}\right)}^{\text{-1}}{\text{=1+x+x}}^{\text{2}}{\text{+x}}^{\text{4}}\text{+}\mathrm{......}$