Chapter 7.6, Problem 17E

Interpretation Introduction

Interpretation:

To show that the average equation becomes $\text{r'}=\frac{1}{4}\text{r sin}\left(2\varphi \right)$, $\varphi \text{'}=\frac{1}{\text{2}}\left(\text{γ}+\frac{1}{2}\cos \left(2\varphi \right)\right)$ for small x. The fixed point $\text{r=0}$ is unstable to exponentially growing oscillations is to be shown. For $\left|\text{γ}\right|{\text{< γ}}_{\text{c}}$, A formula for the growth rate k in terms of $\text{γ}$ is to be written. The behavior of the solution $\left|\text{γ}\right|{\text{> γ}}_{\text{c}}$ is to be determined.

Concept Introduction:

The system equation is in the form of, $\ddot{\text{x}}\text{ + x + h}\left(\text{x,x}\right)\text{=0}$.

The expression for the averaging equation for magnitude is $\text{x}\left(\text{t,ε}\right){\text{ = x}}_{\text{0}}\text{+ O}\left(\text{ε}\right)$.

Here $\text{x}$ is the position, and ${\text{x}}_{\text{0}}$ is the initial position of the system.

The expression of initial displacement ${\text{x}}_{\text{0}}$ is, ${\text{x}}_{\text{0}}\text{= r}\left(\text{T}\right)\text{ cos}\left(\varphi \left(\text{T}\right)\text{+ τ}\right)$. Here $\text{r}\left(\text{T}\right)$ is the radius of the polar coordinate system, and $\varphi$ is the angle formed by the radius $\varphi$ in the polar coordinate.

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 expressions of the Taylor series expansion.

Taylor’s series expansion ${\left(1-x\right)}^{-1}$ is

${\left(\text{1- x}\right)}^{\text{-1}}{\text{=1+x+x}}^{\text{2}}{\text{+x}}^{\text{4}}\text{+}\mathrm{......}$