Chapter 7.6, Problem 4E

Interpretation Introduction

Interpretation:

To calculate the averaged equations and to analyze the long-term behavior of the system $\ddot{\text{x}}\text{ + x + ε h}\left(\text{x, }\dot{\text{x}}\right)\text{ = 0}$, where $\text{h}\left(\text{x, }\dot{\text{x}}\right)\text{ = x}$. To find the amplitude and frequency of any limit cycles for the original system. If possible, to solve the averaged equations explicitly for $\text{x}\left(\text{t, ε}\right)$, given the initial conditions $\text{x}\left(\text{0}\right)\text{ = a}$, $\dot{\text{x}}\left(\text{0}\right)=0$.

Concept Introduction:

The averaged or slow-time equations are given by ${\text{r}}^{\prime }\text{ = }\langle \text{h sin θ}\rangle$, $\text{r}{\varphi }^{\prime }\text{ = }\langle \text{h cos θ}\rangle$.

The overall solution of the system is given by ${\text{x}}_0\text{ = r(T) cos}\left(\varphi \left(\text{T}\right)+\text{τ}\right)$.

The explicit solution is given by $\text{x}\left(\text{t, ε}\right){\text{ = x}}_{\text{0}}\text{+ O(ε)}$.