Chapter 4.4, Problem 32P

If the restoring force of a nonlinear spring satisfies the relation

$F_s\left(\Delta x\right)=-k\Delta x-\epsilon {\left(\Delta x\right)}^3$,

where $k>0$, then the differential equation for the displacement $x\left(t\right)$ of the mass from its equilibrium position satisfies the differential equation (see Problem $17$, section $4.1$ )

$mx\text{'}\text{'}+\gamma x\text{'}+kx+\epsilon x^3=0$.

Assume that the initial conditions are,

$\begin{array}{cc}x\left(0\right)=0,& x\text{'}\left(0\right)=1\end{array}$.

Find $x\left(t\right)$ when $\epsilon =0$ and also determine the amplitude and period of the motion.

Let $\epsilon =0.1$. Plot a numerical approximation to the solution. Does the motion appear to be periodic? Estimate the amplitude and period.

Repeat part (c) for $\epsilon =0.2$ and $\epsilon =0.3$.

Plot your estimated values of the amplitude $A$ and the period $T$ versus $\epsilon$. Describe the way in which $A$ and $T$, respectively depend on $\epsilon$.

Repeat parts (c),(d), and (e) for negative values of $\epsilon$.