Chapter 7.2, Problem 27P

A generalization of the damped pendulum equation discussed in the text, or a damped spring-mass system, is the Lienard equation

$\frac{d^{2}x}{d{t}^2}+c\left(x\right)\frac{dx}{dt}+g\left(x\right)=0$

If $c\left(x\right)$ is a constant and $g\left(x\right)=kx$, then this equation has the form of the linear pendulum equation [replace $\sin \theta$ with $\theta$ in Eq. $\left(13\right)$ of Section $7.11$; otherwise, the damping force

$c\left(x\right)dx/dt$ and the restoring force $g\left(x\right)$ are nonlinear. Assume that $c$ is continuously differentiable, $g$ is twice continuously differentiable, and $g\left(0\right)=0$.

(a) Write the Lienard equation as a system of two first order equations by introducing the variable $y=dx/dt$.

(b) Show that $\left(0,0\right)$ is a critical point and that the system is almost linear in the neighbourhood of $\left(0,0\right)$.

(c) Show that if $c\left(0\right)>0$ and $g\text{'}\left(0\right)>0$, then the critical point is asymptotically stable, and that if $c\left(0\right)<0$ or $g1\left(0\right)<0$, then the critical point is unstable.

Hint: Use Taylor series to approximate $c$ and $g$ in the neighbourhood of $\text{x=0}$.