Moon and $\mathrm{Li}$ (1985) suggested a way to estimate Lyapunov exponents. The method is based on the fact that at a saddle point the stable and unstable trajectories meet and deline directions of convergence and divergence of trajectories, respectively. The rates of these behaviors approximate the respective Lyapunov exponents. To study the behavior of the trajectories in the saddle point region the equation of motion is linearized at the saddle point and then solved. Furthermore it is only necessary to solve the homogeneous part of the differential equation because the saddle point is a fixed point in any Poincaré section taken at frequency $\omega_{\mathrm{D}}$. An examin- ation of Poincaré sections taken at various phases shows that $\theta= \pm \pi$ appears to be the $\theta$ saddle point coordinate and therefore the required homogeneous, linearized equation is
$$
\mathrm{d}^2 \theta / \mathrm{dt}^2+(\mathrm{d} \theta / \mathrm{d} t) / q-\theta=0 .
$$
Substitute the solution $\theta=e^{m t}$ into the differential equation to obtain the relation
$$
m=-\frac{1}{2 q} \pm\left(\frac{1}{4 q^2}+1\right)
$$
which has both positive and negative values. These values are the estimates of $\lambda_{+}$and $\lambda_{\text {_ }}$. Check that they sum to $\nabla \cdot F$. Substitute these values into the Kaplan-Yorke relation for $d_1$, thereby obtaining a relation for $d_{\mathrm{L}}$ in terms of $q$. Find $d_{\mathrm{L}}$ in the limiting cases where $q=0$ (infinite damping) and $q=\infty$ (no damping). Calculate $d_1$ in the cases where $q=2$ and $q=5$. Do your results match those given in the text and developed in a previous problem? If not suggest a reason. (Moon and $\mathrm{Li}$ change the value of the damping parameter to an effective value giving numerical results that better match the dimension. If you were to follow a similar procedure. what value of $q$ would you choose for essential agreement with the numerically derived $d_{\mathrm{L}}$ ?)