Lyapunov exponent for chaotic damped system with time-varying potential

Question

Consider the following linear damped system $$\ddot{Z}+3\dot{Z}+M^2(t)Z(t)=0 $$ where the goal is to solve for $Z$ for a time-varying potential $M^2$ independent of $Z$. The above can be expressed as a system of first-order ODE as $$\dot{X}=G(X)$$ where now $J=\partial_X G(X)$ is the Jacobian. Now the key is that the potential $M^2$ has an analytical form such that it is chaotic/unstable for a brief period of time say from $t_1$ to some $t_2$ (in other words, the Jacobian J has non-negative real eigenvalues). Later for some $t>t_3$ the potential vanishes and thus $\lim_{t\rightarrow\infty}Z$ settles to a constant as per the above second order ODE. Thus, I have following questions regarding the chaotic structure of the above system.

1. Given that I start with $Z(t_0)=1$ and say a perturbed trajectory has $\delta Z(t_0)={10}^{-5}$, the system evolves and enters chaotic region between $t_1$ and $t_2$ and eventually settles. The usual definition of Lyapunov exponent is $$\lambda=\lim_{t\rightarrow\infty}\frac{1}{t}\ln\frac{|\delta Z(t)|}{|\delta Z(t_0)|}$$. Thus even though the two trajectories might yield very different values of $Z(t)$ for some large $t$, the Lyapunov exponent evaluated from the above equation will always tend to $0$ since the $Z$ must eventually settle. That is to say that after the potential vanishes the separation between the two trajectories freezes out. Is it then appropriate to modify the above equation by changing $t$ to some $t_{end}$(say $t_3$) at which the $Z$ is known to settle to a constant? If yes, how do you justify it mathematically?
2. Also, the damping factor in the original equation implies that the value to which $Z$ eventually settles to is much smaller than $Z(t_0)$. For example if $Z(t_0)=1$, the final value is say some $Z(t_{end})={10}^{-7}$. An initial perturbation of $\delta Z(t_0)={10}^{-5}$ leads to a final value of anywhere from $Z(t_{end})=0.5 \times{10}^{-7}$ to $Z(t_{end})=1.5 \times{10}^{-7}$. The final separation is surely significant, but is just about $5\times{10}^{-8}$ at max compared to initial perturbation of ${10}^{-5}$. Thus, the exponent turns out as negative contrary to a positive value for a chaotic system. Hence, shouldn't the $\delta Z$ used for evaluating the exponent be normalized to the final $Z$ values? While it seems logical, I haven't found any text saying that the $\delta Z$ must be normalized.