Approximating the solution to a system of 3 oscillatory ODEs? ODE System
I have the following system of ODEs:
$x'(t)=x(t)\frac{z(t)}{Z}-x(t)\frac{x(t)+y(t)}{J}$
$y'(t)=y(t)\left(1-\frac{z(t)}{Z}\right)(1-q)-y(t)\frac{x(t)+y(t)}{J}$
$z'(t)=y(t)\left(1-\frac{z(t)}{Z}\right)(1-q)-mz(t)$,
where all variables and parameters are positive and $0<q<1$.
Background
The system exhibits an equilibrium, $\left(\overline{x},\overline{y},\overline{z}\right)$, where $\overline{x}$, $\overline{y}$, and $\overline{z}$ are positive when $J$ is greater than a critical value $J_{Crit}$.
The real parts of the eigenvalues $\left(\lambda_1,\lambda_2,\lambda_3\right)$ corresponding to $\left(\overline{x},\overline{y},\overline{z}\right)$ are all negative when $J>J_{Crit}$.
Upon further analysis of the eigenvalues, one finds that $\left(\overline{x},\overline{y},\overline{z}\right)$ always exhibits oscillations (i.e., two of the eigenvalues are complex).
The answer to this post showed that the solution to such a 3-dimensional system is well-approximated by
$\overline{V} + Ae^{\sigma t}\cos{\left(\omega t+ \phi\right)} + be^{\lambda_3t} \ \forall \ \overline{V} \in \left(\overline{x},\overline{y},\overline{z}\right)$,
where $A$, $\sigma$, $\omega$, and $\phi$ give the oscillations' amplitude, decay rate, frequency, and initial phase, respectively.
Both $A$ and $b$ are functions of initial conditions, which are $x(0)$, $y(0)$, $z(0)$.
Question
In general, how does one calculate $b$ for a 3-dimensional ODE system like the one presented here?

You want the solution
$$
\overline{V} + Ae^{\sigma t}\cos{\left(\omega t+ \phi\right)} + be^{\lambda_3t} 
$$
to be equal to $V_0=(x(0),y(0),z(0))$ at $t=0$. So
$$
\overline{V} + A\cos{\left(\phi\right)} + b = V_0.
$$
Next it needs to be taken into account that $b$ is a multiple of the eigenvector associated with $\lambda_3$ and $A$ is a linear combination of the eigenvectors associated with $\lambda_1$ and $\lambda_2$.
