Let $\xi^*,\varsigma,\eta:\mathbb{R}\rightarrow (0,\infty)$. You may assume any degree of smoothness required of these functions.
Assume that $\xi^{*}(t)$ and $\varsigma(t)$ have finite positive limits as $t\rightarrow\infty$ and as $t\rightarrow -\infty$.
Define $\delta:\mathbb{R}\rightarrow\mathbb{R}$ by: $$\delta(t)=\frac{\dot{\xi}^{*}(t)}{\xi^{*}(t)}-\varsigma(t),$$ for all $t\in\mathbb{R}$.
Finally, assume that for all $\tau\in\mathbb{R}$, $\xi_{\tau}:[\tau,\infty)\rightarrow [0,\infty)$ satisfies: $$\xi_{\tau}(\tau)=\varsigma(\tau),$$ and: $$\dot{\xi}_{\tau}(t)={\xi_{\tau}(t)}^2+\left(\delta(t)-\eta(t) {\delta(t)}^2 \right) \xi_{\tau}(t)+\eta(t) {\delta(t)}^2 \xi^{*}(t).$$
In particular, I need that $\xi_{\tau}(t)<\infty$ for all $t\ge\tau$, $\tau\in\mathbb{R}$, so $\xi_{\tau}$ does not explode to infinity in finite time for any $\tau$.
This will hold for example, if $\eta(t)=0$ and $\varsigma(t)=\xi^{*}(t)$ for all $t\in\mathbb{R}$, in which case a solution is $\xi_{\tau}(t)=\xi^{*}(t)$. I.e., if we restrict BOTH $\varsigma$ and $\eta$ then we can definitely ensure that $\xi_{\tau}$ does not explode to infinity in finite time.
My question is the following: Can you ensure that $\xi_{\tau}$ does not explode to infinity in finite time (for all $\tau$) by only restricting ONE of $\varsigma$ and $\eta$, not both?
(You may nonetheless impose restrictions on the limits of $\varsigma(t)$ as $t\rightarrow\pm\infty$ if necessary, even if you are restricting $\eta$ not $\varsigma$.)
Constant coefficient case
In the case in which $\xi^*$, $\varsigma$ and $\eta$ are all constant over time, Maple gives the solution as: $$2\xi_{\tau}(t)=\varsigma(\eta\varsigma+1)-\omega\tanh{\left[(t-\tau)\frac{\omega}{2}+\operatorname{arctanh}{\left(\frac{\varsigma(\eta\varsigma-1)}{\omega}\right)}\right]},$$ where $\omega = \varsigma\sqrt{\eta^2\varsigma^2+2\eta(\varsigma-2\xi^{*})+1}$.
So, in this case, for the solution to be bounded, we need $\varsigma>\max{\left\{\frac{1}{\eta},\frac{2\sqrt{\eta\xi^{*}}-1}{\eta}\right\}}$ OR $\varsigma\in\left(\xi^{*},\frac{1}{\eta}\right)$ so at least in this case, we only need to restrict $\varsigma$, not $\eta$.
Another special case
Suppose: $$\eta(t)=\frac{\delta(t)+2\xi^{*}(t)+2\sqrt{\xi^{*}(t){\left[\xi^{*}(t)+\delta(t)\right]}}}{{\delta(t)}^2},$$ then the two roots of the right hand side of the ODE are identical, meaning that $\dot{\xi}_{\tau}(t)\ge 0$, and making the algebra much neater.
This is positive and real, as required, as long as $\xi^{*}(t)+\delta(t)\ge 0$, i.e. $\varsigma(t)\le \xi^{*}(t)+\frac{\dot{\xi}^{*}(t)}{\xi^{*}(t)}$.
With this particular $\eta(t)$, the root of the right hand side of the ODE at $t$ is given by: $$r(t)=\xi^{*}(t)+\sqrt{\xi^{*}(t){\left[\xi^{*}(t)+\delta(t)\right]}}.$$
Using the results from Kilicaslan and Banks (2010), we have that the system does not explode to infinity if and only if for all $t\in\mathbb{R}$, $r(t)\ge\varsigma(t)$, for which it is sufficient that $\varsigma(t)\le\xi^{*}(t)$ and $\varsigma(t)\le \xi^{*}(t)+\frac{\dot{\xi}^{*}(t)}{\xi^{*}(t)}$ (as already assumed).
The formulas given in Kilicaslan and Banks (2010) imply that in this case, the solution is given by: $$\xi_{\tau}(t)=r(\infty)-\frac{r(\tau)-\varsigma(\tau)}{1+(t-\tau)\left[r(\tau)-\varsigma(\tau)\right]},$$ where $r(\infty):=\lim_{t\rightarrow\infty}{r(t)}$, however, this does not appear to solve the original ODE, so something must be wrong (perhaps in the paper??).
Yet another special case
Suppose: $$\eta(t)=\frac{\bar{\xi}\left[\bar{\xi}+\delta(t)\right]}{{\delta(t)}^2\left[\bar{\xi}-\xi^{*}(t)\right]},$$ where $\bar{\xi}$ is a constant with $\bar{\xi}+\delta(t)>0$ and $\bar{\xi}-\xi^{*}(t)>0$ for all $t\in\mathbb{R}$.
Define: $$r(t)=\frac{\xi^{*}(t)\left[\bar{\xi}+\delta(t)\right]}{\bar{\xi}-\xi^{*}(t)}>0,$$ then the original ODE is: $$\dot{\xi}_{\tau}(t)=\left[\xi_{\tau}(t)-\bar{\xi}\right]\left[\xi_{\tau}(t)-r(t)\right].$$ For this not to explode to infinity in finite time, it is sufficient that $\xi_{\tau}(\tau)=\varsigma(\tau)<\bar{\xi}$, so we need $\varsigma(t)<\bar{\xi}$ for all $t\in\mathbb{R}$.