Consider the nonlinear system described by \begin{equation*} \dot{z_1}=-z_1, \quad \dot{z_2}=z_1^2+2z_1\gamma-z_2, \end{equation*} where $\gamma\in\Gamma=[\gamma_{min},\gamma_{max}]\subset(0,\infty)$ fixed.
I am interested in finding $\epsilon_0>0$, $\lambda>0$ and $m\geq 1$ such that the following holds for all $\gamma\in\Gamma$: \begin{equation*} \text{If} \quad \|z(0)\|\leq \epsilon_0, \quad \text{then} \quad \|z(t)\|\leq m e^{-\lambda t}\|z(0)\|. \end{equation*}
As a starting point, I thought of solving the first equation, which trivially gives \begin{equation} z_1(t)=z_1(0)e^{-t}. \end{equation} Then, substitute this in the second equation, yelding \begin{equation} \dot{z_2}=z_1(0)^2e^{-2t}+2z_1(0)e^{-t}\gamma-z_2(t)\leq \epsilon_0^2e^{-2t}+2\epsilon_0e^{-t}\gamma_{max}-z_2(t). \end{equation} But, at this point, I have no idea on how to proceed.