Stability of parameterized Morse-Smale gradient-like flows

In this paper, we are concerned with the structural stability of a Morse-Smale gradient-like flow $\varphi^t$ and show that if $\{\varphi_\epsilon^t\}$ is a smooth one-parameter family of $C^3$ flows with $\varphi_0^t=\varphi^t$, and {$\psi_\epsilon^t$} is another one-parameter family of $C^3$ flows such that $\psi_\epsilon^t$ is $C^0$ $O(\epsilon^3)$-close and $C^1$ $O(\epsilon^2)$-close to $\varphi_\epsilon^t$, then for all small $|\epsilon|$, there is a homeomorphism $h_\epsilon$, which is $C^0$ $O(\epsilon^2)$-near the identity map, such that $h_\epsilon$ takes the trajectories of $\varphi^t_\epsilon$ to the ones of $\psi^t_\epsilon$.