On the Deformation Method of Study of Global Asymptotic Stability

We consider the one-parameter family of systems
$$ x'=F(x,\lambda),\qquad x\in\mathbb R^n, \quad 0\le\lambda\le1, $$
where $F\colon \mathbb R^n\times[0,1] \to \mathbb R^n$ is a continuous vector field. The solution $x(t)=\varphi(t,y,\lambda)$ is uniquely determined by the initial condition $x(0)=y=\varphi(0,y,\lambda)$ and can be continued to the whole axis $(-\infty,+\infty)$ for all $\lambda\in[0,1]$. We obtain conditions ensuring the preservation of the property of global asymptotic stability of the stationary solution of such a system as the parameter $\lambda$ varies.