On the deformation method of study of global asymptotic stability

We consider the one-parameter family of systems $$x' = F(x,\lambda ), x \in \mathbb{R}^n , 0 \leqslant \lambda \leqslant 1,$$ where F: ? ⁿ × 0, 1] → ? ⁿ is a continuous vector field. The solution x(t) = φ(t, y, λ) is uniquely determined by the initial condition x(0) = y = φ(0, y, λ) and can be continued to the whole axis (?∞, +∞) for all λ ∈ 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 λ varies.