The definitions of asymptotic stability and stability in the sense of Lyapunov can apply locally if S(λ) is small, or globally if S(λ) includes the entire state space. When applied globally, any trajectory starting within the region S(λ) will converge to the equilibrium point as time increases.

10. The definition of asymptotic stability & stability in the sense of lyapunov apply in
a local sense (stability in small) if S (λ) is small. When S (λ) become large (include
entire state space), the both above definition of stability are said to apply in a
global sense (stability in large)
Each trajectory starting within S (λ) converges to the equilibrium point as t
increases.