Stability
DeÖnition
An equilibrium pointxof a dynamic systemXis called attractive if for
any initial valuex 0 the solutionX(t,x 0 )is convergent toxthat is
tlim!∞X(t,x^0 )=x,^8 x^0.
If it is true only for some neighborhood ofxthenxis called locally
attractive.
DeÖnition
An equilibrium pointxof a dynamic system is (Lyapunov) stable if for
anyε>0 there is aδ>0 such that for anykx 0 xk<δ
kX(t,x 0 ) xk<ε.The system is (Lyapunov) asymptotically stable if it
is stable and it is attractive. If it is true only for some neighborhood ofx
thenxis called locally asymptotically stable.
