# Asymptotic Stability of Dynamical System by Lyapunov's Direct Method

Asymptotic Stability of Dynamical System by Lyapunov's Direct Method

Consider the two-dimensional autonomous dynamical system:

x

t

y

t

2

There are two steady states, and , shown by the green and red points.

S=(0,0)

1

S=(-2b/3,0)

2

Restricting the parameters to and , we compute and plot the region of asymptotic stability, shown in orange. All trajectories originating at an initial condition (IC) chosen in this orange region end at the . Trajectories diverge otherwise. The blue region is also a basin of attraction for . This region is obtained using Lyapunov's direct method and the Lyapunov function . The blue region cannot contain because it is an unstable steady state. defines a closed curve containing the origin if is a positive constant. The largest such region can be obtained by having at its frontier, or . You can drag the locator to change the IC.

a=1

b=1

S

1

S

1

V(x,y)=bx+x+y/2

2

3

2

S

2

V=K

K

S

2

K=4b/27

3