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
x
There are two steady states, =(0,0) and =(-2b/3,0), shown by the green and red points.
S
1
S
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)=b++2
2
x
3
x
2
y
S
2
V=K
K
S
2
K=4/27
3
b