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Problem 9.31 PP
Consider the nonlinear autonomous system
j r 1 r 1
(a) Find the equilibrium point(s).
(b) Find the linearized system about each equilibrium point.
(c) For each case in part (b), what does Lyapunov theory tell us about the stability of the 
nonlinear system near the equilibrium point?
nonlinear system near the equilibrium point?
Step-by-step solution
step 1 of 7
(a)
The non-linear autonomous system is,
* l ' X ,(2 1 i-2 C ,)
•«2 =
- X , ! ,
The non- linear autonomous system can be written as follows:
x ,= x ,{x ,-x ,) (1)
i i - j f - l ...... (2)
= ...... - ^
step 5 of 7
Recall equation (1).
x ,= x ,(x ,-x ,)
Substitute for i | , J>|-1 for x^, for jtj,and y^ for Xy.
.Vi =3’2[3’3 - ( j ' . - 0 ]
= y 2 y 2 -y iy i+ y i
Recall equation (2).
Substitute for i^ ,and - I f o r a:|.
= yt-2y,
Recall equation (3). 
iy^-JC^X,
Substitute y^ for jfcj, _yj for jt j, and ^ , - 1 for a:|.
— y , y , + y .
The linearized system is,
y ^ F y
0 I O'
= -2 0 0 y 
0 0 1
Therefore, the linearized system about the equilibrium, point is.
0 1 0
y= -2 0 0 
0 0 1
y
Step 6 of 7
(c)
(i)
The linearization is developed to determine the stability of the system near the equilibrium 
conditions.
Calculate the characteristic equation.
| d - F | = 0 
s 0 0
0 1 0 
0 0 s 
3 I 0 
-2 » 0 
0 0 3 + 1
0 -1 o '
- 2 0 0
0 0 -1
= 0
= 0
3 [ s (s + 1 ) - 0 ] - 1 [ ( - 2 ) ( j + 1 ) - 0 ] + 0 = 0 
3*(s + 1 )+ 2 ( s + 1) = 0 
(3 * + 2 )( j + 1) = 0
From the characteristic equation observe that the system has two poles on the ja> axis. These 
poles make the system neutrally or marginally stable. Hence, the Lyapunov theory does not tell 
whether the system is stable or not.
Therefore, the nonlinear terms affect the stability at the equilibrium point [1. 0. o f
Step 7 of 7
(ii)
Calculate the characteristic equation.
|5 l- F | = 0
s 0 0
0 4 0
0 0 4
s -1
0 I 0 
-2 0 0 
0 0 1
= 0
0 0 s - 
j[s (s - l) -0 ]- ( - l) [ (2 ) ( j- l) -0 ]+ 0 = 0 
3*(j - 1 ) + 2 ( j - 1 ) = 0 
(j “+ 2 ) ( 3 - 1 ) = 0
From the characteristic equation, the system has two poles on the Ja> axis and one on the real
Therefore, the system at equilibrium point - 1, 0, o f