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y
1
Dx
1
D ˛
11
y
1
C ˛
12
y
2
C h.o.t.
y
2
Dx
2
D ˛
21
y
1
C ˛
22
y
2
C h.o.t.
(2.56)
By omitting the higher order terms one can approximate the initial nonlinear system
with its linearized equivalent
y
1
D ˛
11
y
1
C ˛
12
y
2
y
2
D ˛
21
y
1
C ˛
22
y
2
(2.57)
which in matrix form is written as
(2.58)
y D
Ay
where
@f
1
@x
1
!
˛
11
˛
12
˛
21
˛
22
@f
1
@x
2
@f
@x
j
xDp
A D
D
j
xDp
D
(2.59)
@f
2
@x
1
@f
2
@x
2
@f
@x
is the Jacobian matrix of the system that is computed at point
.x
1
;x
2
/ D .p
1
;p
2
/. It is anticipated that the trajectories of the phase diagram of the
linearized system in the vicinity of the equilibrium point will be also close to the
trajectories of the phase diagram of the nonlinear system.
Therefore, if the origin (equilibrium) in the phase diagram of the linearized
system is (1) a stable node (matrix A has stable linear eigenvalues), (2) a stable focus
(matrix A has stable complex eignevalues), (3) a saddle point (matrix A has some
eigenvalues with negative real part while the rest of the eigenvalues have positive
real part), then one concludes that the same properties hold for phase diagram of the
nonlinear system.
Matrix A D
Example.
A nonlinear oscillator of the following form is considered:
x D f.x/)
x
1
x
2
x
2
sin.x
1
/ 0:5x
2
(2.60)
D
The associated Jacobian matrix is
0 1
cos.x
1
/ 0:5
@f
@x
D
(2.61)
There are two equilibrium points .0;0/ and .;0/. The linearization round the
equilibria gives
