Information Technology Reference
In-Depth Information
01
1 0:5
01
1 0:5
A
1
D
A
2
D
(2.62)
with eigenvalues
with eigenvalues
1;2
D0:25˙j0:97
1
D1:28
2
D 0:78
Consequently, the equilibrium .0;0/ is a stable focus (matrix A
1
has stable complex
eignevalues) and the equilibrium .;0/ is a saddle point (matrix A
2
has an unstable
eigenvalue).
2.4.3
Limit Cycles
A dynamical system is considered to exhibit limit cycles when it admits the
nontrivial periodic solution
x.t C T/D x.t/ 8 t0
(2.63)
for some T>0(the trivial periodic solution is the one associated with x.t/ =
constant). An example about the existence of limit cycles is examined in the case of
the Van der Pol oscillator (actually this is a model that approximates the functioning
of neural oscillators such as Central Pattern Generators). The state equations of the
oscillator are
x
1
D x
2
x
2
Dx
1
C .1 x
Š
/x
2
(2.64)
Next the phase diagram of the Van der Pol oscillator is designed for three different
values of parameter , namely D 0:2, D 1, and D 5:0.InFigs.
2.15
and
2.16
,
it can be observed that in all cases there is a closed trajectory to which converge all
curves of the phase diagram that start from points far from it.
To study conditions under which dynamical systems exhibit limit cycles, a
second order autonomous nonlinear system is considered next, given by
x
1
x
2
f
1
.x
1
;x
2
/
f
2
.x
1
;x
2
/
D
(2.65)
Next, the following theorem defines the appearance of limit cycles in the phase
diagram of dynamical systems [
92
,
209
].
