Information Technology Reference
In-Depth Information
x
is a hyperbolic equilibrium point if the real parts of all its eigenvalues are
non-zero. One can have stable or unstable bifurcations of hyperbolic equilibria. A
saddle-node bifurcation occurs when there are eigenvalues of the Jacobian matrix
with negative real part and other eigenvalues with non-negative real part. A Hopf
bifurcation appears when the hyperbolicity of the equilibrium point is lost due to
variation of the system parameters and the eigenvalues become purely imaginary. By
changing the values of the parameters at a Hopf bifurcation, an oscillatory solution
appears.
The stages for finding bifurcations in nonlinear dynamical systems are described
next. The following autonomous differential equation is considered:
dx
dt
D f
1
.x;q/
(3.15)
where x is the state vector x2R
n
and q2R
m
is the vector of the system parameters.
In Eq. (
3.15
) a point x
satisfying f
1
.x
/ D 0 is an equilibrium point. Therefore
from the condition f
1
.x
/ D 0 one obtains a set of equations which provide
the equilibrium point as function of the bifurcating parameter. The stability of
the equilibrium point can be evaluated by linearizing the system's dynamic model
around the equilibrium point and by computing eigenvalues of the Jacobian matrix.
The Jacobian matrix at the equilibrium point can be written as
@f
1
.x/
@x
(3.16)
J
f
1
.x
/
D
j
xDx
and the determinant of the Jacobian matrix provides the characteristic equation
which is given by
det.
i
I
n
J
f
1
.x
/
/ D c
n
1
C c
n1
n1
CCc
1
i
C c
0
D 0;
(3.17)
i
where I
n
is the nn identity matrix,
i
with i D 1;2; ;ndenotes the eigenvalues
of the Jacobian matrix J
f
1
.x
/
. Using the above characteristic polynomial one can
formulate conditions for the system to have stable or unstable fixed points or saddle-
node fixed points. From the requirement the eigenvalues of the system to be purely
imaginary one obtains a condition, i.e. values that the bifurcating parameter should
take, so as Hopf bifurcations to appear.
3.4
Bifurcation Analysis of the FitzHugh-Nagumo Neuron
3.4.1
Equilibria and Stability of the FitzHugh-Nagumo
Neuron
The model of the dynamics of the FitzHugh-Nagumo neuron is considered
dV
dt
D V.V a/.1 V/
w
C I
d
w
(3.18)
dt
D .V
w
/
