Systems Theory for the Analysis of Biological Neuron Dynamics - Advanced Models of Neural Networks - page 49

2.5.4 The Hopf Bifurcation

Bifurcation of the equilibrium point means that the locus of the equilibrium on the
phase plane changes due to variation of the system's parameters. Equilibrium x is
a hyperbolic equilibrium point if the real parts of all its eigenvalues are non-zero.
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 [ 131 ].
The stages for finding Hopf bifurcations in nonlinear dynamical systems are
described next. The following autonomous differential equation is considered:

dx
dt

(2.66)

D f 1 .x;/

where x is the state vector x2R n and 2R 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

(2.67)

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 1 C ˛ 1 n1 CC˛ n1 i C ˛ n D 0 (2.68)

i

where I n is the nn identity matrix, i with i D 1;2; ;ndenotes the eigenvalues
of the Jacobian matrix Df 1 .x /. 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, for the appearance of Hopf bifurcations.
As example, the following nonlinear system is considered:

x 1 D x 2 x 2 Dx 1 C .m x 1 /x 1

(2.69)

Setting x 1 D 0 and x 2 D 0 one obtains the system's fixed points. For m1 the
system has the fixed point .x 1 ;x 2 / D . 0;0/.F orm>1the syste m has t he fixed

points .x 1 ;x 2 / D .0;0/, .x 1 ;x 2 / D . p m 1;0/, .x 1 ;x 2 / D . p m 1;0/.The
system's Jacobian is

0 1
1 C m 3x 1 0

J D (2.70)