Systems Biology (Methods in Molecular Biology)

J¼

∂f 1

∂x

∂f 1

∂y

∂f 2

∂x

∂f 2

∂y

0

B

B

@

1

C

C

A ð^34 Þ

A.5 Hopf Bifurcation Using the following procedure it is possible to compute the Hopf
bifurcation parameter value of two or three-dimensional dynamical
systems. Since the two-dimensional procedure may be obtained by
a simple reduction, the three-dimensional procedure is only pre-
sented. A Hopf bifurcation occurs when a complex conjugate pair
of eigenvalues crosses the imaginary axis.
The phase portrait with the vector field of directions around the
critical point (xS,yS) may be simply obtained. In addition, the
eigenvalues ofJ, the tracetrace(J), the determinant | J|, and
Δ¼traceðJÞ^2  4 jJjmay be computed as far as the time series
corresponding tox(t) andy(t), respectively and the real and imagi-
nary parts of the two eigenvaluesλ 1 andλ 2. Then to observe a Hopf
bifurcation may be defined the parameters.

A.6 Nonlinear
Equations System

Newton’s method can be used to solve systems of nonlinear equa-

tions. Newton-Raphson Method for two-dimensional Systems.

To solve the nonlinear systemF(X) ¼0, given one initial

approximationP 0 , and generating a sequencePkwhich converges

to the solutionPii.e.F(X)¼0.

Suppose thatPkhas been obtained, use the following steps to

obtainPk+1.

1. Evaluate the function

FðPkÞ¼

f 1 ðpk,qkÞ

f 2 ðpk,qkÞ



ð 35 Þ

2. Evaluate the Jacobian

JPðÞ¼k

δ

δx

f 1 pk,qk

δ

δx

f 1 pk,qk



δ

δy

f 2 pk,qk

δ

δy

f 2 pk,qk



0

B

B

@

1

C

C

A ð^36 Þ

3. Solve the linear system

JðPkÞΔPk¼FðPkÞ for ΔP ð 37 Þ

4. Compute the next approximation

Pkþ 1 ¼PkþΔPk ð 38 Þ

Inverse Problems in Systems Biology 89