Systems Biology (Methods in Molecular Biology)

We now consider each of the two solutions forr>0, and

examine their linear stability in the usual way. First, we add a

small perturbation:

x¼xþE:

Substituting this into the equation yields

dE

dt

¼ rx^2



 2 xEE^2 , ð 28 Þ

and since the term in brackets on the RHS is trivially zero,

therefore

dE

dt

¼ 2 xE,

which has the solution

EðtÞ¼Aexpð 2 xtÞ:

From this, we see that forx¼þ

ffiffiffi

r

p

|x|!0ast!1(linear

stability); forx¼

ffiffiffi

r

p

jxj!0ast!1(linear instability).

In a typical “bifurcation diagram,” therefore, the saddle

node bifurcation atr¼0 corresponds to the creation of two

new solution branches. One of these is linearly stable, the other

is linearly unstable.

Let’s do the same for the next

l transcritical:dx/dt¼rxx^2

l supercritical pitchfork super:dx/dt¼rxx^3

l subcritical pitchfork sub:dx/dt¼rx+x^3

and easily we find the relative stability and instability.

2. Linear stability analysis for systems
   Consider the two-dimensional nonlinear system

x_¼fðx,yÞ,

y_¼gðx,yÞ,

ð 29 Þ

and suppose thatðx,yÞis a steady state (equilibrium point), i.e.,

fðx,yÞ¼0 andgðx,yÞ¼ 0 :Now let’s consider a small pertur-

bation from the steady stateðx,yÞ

x¼xþu,

y¼yþv,

ð 30 Þ

whereuandvare understood to be small asu1 andv1. It

is natural to ask whetheruandvare growing or decaying so

thatxandywill move away form the steady state or move

towards the steady states. If it moves away, it is called unstable

equilibrium point, if it moves towards the equilibrium point,

86 Rodolfo Guzzi et al.