Systems Biology (Methods in Molecular Biology)

A.2 Bifurcation
Concepts

A bifurcation occurs when a small smooth change made to the

parameter values (the bifurcation parameters) of a system causes a

sudden qualitative or topological change in its behavior. Generally,

at a bifurcation, the local stability properties of equilibria, periodic

orbits or other invariant sets changes. It has two types;Local

bifurcations, which can be analyzed entirely through changes in

the local stability properties of equilibria, periodic orbits or other

invariant sets as parameters cross through critical thresholds; and

Global bifurcations,which often occur when larger invariant sets of

the system collide with each other, or with equilibria of the system.

They cannot be detected purely by a stability analysis of the equili-

bria (fixed or equilibrium points, see the next section).

In dynamical systems, only the solutions of linear systems may

be found explicitly. Unfortunately, real life problems can generally

be modelled only by nonlinear systems The main idea is to approxi-

mate a nonlinear system by a linear one (around the equilibrium

point).

A.3 Linear Stability
Analysis

Bifurcations indicate qualitative changes in a system’s behavior. For

a dynamical systemdydt¼fyðÞ,λ, bifurcation points are those equi-

librium points at which the Jacobian∂∂fyis singular. For definition

consider a nonlinear differential equation

x_ðtÞ¼fðxðtÞ,uðtÞÞ, ð 19 Þ

wherefis a function mappingRR^3 !Rn. A pointxis called an

equilibrium point if there is a specificu∈Rmsuch that

fðxðtÞ,uðtÞÞ¼ (^0) n: ð 20 Þ
Supposexis an equilibrium point (with the inputu). Consider the
initial conditionxð 0 Þ¼x, and applying the inputuðtÞ¼ufor all
tt 0 , then resulting solutionx(t) satisfies
xðtÞ¼x, ð 21 Þ
for alltt 0. That is why it is called an equilibrium point or
solution.
Linear stability of dynamical equations can be analyzed in two
parts: one for scalar equations and the other for two dimensional
systems

1. Linear stability analysis for scalar equations
   To analyze the Ordinary Differential Equations (ODE)

x_¼fðxÞð 22 Þ

locally about the equilibrium point x¼x, we expand the

functionf(x) in a Taylor series about the equilibrium pointx.

To emphasize that we are doing a local analysis, it is customary

84 Rodolfo Guzzi et al.