Systems Biology (Methods in Molecular Biology)

l Stable (or neutrally stable). Each trajectory move about the

critical point within a finite range of distance.

l Definition(Hyperbolic point). The equilibrium is said to be

hyperbolic if all eigenvalues of the jacobian matrix have

nonzero real parts.

l Hyperbolic equilibria are robust (i.e., the system is structur-

ally stable). Small perturbations of order do not change

qualitatively the phase portrait near the equilibria. More-

over, local phase portrait of a hyperbolic equilibrium of a

nonlinear system is equivalent to that of its linearization.

This statement has a mathematically precise form known as

the Hartman-Grobman. This theorem guarantees that the

stability of the steady stateðx,yÞof the nonlinear system is

the same as the stability of the trivial steady state (0, 0) of

the linearized system.

l Definition(Non-Hyperbolic point). If at least one eigen-

value of the Jacobian matrix is zero or has a zero real part,

then the equilibrium is said to be non-hyperbolic. Non-hy-

perbolic equilibria are not robust (i.e., the system is not

structurally stable). Small perturbations can result in a local

bifurcation of a non-hyperbolic equilibrium, i.e., it can

change stability, disappear, or split into many equilibria.

Some refer to such an equilibrium by the name of the

bifurcation.

A.4 Applications
to Two Nonlinear
Equations System

In the study of nonlinear dynamics, it is useful to first introduce a

simple system that exhibits periodic behavior as a consequence of a

Hopf bifurcation. The two-dimensional nonlinear and autonomous

system given by

x_ ¼ f 1 ðx,yÞ¼xþayþx^2 y,

y_ ¼ f 2 ðx,yÞ¼bayx^2 y

ð 33 Þ

has this feature. These equations describe the autocatalytic reaction

of two intermediate speciesxandyin an isothermal batch reactor,

when the system is far from equilibrium. In this context, the steady

state referred to below is a pseudo steady state, and is applicable

when the precursor reactant is slowly varying with time.

The unique steady state is given byxS¼bandyS¼b/(a+b^2 ).

This steady state is at the position of the green dot in the phase

portrait diagram. It appears as the intersection of the dotted blue

and green curves, which are the level curves given byf 1 (x,y)¼

0 andf 2 (x,y)¼0.

The stability of steady state to small disturbances can be

assessed by determining the eigenvalues of the JacobianJ

88 Rodolfo Guzzi et al.