Bruselator

Introduction to bifurcations
Marc R. Roussel
September 16, 2005

1 Introduction
Most dynamical systems contain parameters in addition to variables. A general system of ordinary
differential equations (ODEs) could therefore be written
˙
x = f(x; k),
where k is a set of parameters on which the equations, and thus their solutions, depend.
If we solve a set of differential equations atdifferent parameter values, we often find that,
qualitatively, not much changes. However, in some models, we can find sets of parameter values
which are close to each other but where the behavior of the model is in some way qualitatively
different for one set or the other. For instance, a stable equilibrium point might have become
unstable. We then say that the system has undergone a bifurcation.Bifurcations often change the
attractors of a dynamical system. Informally, an attractor is a solution which is approached at
long times. Stable equilibrium points are attractors, but they are not the only possibility.
In today’s lecture, we will engage in some simple numerical discovery exercises in which we
will see a few important bifurcations and their consequences. We will be using thedynamical
systems software xppaut (xpp for short). Full instructions for using this software will not be given
here. You are directed to consult the xpp documentation for details.

2 Andronov-Hopf bifurcations
One very important kind of bifurcation is the Andronov-Hopf bifurcation (formerly known in
the West as a Hopf bifurcation). In an Andronov-Hopf bifurcation, a stable focus becomes anunstable focus as a parameter is varied, and the attractor becomes a limit cycle. A limit cycle
is an asymptotically stable, periodic solution which can be pictured as a closed curve in phase
space. Limit cycles differ from conservative oscillations in mechanical systems in that the former
have fixed shapes and sizes at given parameter values, while the corresponding quantities in a
conservativemechanical oscillator depend on the total energy, i.e. on the initial conditions. There
are two qualitatively different kinds of Andronov-Hopf bifurcations, sketched below.
In a supercritical Andronov-Hopf bifurcation, the limit cycle grows out of the equilibrium
point. In other words, right at the parameters of the Andronov-Hopf bifurcation, the limit cycle
1

has zero amplitude, and thisamplitude grows as the parameters move further into the limit-cycle
regime. Pictorially, think of it this way:

In a subcritical Andronov-Hopf bifurcation, there is an unstable limit cycle surrounding the
equilibrium point, and a stable limit cycle surrounding that. The unstable limit cycle shrinks down
to the equilibrium point, which becomes unstable in the process. For systems started nearthe
equilibrium point, the result is a sudden change in behavior from approach to a stable focus, to
large-amplitude oscillations. Here is the corresponding picture:

Because the stable limit cycle exists even when the equilibrium point is stable, if we imagine slowly
varying a system parameter back-and-forth across the Andronov-Hopf bifurcation, we wouldn’t
expect to jump back to theequilibrium point at the same parameter value of the parameter from
which this point lost stability. This is called hysteresis, and is associated with bistability, the fact
that the system actually has two attractors over a range of parameters.
In both cases, Andronov-Hopf bifurcations occur when an equilibrium point changes from
being a stable to an unstable focus. We therefore detect Andronov-Hopfbifurcations through
linear stability analysis.1
Example 2.1 The Brusselator is a historically important model of an oscillating chemical reaction. The Brusselator is an abstract model which was used to show that chemical systems
could oscillate, but it does not describe any particular reaction. As you can guess from the
1 There

are additional technical conditions which are required to...