Dynamics of a Forced Brusselator

time t

FE`time$$4831636222057695441558866568940002723202

forcing amplitude f

FE`f$$4831636222057695441558866568940002723202

parameter a

FE`a$$4831636222057695441558866568940002723202

forcing frequency ω

FE`\:03c9$$4831636222057695441558866568940002723202

parameter b

FE`b$$4831636222057695441558866568940002723202

This Demonstration shows the dynamics of a forced Brusselator model. The Brusselator is an example of an autocatalytic chemical reaction. The model can represent a limit cycle, Andronov–Hopf bifurcation, and also chaotic behavior when a sinusoidal force acts on the system [1]. This force could be heat convection, microwave radiation, or some other force that varies sinusoidally with small intensity.

The chemical reactions of the Brusselator are

AX;B+XY+D;2X+Y3X;XE

. The model describes a chemical system that converts reactant

A

to a final product

E

through four steps. For simplicity, the concentrations

a

and

b

of

A

and

B

are maintained constant, and all reaction rates are set equal to one. The system of differential equations that describe the dimensionless concentrations

x

and

y

of species

X

and

Y

in the forced Brusselator are



x

=a+

2

x

y-(b+1)x+fcos(ωt)

and



y

=bx-a

2

x

y

, where

f

is the force amplitude and

ω

is the force frequency. You can vary the parameters

a

,

b

,

f

, and

ω

to see the trajectories of the variables

x

and

y

.