455 HW#7 Problem 1(a)
The matrix of the linear equations for the solution amplitudes:

> restart:
thematrix:=evalm([[-w+B-1-D1*k^2,A^2],[-B,-w-A^2-D2*k^2]]);

Expand the determinant, and collect by powers of w:

> determ:=linalg[det](thematrix);
determ:=collect(determ,w);

Plug in parameter values of interest:

> A:=2.0; B:=4.6; L:=1; D1:=1.6e-3; D2:=6.0e-3; k:=n*evalf(Pi)/L;

Find roots of quadratic for these parameter values:

> ws:=solve(determ=0,w);

Make table of roots for a range of wave numbers:

> my_root_table:=matrix([["n","w_plus","w_minus"],
seq([n,ws[1],ws[2]],n=1..12)]);

Observe that there are roots with positive real part (corresponding to growth) for n between 6 and 11.
The roots for n between 1 and 4 are complex (corresponding to oscillation) with negative real part (decay).
Also note all the "w_" solutions decay.

455 HW#7 Problem 1(b)
Grader will expect some kind of description or sketch of the following animations:

> restart:
L:=1;
form_component:=n->alpha[n]*exp(Re(omega[n])*t)*cos(Im(omega[n])*t)*sin(n*Pi*x/L);

omega[1]:=-.2375044967+1.947949885*I;
alpha[1]:=1;

omega[7]:=.2553005850;
alpha[7]:=.01;

omega[12]:=-.161985586;
alpha[12]:=.01;

dt:=.25: nt:=60: tvals:=[seq(i*dt,i=0..nt)]:
pure_mode:=form_component(1);
plots[display]( seq( plot(pure_mode,x=0..L,color=blue,
title=convert(t,string)),t=tvals),
insequence=true);

> dt:=.25: nt:=60: tvals:=[seq(i*dt,i=0..nt)]:
pure_mode:=form_component(7);
plots[display]( seq( plot(pure_mode,x=0..L,color=blue,
title=convert(t,string)),t=tvals),
insequence=true);

> dt:=.25: nt:=60: tvals:=[seq(i*dt,i=0..nt)]:
pure_mode:=form_component(12);
plots[display]( seq( plot(pure_mode,x=0..L,color=blue,
title=convert(t,string)),t=tvals),
insequence=true);

455 HW#7 Problem 1(c)

As I said in class, the behavior of the Brusselator, with the default parameters and the initial conditions

obtained by clicking "reset" to me seem predominantly a combination of the oscillatory decay of an n=1 mode,

and the monotonic growth of an n=8 mode, something like the following. There's probably also a bit of decaying

n=2 in there.

> omega[2]:=-.3500179870+1.814490975*I;
alpha[2]:=1;

omega[8]:=.320733683;
alpha[8]:=-.01;

components:=[form_component(1),form_component(2),form_component(8)];
combo:=add(components[k],k=1..nops(components));

dt:=.5: nt:=30: tvals:=[seq(i*dt,i=0..nt)]:
plots[display]( seq( plot([seq(components[k],k=1..nops(components))],x=0..L,color=blue),t=tvals),
title="Individual components",insequence=true);
plots[display]( seq( plot(combo,x=0..L),t=tvals),title="Sum of components",insequence=true);

Problem 1d
We got pictures of clouds, water ripples, sand ripples, fossilized sand ripples, fingerprints, fruit-fly embryo, tree bark, and many others. Here are the pictures you sent me.

Problem 2.

Problem 3. Here are distinct persistent states I found all at the same paramter values. I found several steady states: 1 with n=6, 2 different ones with n=7, and 1 with n=8. (Notice that since our equations and BCs are symmetric about the mid-point of the reactor, for any of the assymetric steady solutions, there must be another which is the mirror image (in the point x=L/2). Also 2 radically different periodic states.