Pattern Homework for Math 605
Due: March 31, 2005
1. Consider the problem
ut = uxx + λ(eu − 1), 0 < x < π, u(0, t) = u(π, t) = 0
For the steady state problem (ut = 0), use perturbation expansions in 1 to determine
the solutions which bifurcate from u ≡ 0, as follows.
a) By considering the linearized problem, find the possible bifurcation points are λ = λn ,
n = 1, 2, . . ..
b) Find the solvability condition at O(2 ), which gives a nontrivial correction to λn for n odd.
c) Find the contribution to u at O(2 ), and determine the solvability condition at O(3 ), which
gives a nontrivial correction for λn for n even.
d) Sketch the bifurcation diagram for the amplitude of u vs. λ.
2. Determine the stability for the steady states you found in Problem 1.
3. Consider the following equation which arises in the buckling of a column (steady state
solutions),
uxxxx + λuxx + αu − βu3 − γu2 uxx = 0,
u = uxx = 0 at x = 0, π
The parameter λ is the bifurcation parameter, related to the pressure applied at the ends of
the column.
a) What is the basic solution?
In the following, take α = 4.
b) Linearize about the basic solution. Find the first bifurcation point λ0 . Show that there are
two possible eigenfunctions (modes), w1 (x) and w2 (x), at this value of λ.
Perform a bifurcation analysis using the perturbation expansions u ∼ u1 + 2 u2 + 3 u3 ,
λ ∼ λ0 + λ1 + 3 λ2 + . . ..
c) Show that u1 = Aw1 + Bw2 and λ1 = 0.
d) Derive a system of equations for A,B, (both constants) and λ2 . Determine the three possible
solutions, two “pure” mode solutions (either A = 0 or B = 0) and one√“mixed” mode solution
( A 6= 0 and B 6= 0). Sketch the corresponding bifurcation diagram, A2 + B 2 vs. λ. (Note:
Choose β and γ so that the solutions are real.)
4. Consider the reaction diffusion problem
ut = D1 uxx + a1 u(1 − u2 ) − b1 v
vt = D2 vxx − a2 v(1 − v 2 ) + b2 u
on an infinite domain.
a) Use a linear stability analysis about the basic (zero) state to determine the threshold
values of the parameters b1 and b2 ; that is, beyond bc1 and bc2 , the basic state loses stability to
spatially periodic states. (Assume Dj and aj are constants.)
b) Derive the Ginzburg-Landau equation for the modulating amplitude of these periodic states
for b1 = bc1 + B1 2 and b2 = bc2 + B2 2 .