Busse balloons - from the reversible Gray-Scott model ... - Eurandom

Introduction
Rise of Patterns
The Busse balloon
Conclusions and ongoing research
The Eckhaus band: symmetric (Gray-Scott) case
Let A be the amplitude of the emerging pattern. Then the pattern can
be written out as
U ∝ U0 + εA(ξ, τ)e ikc x + c.c. + h.o.t.
In the case of the Gray-Scott system, e.g., we find for A a real GLE:
Aτ = 2
√ b A + 2 √ 2Aξξ − 2
9 (10√ 2 − 7)|A| 2 A
A band of spatially periodic patterns bifurcates for
k ∈ (kc − ε √ r, kc + ε √ r)
Stable patterns only exist in the Eckhaus band
r
k ∈ (kc − ε
3 , kc
r
+ ε
3 )
Sjors van der Stelt Busse balloons