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

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$Continuous-time random walks, fractional calculus and ... - Eurandom$

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
Introduction Rise of Patterns The Busse balloon Conclusions and ongoing research Let k be the wavenumber of a pattern born at the Turing bifurcation. For |A − Ac| = O(ε), the Eckhaus band describes a region in (A, k)-space where stable patterns exist. A Eckhaus band Existence band k Figure: The Eckhaus band. Sjors van der Stelt Busse balloons A=A Turing
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Busse balloons - from the reversible Gray-Scott model ... - Eurandom

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