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Matthen

Turing Stripes

May 17th, 2013
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  1. window[x_] := (Cos[Pi x] + 1)/2;
  2. n = 3;
  3. ps = {0.6, -0.3, 0.1};
  4. as = {0.1, 0.5, 0.8};
  5. initu[x_] := Sum[2 as[[i]] Exp[-20 (x - ps[[i]])^2], {i, n}];
  6. Tmax = 80;
  7. \[Tau] = 0.1; du2 = 0.00028; dv2 = 0.005; \[Kappa] = -0.005;
  8. soln = NDSolve[{
  9. D[u[x, t], t] ==
  10. du2 D[u[x, t], {x, 2}] + u[x, t] - u[x, t]^3 -
  11. v[x, t] - \[Kappa],
  12. \[Tau] D[v[x, t], t] ==
  13. dv2 D[v[x, t], {x, 2}] + u[x, t] - v[x, t],
  14. u[x, 0] == 0.5 window[x] x,
  15. v[x, 0] == 0.1 window[x],
  16. Derivative[1, 0][u][-1, t] == 0,
  17. Derivative[1, 0][u][1, t] == 0,
  18. Derivative[1, 0][v][-1, t] == 0,
  19. Derivative[1, 0][v][1, t] == 0
  20. }, {u, v}, {x, -1, 1}, {t, 0, Tmax}];
  21. frame[tt_] := Show[
  22. ArrayPlot[{
  23. Table[
  24. Tanh[
  25. 3 Max[0, (First[(u[x, t] - v[x, t]) /. soln] /. {x -> xx,
  26. t -> tt})]]^2,
  27. {xx, -1, 1, 2/100}]}
  28. , AspectRatio -> 1, ColorFunction -> "DeepSeaColors",
  29. Frame -> False]
  30. ,
  31. Plot[
  32. Max[0, (1 + First[(u[x, t] /. soln)] /. {x -> (xx - 50)/50,
  33. t -> tt})/2]
  34. , {xx, 0, 101}, PlotRange -> {-1, 1},
  35. PlotStyle -> Directive[Thick, Red], Axes -> None],
  36. Plot[
  37. Max[0, (1 + First[(v[x, t] /. soln)] /. {x -> (xx - 50)/50,
  38. t -> tt})/2]
  39. , {xx, 0, 101}, PlotRange -> {-1, 1},
  40. PlotStyle -> Directive[Thick, Yellow]],
  41. ImageSize -> 300];
  42. Manipulate[
  43. frame[Exp[tt] - 0.99],
  44. {tt, 0, 3.5}
  45. ]
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