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I am working at Rossler system and I would like to ask you for some help. This system is described by equations: \begin{array}{ll} \dot{x} = -y - z \\ \dot{y} = x + 0.2y \\ \label{eq:UkladRosSP} \dot{z} = 0.2 + z(x-c) \end{array} I am trying to plot a graph, on which a maximum Lyapunov Exponent would be shown. I have followed the way which was suggested in file : http://www.mathematica-journal.com/issue/v6i3/article/sandri/contents/63sandri.pdf . The problem is that the results does not converge to what it should be. For example, for c=1.1, we should get a 1-period cycle, while the MLE is non-negative (that means the chaos). Also the procedure of plotting takes a lot of time. Is there any way to get a better results? Code:

 RKStep[f_, y_, y0_, dt_] :=
 Module[{k1, k2, k3, k4},
  k1 = dt N[f /. Thread[y -> y0] ];
  k2 = dt N[f /. Thread[y -> y0 + k1/2] ];
  k3 = dt N[f /. Thread[y -> y0 + k2/2] ]; 
  k4 = dt N[f /. Thread[y -> y0 + k3] ];
  y0 + (k1 + 2*k2 + 2*k3 + k4)/6 ]

    IntVarEq[F_List, DPhi_List, x_List, Phi_List, x0_List, 
  Phi0_List, {t1_, dt_}] :=
 Module[{n, f, y, y0, yt},
  n = Length[x0];
  f = Flatten[Join[F, DPhi]];
  y = Flatten[Join[x, Phi]];
  y0 = Flatten[Join[x0, Phi0]];
  yt = Nest[RKStep[f, y, #, N[dt]] &,
    N[y0], Round[N[t1/dt]] ];
  {First[#], Rest[#]} & @Partition[yt, n] ]

JacobianMatrix[funs_List, vars_List] := Outer[D, funs, vars]

Normi[x_] := Sqrt[x.x]

Lyap[c_] := Module[{},
  F[{x_, y_, z_}] := {-z - y, x + 0.2*y, 0.2 + z*(x - c)};
  x0 = {-1, 0, 0};
  T = 40;
  stepsize = 0.01;
  n = Length[x0];
  x = Array[a, n];
  Phi = Array[b, {n, n}];
  DPhi = Phi.Transpose[JacobianMatrix[F[x], x]];
  Phi0 = IdentityMatrix[n];
  {xT, PhiT} = IntVarEq[F[x], DPhi, x, Phi, x0, Phi0, {T, stepsize}];
  xT
  ]

My aim is to achieve a plot similar to this one :enter image description here

Any ideas?

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