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The delayed Lorenz system is as follows: (Bifurcation parameter as $\tau$)

$$\begin{align*} D^\alpha x_1(t) = & a_1(x_2(t-\tau)-x_1(t)) \\ D^\alpha x_2(t) = & a_2x_1(t)-x_2(t)-x_1(t)x_3(t) \\ D^\alpha x_3(t) = &-a_3x_3(t-\tau)+\frac{1}{2}(x_1(t)x_2(t)) \end{align*}$$

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closed as off-topic by m_goldberg, user9660, MarcoB, Yves Klett, RunnyKine Mar 20 '16 at 20:57

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  • "This question cannot be answered without additional information. Questions on problems in code must describe the specific problem and include valid code to reproduce it. Any data used for programming examples should be embedded in the question or code to generate the (fake) data must be included." – m_goldberg, Community, MarcoB, Yves Klett, RunnyKine
If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ If you are looking for a numerical solution to this, you to need to specify the values of a's? It seems that \alpha is a fractional derivative parameter, so again you need to assign a certain value to it. $\endgroup$ – zhk Mar 20 '16 at 8:56
  • $\begingroup$ a_1=10, a_2=28, a_3=8/3, \alpha=1, That is ordinary delay differential equations. $\endgroup$ – G Velmurugan Mar 20 '16 at 9:43
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    $\begingroup$ This question seems incomplete. It states a system of equations, but does not tell what kind of results are wanted. Please supply more information. Also, give the values for constants in the main post, not in a comment. $\endgroup$ – m_goldberg Mar 20 '16 at 12:57
  • $\begingroup$ This question was put here last year already once $\endgroup$ – user36273 Mar 20 '16 at 17:21
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Well, to be honest, you did not provide any information as also mentioned by @m_goldberg. Any ways, I choose random initial conditions and a random value for the delay tau = 1.

a1 = 10; a2 = 28; a3 = 8/3; alpha = 1; \[Tau] = 1;

sol = First[
   NDSolve[{x1'[t] == a1*(x2[t - \[Tau]] - x1[t]), 
     x1[t /; t <= 0] == 3,
     x2'[t] == a2*x1[t] - x2[t] - x1[t]*x3[t], x2[t /; t <= 0] == 6,
     x3'[t] == -a3*x3[t - \[Tau]] + 1/2*x1[t]*x2[t], 
     x3[t /; t <= 0] == 3}, {x1, x2, x3}, {t, 0, 20}]];

Plot[Evaluate[{x1[t], x2[t], x3[t]} /. sol], {t, 0, 200}, 
 PlotStyle -> {Thick}, Frame -> True]

enter image description here

Note: Next time provide complete information, not just a bunch of equations.

Addition (Space curve for:{x1[t],x2[t],x3[t]})

ParametricPlot3D[{x1[t], x2[t], x3[t]} /. sol, {t, 0, 150}, 
 PlotStyle -> {Orange, Thickness[0.015]}, BoxRatios -> {1, 1, 1}, 
 AxesLabel -> {x1, x2, x3}]
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  • $\begingroup$ a_1=10, a_2=28, a_3=8/3, \alpha=1, That is ordinary delay Lorenz system. I really want to draw the bifurcation diagram with varying delay parameter (\tau=0.1 to 1.0). $\endgroup$ – G Velmurugan Mar 21 '16 at 2:32

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