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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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  • $\begingroup$ 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. $\endgroup$ Commented Mar 20, 2016 at 8:56
  • $\begingroup$ a_1=10, a_2=28, a_3=8/3, \alpha=1, That is ordinary delay differential equations. $\endgroup$ Commented Mar 20, 2016 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. $\endgroup$ Commented Mar 20, 2016 at 12:57
  • $\begingroup$ This question was put here last year already once $\endgroup$ Commented Mar 20, 2016 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). $\endgroup$ Commented Mar 21, 2016 at 2:32

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