# Plotting a Bifurcation diagram

I have the following system equation

v'(t)=2*G*J1[v(t-τ)]cos(w*τ)-v(t)


How do you plot the bifurcation diagram, τ in the x axis, Vmax in the y axis? I have written these lines but how can one plot using the following

Table[NDSolve[{v'[t] ==
2*G*BesselJ[1, v[t - τ + i]]*Cos[ω*(τ + i)] -
v[t], v[0] == 0.001}, v, {t, 0, 500}], {i, 0, 4, 0.01}]


τ is varied from 1 to 4 using step 0.01,G=3.55, ω=2*Pi*12*10^6

• Some numerical values (\Tau, \Omega) missing?? Oct 2, 2015 at 6:34
• Please include all definitions for symbols, so that the code runs without errors when copy-pasted into a fresh notebook (the definition for G is also missing, as well as tau and omega). Oct 2, 2015 at 6:43
• Related documentation example. Oct 2, 2015 at 7:00
• Oct 2, 2015 at 10:03
• do you mean tau + i is varied..? (you realise you have tau + i one place and tau - i another) Oct 2, 2015 at 11:40

## 4 Answers

An alternative representation is

G = 3.55; ω = 2*Pi*12*10^6;
s = ParametricNDSolveValue[{v'[t] == 2*G*BesselJ[1, v[t - τ]] Cos[ω*τ] - v[t],
v[t /; t <= 0] == 0.001}, {v, v'}, {t, 0, 120}, {τ}];
Manipulate[ParametricPlot[{s[τ][[1]][t], s[τ][[2]][t]}, {t, 60, 120},
AxesLabel -> {v, v'}, AspectRatio -> 1], {{τ, 2}, 1, 4}]


Note that the diagram becomes progressively more complex as τ is increased, and the run time increases correspondingly.

Addendum

The bifurcations can be seen even more clearly from a return map, for instance,

tab = Table[{sol, points} = Reap@NDSolveValue[{v'[t] ==
2*G*BesselJ[1, v[t - τ]] Cos[ω*τ] - v[t], v[t /; t <= 0] == 0.001,
WhenEvent[v'[t] > 0, If[t > 150, Sow[v[t]]]]}, {v, v'}, {t, 0, 250}];
{τ, #} & /@ Union[Flatten[points], SameTest -> (Abs[#1 - #2] < .05 &)],
{τ, 1.7, 2.4, .01}];
ListPlot[Flatten[tab, 1]]


where v is sampled whenever v' passes from positive to negative values. A blow-up of the map near the transition to chaos is (with SameTest deleted)

It is anyone's guess precisely where the transition to chaos occurs. Perhaps, very near τ = 2.32.

Additional Material in Response to Comments

Recent comments by udichi, the OP, and by Chris K prompted me to consider this problem further. Stability windows typically occur within the chaotic region, udichi now wanted to see them. A straightforward three-hour computation produced interesting results, but no windows. (Note that WorkingPrecision -> 30 is used to reduce the chance that numerical inaccuracies might corrupt the results.)

tab = ParallelTable[{sol, points} =
Reap@NDSolveValue[{v'[t] == 2*G*BesselJ[1, v[t - τ]] Cos[ω*τ] - v[t],
v[t /; t <= 0] == 10^-3, WhenEvent[v'[t] > 0, If[t > 500, Sow[v[t]]]]}, {v, v'},
{t, 0, 1000}, WorkingPrecision -> 30, MaxSteps -> 10^6]; {τ, #} & /@
Union[Flatten[points]], {τ, 1, 15, 1/100}];
ListPlot[Flatten[tab, 1], AspectRatio -> .75/GoldenRatio,
ImageSize -> Full, PlotStyle -> PointSize[Tiny]]


Here are diagrams for interesting values of τ. Typical plots for τ > 8 are

f[τ_] := Module[{},
ss = NDSolveValue[{v'[t] == 2*G*BesselJ[1, v[t - τ]] Cos[ω*τ] - v[t],
v[t /; t <= 0] == 10^-3}, {v, v'}, {t, 0, 1000},
WorkingPrecision -> 30, MaxSteps -> 10^6];
GraphicsRow[{ParametricPlot[Through[ss[t]], {t, 500, 1000},
AxesLabel -> {v[t], v'[t]}, AspectRatio -> 1, PlotPoints -> 200],
ParametricPlot[First[ss][#] & /@ {t, t - τ}, {t, 500, 1000},
AxesLabel -> {v[t], v[t - τ]}, AspectRatio -> 1, PlotPoints -> 200]},
ImageSize -> Large]]

f[15]


The left plot depicts v' vs. v, similar to some of the earlier plots although much more chaotic. The solution appears to move randomly between two chaotic attractors. The right plot depicts v[t - τ] vs. v[t], as suggested here. The advantage of this alternative representation will soon become evident. Typical plots from the transition region, centered around τ == 7, are

f[15/2]


while typical plots from smaller but chaotic values of τ look much different.

f[3]


Finally, plots for τ = 2.285, the approximate onset of chaos (as determined by Chris K) are

Plots for τ as large as 2.4 are qualitively similar, although obviously chaotic. This suggests computing a return map based on v[t - τ] == 2.5.

tab = ParallelTable[{sol, points} =
Reap@NDSolveValue[{v'[t] == 2*G*BesselJ[1, v[t - τ]] - v[t],
v[0] == 10^-3, tem[0] == 1500, WhenEvent[v[t] > 5/2, tem[t] -> t],
WhenEvent[t > tem[t] + τ, If[t > 1500, Sow[v[t]]]]}, {v[t], tem[t]}, {t, 0, 2200},
DiscreteVariables -> {tem}, WorkingPrecision -> 30, MaxSteps -> 10^6];
{τ, #} & /@ Flatten[points], {τ, 225/100, 240/100, 1/2000}];
ListPlot[Flatten[tab, 1], AspectRatio -> .75/GoldenRatio, ImageSize -> Full,
PlotStyle -> PointSize[Tiny]]


It shows the transition to chaos (at about τ = 2.286) as well as the first three windows of stability within the region of chaos. Note that a comparatively long run-time in t is necessary to allow solutions near bifurcation points to reach asymptotic states. High resolution in τ is, of course, also needed. Incidentally, this last computation throws the warning message described in the second section of question 157889, but it can be ignored.

Plots in Windows of Stability

As suggested by Chris K, it may be useful to provide plots in the three windows of stability shown in the last figure.

f[2303/1000]


f[2330/1000]


f[2348/1000]


These plots differ strikingly from their chaotic neighbors, say τ == 3, above.

• The x-axis is τ (tau), and the y-axis is the value of v at which v' passes from positive to negative values. Oct 3, 2015 at 1:17
• @Udichi Use the 'PlotRange' option. Apr 7, 2017 at 18:14
• @Udichi Please refer to the documentation of Plot or ListPlot for how to use PlotRange. By the way, if any of the answers above met your needs, please accept that answer. Best wishes. Apr 7, 2017 at 18:41
• @Udichi Using a new criterion for the return map produces the plot that I think you have been seeking, complete with four levels of bifurcation and three windows of stability. By the way your model may be similar to the Maclay-Glass model at smaller values of τ. Oct 16, 2017 at 22:31
• @bbgodfrey I'd +1 your "Additional Material in Response to Comments" if it were possible. Nice job! I did take a look at the first periodic window you found and verified that τ = 2.302 is periodic with period 75.13 and τ = 2.303 has period 150.33. Oct 16, 2017 at 22:37

Please note that though cosmetically appealing this is not rigorous or the best insight into basin of attraction of system as pointed out by bbgodfrey in comment below. I leave it as, perhaps, a road to refinement after exploring the behaviour of equilibrium points in system and apologise for any misconceptions.

Inspired by bbgodfrey's answer (which I have upvoted) but not as clean (but fun...1 am brain: no guarantees):

Using:

G = 3.55; ω = 2*Pi*12*10^6;
s = ParametricNDSolveValue[{v'[t] == 2*G*BesselJ[1, v[t - τ]] Cos[ω*τ] - v[t],
v[t /; t <= 0] == 0.001}, {v, v'}, {t, 0, 120}, {τ}];


then

coll = {};
Table[pp =
ParametricPlot[Through[s[a][t]], {t, 0, 120},
PlotRange -> {{0, 4}, {-2, 2}}, PerformanceGoal -> "Quality",
MeshFunctions -> (#2 &), Mesh -> {{0.}},
MeshStyle -> {Red, PointSize[0.02]}];
pts = pp[[1, 1]];
AppendTo[
coll, {a, #[[1]]} & /@
pts[[First@Cases[pp[[1]], Point[x__] :> x, -1]]]];
, {a, 0.2, 3, 0.01}];
lp = ListPlot[Join @@ coll, Frame -> True, PlotStyle -> Red];
Manipulate[
Column[{
ParametricPlot[Through[s[par][t]], {t, 0, 120},
PlotRange -> {{0, 4}, {-2, 2}}, PerformanceGoal -> "Quality",
MeshFunctions -> (#2 &), Mesh -> {{0.}},
MeshStyle -> {Red, PointSize[0.02]}, Frame -> True,
FrameLabel -> {"v[t]", "v'[t]"}],
Show[lp, Graphics[{Gray, Line[{{par, 0}, {par, 4}}]}]]
}], {par, 0.2, 3, 0.01}]


• Your second plot, although very attractive, contains many points that are not on the attractor, especially for τ < 1. Oct 5, 2015 at 3:38
• @bbgodfrey yes you are absolutely correct...I should have been cleaner as you were in longer term behaviour and deleting duplicates (relating to numerical issues)...I think you have made a very important point and if I have time I may run as you did late behaviour...i guess I wanted to illustrate the equilibrium points and the beginning...but laziness and time...I will make a comment...I may not have time to come back to...not near a computer at present... Oct 5, 2015 at 4:10

This could be "close" to what you are looking for

G = 3.55;

ω = 2*Pi*12*10^6;

Manipulate[
sol =
First[
NDSolve[{
Derivative[1][v][t] == 2*G*BesselJ[1, v[t - τ]]*Cos[ω*τ] - v[t],
v[t /; t <= 0] == 0.001},
v, {t, 0, 50}]];
Plot[Evaluate[{v[t]} /. sol], {t, 0, 50}],
{{τ, 1.5}, 1, 4}]


Just increase t = 50 to t = 500, and adjust the tau value accordingly.

The effect of G and \Omega on the bifurcation point can also be seen using

 \[Omega] = Pi*12*10^6;

Manipulate[
sol = First[
NDSolve[{Derivative[1][v][t] ==
2*G*BesselJ[1, v[t - \[Tau]]]*Cos[\[Omega]*a*\[Tau]] - v[t],
v[t /; t <= 0] == 0.001}, v, {t, 0, 150}]];
ParametricPlot[
Evaluate[{v[t], Derivative[1][v][t]} /. sol], {t, 0,
150}], {{\[Tau], 1.5}, 1, 4},
{{G, 3.5}, 3, 4}, {{a, 2}, 1, 3}]


The factor a is linked to \Omega, and the graph shows the sensitivity of G

This is an extended comment on the discussion under @bbgodfrey's answer about period finding and period doubling.

How about finding the period by storing local maxima of v, then looking back to find when the last maximum is repeated.

G = 3.55; ω = 2*Pi*12*10^6;
tmax = 4000;

τ = 1.7;

maxs = {};
sol = NDSolve[{v'[t] == 2*G*BesselJ[1, v[t - τ]] Cos[ω*τ] - v[t],
v[t /; t <= 0] == 0.001, WhenEvent[v'[t] < 0, PrependTo[maxs, {t, v[t]}]]},
v, {t, 0, tmax}][[1]];

Plot[v[t] /. sol, {t, tmax - 100, tmax}]

(* find period *)
SelectFirst[Map[maxs[[1]] - # &, maxs[[2 ;;]]], Abs[#[[2]]] < 10^-6 &][[1]]


4.83605

Past the first period doubling:

τ = 1.8;


10.1394

Notice when "shoulder" maximum appears around v[t]=3 the period doesn't jump:

τ = 2.0;


11.1417

Past the second period doubling:

τ = 2.2;


24.1686

Another period doubling:

τ = 2.28;


49.6606

And one more, starting to pile up in τ:

τ = 2.285;


99.493

Note that the last two period doublings are hard to see from the time series, but looking at maxs gives me some confidence that they're accurate.