Converting 1D map into first order differential equation for bifurcation diagram code

I have existing code to plot bifurcation diagram for coupled first order differential equations. It seems to work for coupled differential equations. To test the code further, I use the simplest case which is a logistic map. However, it is in iteration format. Therefore, it needs to be converted to first order differential equation form. The plot shows me incorrect results. I still want to use the same code because I can use it for coupled differential equations. I do not understand why it does not work for single differential equation.

 Remove[x, data1, dsol, xt];

LogisticODE[r_] := x'[t] - r*x[t] (1 - x[t]) + x[t] == 0;

solution2[r_] :=
NDSolve[{LogisticODE[r], x[0.] == 0.5}, x, {t, 0, tmax},
MaxSteps -> Infinity];

(* Bifurcation Plot *)

tmax = 1000 T; tmin = tmax - 50 T;
w = 7;
T = 2 Pi/w;

data1 = {};

For[A1 = 0.0, A1 <= 4.0, A1 += 0.01,
dsol = solution2[A1];
xt = x[t] /. dsol[[1]];
For[t = tmin, t <= tmax, t += T, AppendTo[data1, {A1, N[xt]}]];
Clear[t]] // AbsoluteTiming

{8.58766, Null}

ListPlot[data1, PlotRange -> Automatic, Frame -> True, Axes -> False,
PlotStyle -> {PointSize[0.006]}]


I have tried $$x'[t]- r x[t] (1 - x[t])$$ and $$x'[t]- (r x[t] (1 - x[t])-x[t])/T^2$$ But still, I do not get a proper bifurction diagram for logistic map because I believe that transformation from map to ODE is incorrect or code need to change slightly for single ODE equation. Why I am not getting a correct bifurcation diagram? What I am missing?

• The code will be different depending on what you want -- the bifurcation diagram for the logistic equation (differential equation) or logistic map (difference equation). As it is, this looks like the correct diagram for the logistic equation with density-independent mortality, $dx/dt=rx(1-x)-x$, which you've got encoded in LogisticODE[r]. Nov 16 '21 at 0:59
• How can one have different bifurcation diagram for the same equation? Both differential and difference represent the same equation but in different format. Then how can one get different results? Nov 16 '21 at 14:35
• The differential and difference equations are related but not the same. One is $dx/dt=rx(1-x)$, the other is $x_{t+1}=x_t+rx_t(1-x_t)$ (or something like it). They have the same equilibria, but the stability differs. Assuming $r>0$, the differential equation has an unstable equilibrium at $x=0$ and a stable one at $x=1$. The difference equation is much more complicated, because the positive equilibrium can undergo a series of period-doubling bifurcations leading to chaos, which is impossible in the differential equation. Nov 16 '21 at 14:44
• see abel.harvard.edu/archive/118r_spring_05/docs/may.pdf for more info Nov 16 '21 at 14:45