Following your comment, I have now amended this code to incorporate the parameter estimates you suggested.
T = 20; c = 10;
y = ParametricNDSolveValue[{Derivative[1][x][t] ==
a*(x[t - T]/(1 + x[t - T]^c)) - x[t]*b,
x[t /; t <= 0] == 0.1}, {x, Derivative[1][x]}, {t, 0, 500}, {a,
b}];
Manipulate[
ParametricPlot[{y[a, b][[1]][t], y[a, b][[2]][t]}, {t, 0, 500},
AxesLabel -> {x, Derivative[1][x]}, AspectRatio -> 1], {{a, 0.1},
0.05, 0.5}, {{b, 0.1}, 0.05, 0.3}]
The x-axis is x[t]
and the y-axis s x'[t]
and there appears to be oscillations at select range of a and b
values.
What is the suitable value for T? The code below may help
Clear[T, a, b, c]
Manipulate[
Module[{a = 0.2, b = 0.1, c = 10, eqs},
eqs = NDSolve[{Derivative[1][x][t] ==
a*(x[t - T]/(1 + x[t - T]^c)) - x[t]*b,
x[t /; t <= 0] == 0.1}, x,
{t, 0, 500}];
Plot[Evaluate[{x[t], Derivative[1][x][t]}] /. eqs, {t, 0, 500},
PlotRange -> All]], {{T, 15}, 10, 50}]
The Lyapunov exponent is a measure of sensitive dependence on conditions at t=0
and is calculated based on how rapidly two nearby states diverge from each other. So evaluation of this exponent depends on the parameter that you have selected. I just show the evaluation of the Lyapunov Exponent for a simpler case where it exist, such as f(x)=a x(b-x)
where a
and b
are parameters. I set b
=1.
lya[a_, b_] :=
Module[{f, ly}, f[x_] := a*x*(b - x);
ly[x_] := Log[Abs[a*(b - 2*x)]]; Mean[ly[NestList[f, 0.05, 100]]]];
Show[Plot[lya[a, 1], {a, 0, 4}, PlotTheme -> "Scientific",
PlotStyle -> Thickness[0.01], PlotRange -> {-3, 1.5}],
ListPlot[
ParallelTable[
Thread[{a, Nest[a*#1*(1 - #1) & , Range[0, 1, 0.01], 500]}], {a,
0, 4, 0.01}]]]
Notice that there are locations where nearby points diverge and the lyapunov exponent swings downwards. This code can be modified to determine the lyapunov
exponent for your model (provided you fix all known parameters except the one you wish to test for sensitivity of evolving states). No doubt there are alternative approaches to determine the lyapunov exponent for higher dimensional cases. I have just shown the one-dimensional case here.