We study the paper "Dynamic analysis and control of a new hyperchaotic finance system"
It is given the following system
$$X'=Z+(Y-\alpha)X$$ $$Y'=1-\beta Y-X^2$$ $$Z'=-X-\gamma Z$$ with initial conditions $(X(0),Y(0),Z(0))=(1,2,3)$.
Where
X: interest rate
Υ:investment demand
Z: price index
$\alpha$: savings, $\beta$: cost per investment, $\gamma$: the absolute value of the elasticity of demand
One of the characteristics of systems with chaotic behavior is the so-called sensitive dependence on initial conditions. So we take now $(X(0),Y(0),Z(0))=(1.001,2.001,3.001)$. We notice that the new intial conditions differ in each coordinate by $10^{-3}$. Now I am trying to compare them on Mathematica but the plots it returns are identical. Is this possible?
MakePlot[{α_, β_, γ_}, {a_, b_, c_}, h_,
max_Integer] :=
Module[{u, if},
f[{x_, y_, z_}] := {z + (y - α)*x,
1 - β*y - x^2, -x - γ*z};
u[0] = {a, b, c};
Do[u[n + 1] = u[n] + h*f[u[n] + h/2*f[u[n]]], {n, 0, max}];
if = Interpolation[Table[{n, u[n]}, {n, 0, max/h}]];
Plot[if[t], {t, 0, 200}, PlotLegends -> Automatic,Frame -> True, ImageSize -> Medium]]
plot1 = MakePlot[{0.9, 0.2, 1.2}, {1, 3, 2}, 0.1, 1000 ]
plot2 = MakePlot[{0.9, 0.2, 1.2}, {1.001, 3.001, 2.001}, 0.1, 1000]
Show[{plot1, plot2}]
MakePlot[{0.13, 0.18, 0.87}, {1, 2, 3}, 0.1, 1000]andMakePlot[{0.13, 0.18, 0.87}, {1.001, 2.001, 3.001}, 0.1, 1000]. $\endgroup$