CITE THIS NOTEBOOK: Studying hyperchaotic finance system by Athanasios Paraskevopoulos. Wolfram Community JAN 29 2023.
We study from the paper Yu, H., Cai, G. & Li, Y. Dynamic analysis and control of a new hyperchaotic finance system. Nonlinear Dyn 67, 2171–2182 (2012). https://doi.org/10.1007/s11071-011-0137-9
The following system:
Z+(Y-α)X
1-βY-
-X-γZ
with initial conditions. Where
X: interest rate
Υ:investment demand
Z: price index
α: savings, β: cost per investment, γ: the absolute value of the elasticity of demand.
We solve this problem for different values of γ.
The following system:
′
X
′
Y
2
X
′
Z
with initial conditions
(X(0),Y(0),Z(0))(1,2,3)
X: interest rate
Υ:investment demand
Z: price index
α: savings, β: cost per investment, γ: the absolute value of the elasticity of demand.
We solve this problem for different values of γ.
When γ > 1
When γ > 1
First case
First case
On interval [0, 1000]
On interval [0, 1000]
In[]:=
Clear["Global`*"];
In[]:=
α=0.9;β=0.2;γ=1.2;
In[]:=
f[x_,y_,z_]:=z+(y-α)*xg[x_,y_,z_]:=1-β*y-x^2p[x_,y_,z_]:=-x-γ*z
In[]:=
Q[a_,b_,c_,h_,N_]:=(u[0]=a;v[0]=b;w[0]=c;Do[{u[n+1]=u[n]+h*f[u[n]+h/2*f[u[n],v[n],w[n]],v[n]+h/2*g[u[n],v[n],w[n]],w[n]+h/2*p[u[n],v[n],w[n]]],v[n+1]=v[n]+h*g[u[n]+h/2*f[u[n],v[n],w[n]],v[n]+h/2*g[u[n],v[n],w[n]],w[n]+h/2*p[u[n],v[n],w[n]]],w[n+1]=w[n]+h*p[u[n]+h/2*f[u[n],v[n],w[n]],v[n]+h/2*g[u[n],v[n],w[n]],w[n]+h/2*p[u[n],v[n],w[n]]]},{n,0,N}]);
In[]:=
Q[1,3,2,0.1,1000]
In[]:=
Table[u[n],{n,0,1000,50}]
Out[]=
{1,-0.165616,0.255358,0.0449641,0.243021,0.244539,0.105703,-0.69058,-0.885384,0.753933,-1.38676,1.06902,1.69705,-1.14129,-0.741492,0.913907,0.212107,-0.510878,-0.251239,0.322815,0.0439706}
In[]:=
X=Interpolation[Table[{n,u[n]},{n,0,1000}]]
Out[]=
InterpolatingFunction
In[]:=
Y=Interpolation[Table[{n,v[n]},{n,0,1000}]]
Out[]=
InterpolatingFunction
In[]:=
Z=Interpolation[Table[{n,w[n]},{n,0,1000}]]
Out[]=
InterpolatingFunction
In[]:=
ParametricPlot3D[{X[t],Y[t],Z[t]},{t,0,1000}]
Out[]=
In[]:=
q1=Plot[{X[t],Y[t],Z[t]},{t,0,1000},PlotLegends->"Expressions"]
Out[]=
Second case
Second case
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 initial conditions differ in each coordinate by
We notice that the new initial conditions differ in each coordinate by
-3
10
On interval [0, 200]
On interval [0, 200]
In[]:=
Clear["Global`*"];
In[]:=
α=0.9;β=0.2;γ=1.2;
In[]:=
f[x_,y_,z_]:=z+(y-α)*xg[x_,y_,z_]:=1-β*y-x^2p[x_,y_,z_]:=-x-γ*z
In[]:=
Q[a_,b_,c_,h_,N_]:=(u[0]=a;v[0]=b;w[0]=c;Do[{u[n+1]=u[n]+h*f[u[n]+h/2*f[u[n],v[n],w[n]],v[n]+h/2*g[u[n],v[n],w[n]],w[n]+h/2*p[u[n],v[n],w[n]]],v[n+1]=v[n]+h*g[u[n]+h/2*f[u[n],v[n],w[n]],v[n]+h/2*g[u[n],v[n],w[n]],w[n]+h/2*p[u[n],v[n],w[n]]],w[n+1]=w[n]+h*p[u[n]+h/2*f[u[n],v[n],w[n]],v[n]+h/2*g[u[n],v[n],w[n]],w[n]+h/2*p[u[n],v[n],w[n]]]},{n,0,N}]);
In[]:=
Q[1.001,3.001,2.001,0.1,2000]
In[]:=
Table[u[n],{n,0,1000,50}]
Out[]=
{1.001,-0.165177,0.255074,0.0461871,0.24179,0.247217,0.103822,-0.680329,-0.864597,0.746368,-1.41386,1.07898,1.70806,-1.16241,-0.664439,0.845914,0.318129,-0.601195,-0.124275,0.33363,0.337253}
In[]:=
X=Interpolation[Table[{n,u[n]},{n,0,2000}]]
In[]:=
InterpolatingFunction
Out[]=
InterpolatingFunction
In[]:=
Y=Interpolation[Table[{n,v[n]},{n,0,2000}]]
Out[]=
InterpolatingFunction
In[]:=
Z=Interpolation[Table[{n,w[n]},{n,0,2000}]]
Out[]=
InterpolatingFunction
In[]:=
ParametricPlot3D[{X[t],Y[t],Z[t]},{t,0,200}]
Out[]=
curve=Plot[{X[t],Y[t],Z[t]},{t,0,200},PlotLegends->"Expressions"]
Out[]=
Now we compare the first the second case
Now we compare the first the second case
On interval [0, 1000]
On interval [0, 1000]
On interval [0, 200]
On interval [0, 200]
We notice that when we choose the interval we cannot distinguish the differences so for our study we have to choose larger intervals
When γ < 1
When γ < 1
Third Case
Third Case
On interval[0, 200]
We choose now (X (0), Y (0), Z (0)) = (1.001, 2.001, 3.001) and we compare the two cases
We notice that the system is not sensitive dependence on initial conditions .