When γ > 1

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)*x​​g[x_,y_,z_]:=1-β*y-x^2​​p[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
Domain: 0.,1.00×
3
10

Output: scalar

In[]:=
Y=Interpolation[Table[{n,v[n]},{n,0,1000}]]
Out[]=
InterpolatingFunction
Domain: 0.,1.00×
3
10

Output: scalar

In[]:=
Z=Interpolation[Table[{n,w[n]},{n,0,1000}]]
Out[]=
InterpolatingFunction
Domain: 0.,1.00×
3
10

Output: scalar

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[]=
200
400
600
800
1000
-2
-1
1
2
3
X(t)
Y(t)
Z(t)

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)*x​​g[x_,y_,z_]:=1-β*y-x^2​​p[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}]]
Out[]=
InterpolatingFunction
Domain: 0.,2.00×
3
10

Output: scalar

In[]:=
Y=Interpolation[Table[{n,v[n]},{n,0,2000}]]
Out[]=
InterpolatingFunction
Domain: 0.,2.00×
3
10

Output: scalar

In[]:=
Z=Interpolation[Table[{n,w[n]},{n,0,2000}]]
Out[]=
InterpolatingFunction
Domain: 0.,2.00×
3
10

Output: scalar

In[]:=
ParametricPlot3D[{X[t],Y[t],Z[t]},{t,0,2000}]
Out[]=
In[]:=
Plot[{X[t],Y[t],Z[t]},{t,0,200},PlotLegends->"Expressions"]
Out[]=
50
100
150
200
-2
-1
1
2
3
X(t)
Y(t)
Z(t)
When γ < 1
oninterval[0,200]
In[]:=
Clear["Global`*"];
In[]:=
α=0.9;​​β=0.2;​​γ=0.4;
In[]:=
f[x_,y_,z_]:=z+(y-α)*x​​g[x_,y_,z_]:=1-β*y-x^2​​p[x_,y_,z_]:=-x-γ*z​​