Hello,
I’m new to this forum. I’m studying physics at Cologne university and for my thesis I want to calculate Lyapunovexponents of several systems. I choosed the Lorentz Attraktor at first to program the way to calculate the Lyapunovexponent and to check if my result is ok.
I tried to use a method by using the variational equation by integrating this equation and using Gram-Schmidt orthonormalization.
My code is below. I didn’t explain everything but maybe some can find my fault.
Thank you very much!
x=X[1]
y=X[2]
z=X[3]
x_=10*(y-x) #16#10
y_=x*(28-z)-y#45.92,28
z_=x*y-8/3*z #4#8/3
return([x_,y_,z_])
end
function rungestep_1(X,h)
#t[i+1]=t[i]+h
K1=lorentz(X)
K2=lorentz(X+h*K1/2)
K3=lorentz(X+h*K2/2)
K4=lorentz(X+h*K3)
return(X+h/6*(K1+2*K2+2*K3+K4))
end
function RungeKutta_2(x0,v_0,N,M,h)
t=zeros(N)
P=zeros(9,N)
P[:,1]=v_0
T=zeros(9,N)
z=M*N
B=zeros(3,z)
R=zeros(9,z)
G=zeros(9,N)
R[:,1]=v_0
G[:,1]=v_0
B[:,1]=x0
s=0
L=zeros(N)
for u in 1:N-3
β=8/3
σ=10
ρ=28
B[:,u+1]=rungestep_1(B[:,u],h)
for i in 1:M
# B[:,s+i+1]=rungestep_1(B[:,i+s],h)
x=B[1,u]
y=B[2,u]
z=B[3,u]
if i==1
R[:,i+s]=rungestep_2(G[:,u],h,x,y,z,σ,ρ,β)
elseif i>1
R[:,i+s]=rungestep_2(R[:,i+s-1],h,x,y,z,σ,ρ,β)
end
end
r=M*u
g=R[:,r]
l=gramschmidt2(matrix(g))[2]
P[:,u+1]=vector(gramschmidt2(matrix(g))[1])
G[:,u+1]=vector(l)
L[u+1]=log(norm( [P[2,u],P[5,u],P[8,u] ]))
s=s+M
end
return sum(L)/(N*π/100)
end
function gramschmidt2(M)
v1=M[:,1]
v2=M[:,2]
v3=M[:,3]
w1=v1
e1=w1/norm(w1)
w2=v2-proj(e1,v2)
e2=w2/norm(w2)
w3=v3-proj(e1,v3)-proj(e2,v3)
e3=w3/norm(w3)
u=[ [w1[1],w2[1],w3[1]] [w1[2],w2[2],w3[2]] [w1[3],w2[3],w3[3]] ]
e=[ [e1[1],e2[1],e3[1]] [e1[2],e2[2],e3[2]] [e1[3],e2[3],e3[3]] ]
return u,e
end