# Computing Lyapunov Spectrum of the Lorenz Attraktor with Gram Schmidt and variational equation

**URL:** <https://discourse.julialang.org/t/computing-lyapunov-spectrum-of-the-lorenz-attraktor-with-gram-schmidt-and-variational-equation/92575>\
**Category:** General Usage\
**Created:** [January 5, 2023, 9:46pm UTC](https://discourse.julialang.org/t/computing-lyapunov-spectrum-of-the-lorenz-attraktor-with-gram-schmidt-and-variational-equation/92575 "2023-01-05T21:46:07Z")\
**Posts on this page:** 2\
**Page:** 1

<div class="post-metadata">

**Author:** ![physician97](https://avatars.discourse-cdn.com/v4/letter/p/a6a055/32.png) [@physician97](https://discourse.julialang.org/u/physician97)\
**Post date:** [January 5, 2023, 9:46pm UTC](https://discourse.julialang.org/t/computing-lyapunov-spectrum-of-the-lorenz-attraktor-with-gram-schmidt-and-variational-equation/92575/1 "2023-01-05T21:46:07Z")

</div>

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!

```plaintext
    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
```

---

<div class="post-metadata">

**Author:** ![stevengj](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/stevengj/32/71_2.png) [@stevengj](https://discourse.julialang.org/u/stevengj)\
**Post date:** [January 6, 2023, 1:21am UTC](https://discourse.julialang.org/t/computing-lyapunov-spectrum-of-the-lorenz-attraktor-with-gram-schmidt-and-variational-equation/92575/2 "2023-01-06T01:21:22Z")

</div>

> [@physician97](#):
>
> My code is below. I didn’t explain everything but maybe some can find my fault.

I can’t comment on the details of your code, but rather than write your own Runge–Kutta integrator you will be _far_ better off using [DifferentialEquations.jl](https://docs.sciml.ai/DiffEqDocs/stable/), rather than implementing your own Gram–Schmidt orthogonalization you are much better off using [QR factorization](https://en.wikipedia.org/wiki/QR_decomposition) via Julia’s [`qr` function](https://docs.julialang.org/en/v1/stdlib/LinearAlgebra/#LinearAlgebra.qr), and rather than implementing your own algorithm to compute Lyapunov exponents you will likely be better off using the [`lyapunovspectrum` function in DynamicalSystems.jl](https://juliadynamics.github.io/DynamicalSystems.jl/dev/chaos/lyapunovs/).

Even if your educational program requires you to re-implement these algorithms yourself, it would still be a good idea to debug your code by validating it against mature libraries.