Lorenz system

 ( Chaotic Maps ) 

1. parameters

1. initial values

 X = 1,
 Y = 1,
 Z = 1
     

Lorenz <- function (t, vars, prm) {

 with(as.list(vars), {
   dX <- prm$sigma*(Y - X)
   dY <- X*(prm$rho - Z) - Y
   dZ <- X*Y - prm$beta*Z
   return(list(c(dX, dY, dZ)))
  })
     

times <- seq(from = 0, to = 100, by = 0.01)

1. call ode solver

          parms = prm)
     

1. to assign color to points

 rep(repArr, ceiling(long/length(repArr)))[1:long]
     

dout <- as.data.frame(out)

dout$color <- gfill(rainbow(10), nrow(dout))

1. Graphics production with Plotly:

 data=dout, x = ~X, y = ~Y, z = ~Z,
 type = 'scatter3d', mode = 'lines',
 opacity = 1, line = list(width = 6, color = ~color, reverscale = FALSE)
     

1. we solve the Lorenz system of the differential equations.
2. Runge-Kutta's method y_{n+1}= y_n + h*(k_1 + 2*k_2+2*k_3+k_4)/6; x_{n+1}=x_n+h
3. k_1=f(x_n,y_n), k_2=f(x_n+h/2, y_n+hk_1/2), k_3=f(x_n+h/2, y_n+hk_2/2), k_4=f(x_n+h, y_n+hk_3)
4. differential equation

   tlist = [a+i*h for i in range(n+1)]
   y = [[0,0,0] for _ in range(n+1)]
   # Taking length of f (number of equations).
   m=len(f)
   # Number of variables in v.
   vm=len(v)
   if m!=vm:
       return("error, number of equations is not equal with the number of variables.")
   for r in range(vm):
       y[0][r]=b[r]
   # making a vector and component will be a list
   # main part of the algorithm
   k1=[0 for _ in range(m)]
   k2=[0 for _ in range(m)]
   k3=[0 for _ in range(m)]
   k4=[0 for _ in range(m)]
   for i in range(1,n+1): # for each t_i, i=1, ... , n
       # k1=h*f(t_{i-1},x_1(t_{i-1}),...,x_m(t_{i-1}))
       for j in range(m): # for each f_{j+1}, j=0, ... , m-1
           k1[j]=f[j].subs(t==tlist[i-1])
           for r in range(vm):
               k1[j]=k1[j].subs(v[r]==y[i-1][r])
           k1[j]=h*k1[j]
       for j in range(m): # k2=h*f(t_{i-1}+h/2,x_1(t_{i-1})+k1/2,...,x_m(t_{i-1}+k1/2))
           k2[j]=f[j].subs(t==tlist[i-1]+h/2)
           for r in range(vm):
               k2[j]=k2[j].subs(v[r]==y[i-1][r]+k1[r]/2)
           k2[j]=h*k2[j]
       for j in range(m): # k3=h*f(t_{i-1}+h/2,x_1(t_{i-1})+k2/2,...,x_m(t_{i-1})+k2/2)
           k3[j]=f[j].subs(t==tlist[i-1]+h/2)
           for r in range(vm):
               k3[j]=k3[j].subs(v[r]==y[i-1][r]+k2[r]/2)
           k3[j]=h*k3[j]
       for j in range(m): # k4=h*f(t_{i-1}+h,x_1(t_{i-1})+k3,...,x_m(t_{i-1})+k3)
           k4[j]=f[j].subs(t==tlist[i-1]+h)
           for r in range(vm):
               k4[j]=k4[j].subs(v[r]==y[i-1][r]+k3[r])
           k4[j]=h*k4[j]
       for j in range(m): # Now x_j(t_i)=x_j(t_{i-1})+(k1+2k2+2k3+k4)/6
           y[i][j]=y[i-1][j]+(k1[j]+2*k2[j]+2*k3[j]+k4[j])/6
   return(tlist,y)
     

1. (Figure 1) Here, we plot the solutions of the Lorenz ODE system.

1. print(tlist)
2. print(y)

1. (Figure 2) Here, we plot every y1, y2, and y3 in terms of time.

1. Runge_Kutta(f,v,0,b,h,n)
2. print(tlist)
3. print(y)

1. (Figure 3) Here, we plot the y and x or equivalently y2 and y1

1. print(tlist)
2. print(y)

1. (Figure 4) Here, we plot the z and x or equivalently y3 and y1

1. print(tlist)
2. print(y)

1. (Figure 5) Here, we plot the z and x or equivalently y3 and y2

1. print(tlist)
2. print(y)

The partial differential equations modeling the system's stream function and temperature are subjected to a spectral Galerkin method: the hydrodynamic fields are expanded in Fourier series, which are then severely truncated to a single term for the stream function and two terms for the temperature. This reduces the model equations to a set of three coupled, Nonlinear system ordinary differential equations. A detailed derivation may be found, for example, in nonlinear dynamics texts from , Appendix C; , Appendix D; or Shen (2016), Supplementary Materials.