Implement solution of differential equations system using exponential matrix in Julia

Question

I try to reproduce the example, in Julia, that I show in the figure and taken from Matrix Exponentiation

I show you how far I have managed to reproduce the exercise in Julia. But I don't know how to introduce the vector t, for a range of interest, for example t = -3: 0.25: 3. in the matrix: [exp (u1 * t 0; 0 exp (u2 * t], u1 u2 eigenvalues.

Julia>A=[0 1;1 0]
2×2 Array{Int64,2}:
 0  1
 1  0

F=eigen(A)
Eigen{Float64,Float64,Array{Float64,2},Array{Float64,1}}
values:
2-element Array{Float64,1}:
 -1.0
  1.0
vectors:
2×2 Array{Float64,2}:
 -0.707107  0.707107
  0.707107  0.707107

D = diagm(exp.(F.values))
2×2 Array{Float64,2}:
 0.367879  0.0
 0.0       2.71828

P = F.vectors
13:06:08->>2×2 Array{Float64,2}:
 -0.707107  0.707107
  0.707107  0.707107

Answer 1

On the site Fabian Dablander code is shown codes in R that implement the solution. These are the scripts brought to Julia:

using Plots
using LinearAlgebra

#Solving differential equations using matrix exponentials
A=[-0.20 -1;1 0] #[-0.40 -1;1 0.45] A=[0 1;1 0]
x0=[1 1]# [1 1] x0=[0.25 0.25] x0=[1 0]
tmax=20
n=1000
ts=LinRange(0,tmax,n)
x = Array{Float64}(undef, 0, 0)
x=x0
for i in 1:n
x=vcat(x,x0*exp(A*ts[i]))
end
plot(x)
plot(x[:,1],x[:,2])


#Solving differential equations finding eigenvalues and eigenvectors
A=[-0.20 -1;1 0] #A=[-0.40 -1;1 0.45] A=[0 1;1 0]
x0=[1 1]# [1 1] x0=[0.25 0.25] x0=[1 0]
tmax=20
n=500
# compute eigenvectors and eigenvalues
  eig = eigen(A)
  E =eig.vectors
  λ=eig.values
# solve for the initial conditon
C =E\x0'
# create time steps
ts=LinRange(0,tmax,n)
x = Array{Float64}(undef,n, size(A,2))
for i in 1:n
   x[i,:]=real.(C'*diagm(exp.(λ*ts[i]))*E)
 end
plot(x)
plot(x[:,1],x[:,2])