Error with a system of coupled ODEs with complex coefficients

General Usage
, ,
1

I was trying to solve a system of coupled differential equations with imaginary coefficients and I ran into a weird trouble.

MWE:

using Symbolics, Plots, DifferentialEquations, ModelingToolkit, LaTeXStrings, Measures, Distributions

@variables z a(z) A(z) B(z)
@parameters Δₐ, Δₘ, Δₚ, Δᵣ, α
d = Differential(z)
E = [d(a) ~ -im*(Δₐ*a + Δₘ*A),
     d(A) ~ -im*(Δₘ*a + im*α*B + Δₚ*A),
     d(B) ~ -im*(-im*α*A + Δᵣ*B)] 

@named model = ODESystem(E, z, [a, A, B], [Δₐ, Δₘ, Δₚ, Δᵣ, α])

The code runs till this last line, running which gives the following error:

MethodError: no method matching ODESystem(::Vector{Any}, ::Num, ::Vector{Num}, ::Vector{Num}; name::Symbol)

Closest candidates are:
  ODESystem(::Any, ::Any, ::Any, ::Any, ::Any, ::Any, ::Any, ::Any, ::Any, ::Any, ::Any, ::Any, ::Any, ::Any, ::Any, ::Any, ::Any, ::Any, ::Any, ::Any, ::Any, ::Any) got unsupported keyword argument "name"
   @ ModelingToolkit C:\Users\shiha\.julia\packages\ModelingToolkit\GFEA2\src\systems\diffeqs\odesystem.jl:143
  ODESystem(::Any, ::Any, ::Any, ::Any, ::Any, ::Any, ::Any, ::Any, ::Any, ::Any, ::Any, ::Any, ::Any, ::Any, ::Any, ::Any, ::Any, ::Any, ::Any, ::Any, ::Any, ::Any, ::Any) got unsupported keyword argument "name"
   @ ModelingToolkit C:\Users\shiha\.julia\packages\ModelingToolkit\GFEA2\src\systems\diffeqs\odesystem.jl:143
  ODESystem(::Any, ::Any, ::Any, ::Any, ::Any, ::Any, ::Any, ::Any, ::Any, ::Any, ::Any, ::Any, ::Any, ::Any, ::Any, ::Any, ::Any, ::Any, ::Any, ::Any, ::Any, ::Any, ::Any, ::Any) got unsupported keyword argument "name"
   @ ModelingToolkit C:\Users\shiha\.julia\packages\ModelingToolkit\GFEA2\src\systems\diffeqs\odesystem.jl:143
  ...


Stacktrace:
 [1] top-level scope
   @ C:\Users\shiha\.julia\packages\ModelingToolkit\GFEA2\src\systems\abstractsystem.jl:1006

I was at a loss at first since I have run similar codes for various other problems (with real coefficients) and I never had this issue - I noticed that if I modify the definition of E and remove all the im from the equations, then the code runs fine and I get a solution.

Any help will be appreciated!

2

I don’t think ModelingToolkit.jl is going to work with Complex like this right now. Check the issues list.

1 Like
3

So, there’s no way of solving this in Julia? Unless of course, I make it a 6 dimensional system and separate the real and imaginary parts by hand?

4

You can use DifferentialEquations.jl. It’s just ModelingToolkit that doesn’t handle complex like this right now.

1 Like
5

Ah, I wasn’t aware of that - sorry - and thanks for the help!

Unfortunately, this results in a new problem. The current MWE is:

using Symbolics, Plots, DifferentialEquations, LaTeXStrings, Measures, Distributions
function HM(du, u, p, z)
    Δₐ, Δₘ, Δₚ, Δᵣ, α = p 
    du[1] = -im*(Δₐ*u[1] + Δₘ*u[2])
    du[2] = -im*(Δₘ*u[1] + im*α*u[3] + Δₚ*u[2])
    du[3] = -im*(-im*α*u[2] + Δᵣ*u[3])
end
p0 = [1.5e-2, 3e-2, 0, 0, rand(-π:π)]
u0 = [0, 1/√2, 1/√2]
prob = ODEProblem(HM, u0, (0, 100), p0)
Sol = solve(prob)

This results in an error (the message is too long so I have attached it separately).

Not sure where the problem is - am I supposed to define the u[] as taking on complex values somehow in the function definition? If so, how do I do this? I am not very familiar with solving complex differential equations numerically.

Thanks again!

6

The error message:

InexactError: Float64(0.0 - 0.021213203435596423im)

Stacktrace:
  [1] Real
    @ .\complex.jl:44 [inlined]
  [2] convert
    @ .\number.jl:7 [inlined]
  [3] setindex!
    @ .\array.jl:969 [inlined]
  [4] HM(du::Vector{Float64}, u::Vector{Float64}, p::Vector{Float64}, z::Float64)
    @ Main .\In[23]:5
  [5] Void
    @ C:\Users\shiha\.julia\packages\SciMLBase\s9wrq\src\utils.jl:473 [inlined]
  [6] (::FunctionWrappers.CallWrapper{Nothing})(f::SciMLBase.Void{typeof(HM)}, arg1::Vector{Float64}, arg2::Vector{Float64}, arg3::Vector{Float64}, arg4::Float64)
    @ FunctionWrappers C:\Users\shiha\.julia\packages\FunctionWrappers\Q5cBx\src\FunctionWrappers.jl:65
  [7] macro expansion
    @ C:\Users\shiha\.julia\packages\FunctionWrappers\Q5cBx\src\FunctionWrappers.jl:137 [inlined]
  [8] do_ccall
    @ C:\Users\shiha\.julia\packages\FunctionWrappers\Q5cBx\src\FunctionWrappers.jl:125 [inlined]
  [9] FunctionWrapper
    @ C:\Users\shiha\.julia\packages\FunctionWrappers\Q5cBx\src\FunctionWrappers.jl:144 [inlined]
 [10] _call
    @ C:\Users\shiha\.julia\packages\FunctionWrappersWrappers\9XR0m\src\FunctionWrappersWrappers.jl:12 [inlined]
 [11] FunctionWrappersWrapper
    @ C:\Users\shiha\.julia\packages\FunctionWrappersWrappers\9XR0m\src\FunctionWrappersWrappers.jl:10 [inlined]
 [12] ODEFunction
    @ C:\Users\shiha\.julia\packages\SciMLBase\s9wrq\src\scimlfunctions.jl:2126 [inlined]
 [13] initialize!(integrator::OrdinaryDiffEq.ODEIntegrator{CompositeAlgorithm{Tuple{Tsit5{typeof(OrdinaryDiffEq.trivial_limiter!), typeof(OrdinaryDiffEq.trivial_limiter!), Static.False}, Rosenbrock23{1, false, LinearSolve.DefaultLinearSolver, typeof(OrdinaryDiffEq.DEFAULT_PRECS), Val{:forward}, true, nothing}}, OrdinaryDiffEq.AutoSwitchCache{Tsit5{typeof(OrdinaryDiffEq.trivial_limiter!), typeof(OrdinaryDiffEq.trivial_limiter!), Static.False}, Rosenbrock23{0, false, Nothing, typeof(OrdinaryDiffEq.DEFAULT_PRECS), Val{:forward}, true, nothing}, Rational{Int64}, Int64}}, true, Vector{Float64}, Nothing, Float64, Vector{Float64}, Float64, Float64, Float64, Float64, Vector{Vector{Float64}}, ODESolution{Float64, 2, Vector{Vector{Float64}}, Nothing, Nothing, Vector{Float64}, Vector{Vector{Vector{Float64}}}, ODEProblem{Vector{Float64}, Tuple{Float64, Float64}, true, Vector{Float64}, ODEFunction{true, SciMLBase.AutoSpecialize, FunctionWrappersWrappers.FunctionWrappersWrapper{Tuple{FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Float64}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{Float64}, Float64}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{Float64}, Vector{Float64}, ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{Float64}, ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}}}, false}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing}, Base.Pairs{Symbol, Union{}, Tuple{}, NamedTuple{(), Tuple{}}}, SciMLBase.StandardODEProblem}, CompositeAlgorithm{Tuple{Tsit5{typeof(OrdinaryDiffEq.trivial_limiter!), typeof(OrdinaryDiffEq.trivial_limiter!), Static.False}, Rosenbrock23{1, false, LinearSolve.DefaultLinearSolver, typeof(OrdinaryDiffEq.DEFAULT_PRECS), Val{:forward}, true, nothing}}, OrdinaryDiffEq.AutoSwitchCache{Tsit5{typeof(OrdinaryDiffEq.trivial_limiter!), typeof(OrdinaryDiffEq.trivial_limiter!), Static.False}, Rosenbrock23{0, false, Nothing, typeof(OrdinaryDiffEq.DEFAULT_PRECS), Val{:forward}, true, nothing}, Rational{Int64}, Int64}}, OrdinaryDiffEq.CompositeInterpolationData{ODEFunction{true, SciMLBase.AutoSpecialize, FunctionWrappersWrappers.FunctionWrappersWrapper{Tuple{FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Float64}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{Float64}, Float64}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{Float64}, Vector{Float64}, ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{Float64}, ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}}}, false}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing}, Vector{Vector{Float64}}, Vector{Float64}, Vector{Vector{Vector{Float64}}}, OrdinaryDiffEq.CompositeCache{Tuple{OrdinaryDiffEq.Tsit5Cache{Vector{Float64}, Vector{Float64}, Vector{Float64}, typeof(OrdinaryDiffEq.trivial_limiter!), typeof(OrdinaryDiffEq.trivial_limiter!), Static.False}, OrdinaryDiffEq.Rosenbrock23Cache{Vector{Float64}, Vector{Float64}, Vector{Float64}, Matrix{Float64}, Matrix{Float64}, OrdinaryDiffEq.Rosenbrock23Tableau{Float64}, SciMLBase.TimeGradientWrapper{ODEFunction{true, SciMLBase.AutoSpecialize, FunctionWrappersWrappers.FunctionWrappersWrapper{Tuple{FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Float64}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{Float64}, Float64}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{Float64}, Vector{Float64}, ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{Float64}, ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}}}, false}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing}, Vector{Float64}, Vector{Float64}}, SciMLBase.UJacobianWrapper{ODEFunction{true, SciMLBase.AutoSpecialize, FunctionWrappersWrappers.FunctionWrappersWrapper{Tuple{FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Float64}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{Float64}, Float64}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{Float64}, Vector{Float64}, ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{Float64}, ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}}}, false}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing}, Float64, Vector{Float64}}, LinearSolve.LinearCache{Matrix{Float64}, Vector{Float64}, Vector{Float64}, SciMLBase.NullParameters, LinearSolve.DefaultLinearSolver, LinearSolve.DefaultLinearSolverInit{LinearAlgebra.LU{Float64, Matrix{Float64}, Vector{Int64}}, LinearAlgebra.QRCompactWY{Float64, Matrix{Float64}, Matrix{Float64}}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, LinearAlgebra.LU{Float64, Matrix{Float64}, Vector{Int64}}, Tuple{LinearAlgebra.LU{Float64, Matrix{Float64}, Vector{Int64}}, Vector{Int64}}, Nothing, Nothing, Nothing, LinearAlgebra.SVD{Float64, Float64, Matrix{Float64}, Vector{Float64}}, LinearAlgebra.Cholesky{Float64, Matrix{Float64}}, LinearAlgebra.Cholesky{Float64, Matrix{Float64}}}, LinearSolve.InvPreconditioner{LinearAlgebra.Diagonal{Float64, Vector{Float64}}}, LinearAlgebra.Diagonal{Float64, Vector{Float64}}, Float64, Bool}, FiniteDiff.JacobianCache{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, UnitRange{Int64}, Nothing, Val{:forward}(), Float64}, FiniteDiff.GradientCache{Nothing, Vector{Float64}, Vector{Float64}, Float64, Val{:forward}(), Float64, Val{true}()}, Float64, Rosenbrock23{1, false, LinearSolve.DefaultLinearSolver, typeof(OrdinaryDiffEq.DEFAULT_PRECS), Val{:forward}, true, nothing}, Nothing}}, OrdinaryDiffEq.AutoSwitchCache{Tsit5{typeof(OrdinaryDiffEq.trivial_limiter!), typeof(OrdinaryDiffEq.trivial_limiter!), Static.False}, Rosenbrock23{0, false, Nothing, typeof(OrdinaryDiffEq.DEFAULT_PRECS), Val{:forward}, true, nothing}, Rational{Int64}, Int64}}}, DiffEqBase.Stats, Vector{Int64}}, ODEFunction{true, SciMLBase.AutoSpecialize, FunctionWrappersWrappers.FunctionWrappersWrapper{Tuple{FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Float64}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{Float64}, Float64}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{Float64}, Vector{Float64}, ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{Float64}, ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}}}, false}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing}, OrdinaryDiffEq.CompositeCache{Tuple{OrdinaryDiffEq.Tsit5Cache{Vector{Float64}, Vector{Float64}, Vector{Float64}, typeof(OrdinaryDiffEq.trivial_limiter!), typeof(OrdinaryDiffEq.trivial_limiter!), Static.False}, OrdinaryDiffEq.Rosenbrock23Cache{Vector{Float64}, Vector{Float64}, Vector{Float64}, Matrix{Float64}, Matrix{Float64}, OrdinaryDiffEq.Rosenbrock23Tableau{Float64}, SciMLBase.TimeGradientWrapper{ODEFunction{true, SciMLBase.AutoSpecialize, FunctionWrappersWrappers.FunctionWrappersWrapper{Tuple{FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Float64}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{Float64}, Float64}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{Float64}, Vector{Float64}, ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{Float64}, ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}}}, false}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing}, Vector{Float64}, Vector{Float64}}, SciMLBase.UJacobianWrapper{ODEFunction{true, SciMLBase.AutoSpecialize, FunctionWrappersWrappers.FunctionWrappersWrapper{Tuple{FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Float64}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{Float64}, Float64}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{Float64}, Vector{Float64}, ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{Float64}, ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}}}, false}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing}, Float64, Vector{Float64}}, LinearSolve.LinearCache{Matrix{Float64}, Vector{Float64}, Vector{Float64}, SciMLBase.NullParameters, LinearSolve.DefaultLinearSolver, LinearSolve.DefaultLinearSolverInit{LinearAlgebra.LU{Float64, Matrix{Float64}, Vector{Int64}}, LinearAlgebra.QRCompactWY{Float64, Matrix{Float64}, Matrix{Float64}}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, LinearAlgebra.LU{Float64, Matrix{Float64}, Vector{Int64}}, Tuple{LinearAlgebra.LU{Float64, Matrix{Float64}, Vector{Int64}}, Vector{Int64}}, Nothing, Nothing, Nothing, LinearAlgebra.SVD{Float64, Float64, Matrix{Float64}, Vector{Float64}}, LinearAlgebra.Cholesky{Float64, Matrix{Float64}}, LinearAlgebra.Cholesky{Float64, Matrix{Float64}}}, LinearSolve.InvPreconditioner{LinearAlgebra.Diagonal{Float64, Vector{Float64}}}, LinearAlgebra.Diagonal{Float64, Vector{Float64}}, Float64, Bool}, FiniteDiff.JacobianCache{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, UnitRange{Int64}, Nothing, Val{:forward}(), Float64}, FiniteDiff.GradientCache{Nothing, Vector{Float64}, Vector{Float64}, Float64, Val{:forward}(), Float64, Val{true}()}, Float64, Rosenbrock23{1, false, LinearSolve.DefaultLinearSolver, typeof(OrdinaryDiffEq.DEFAULT_PRECS), Val{:forward}, true, nothing}, Nothing}}, OrdinaryDiffEq.AutoSwitchCache{Tsit5{typeof(OrdinaryDiffEq.trivial_limiter!), typeof(OrdinaryDiffEq.trivial_limiter!), Static.False}, Rosenbrock23{0, false, Nothing, typeof(OrdinaryDiffEq.DEFAULT_PRECS), Val{:forward}, true, nothing}, Rational{Int64}, Int64}}, OrdinaryDiffEq.DEOptions{Float64, Float64, Float64, Float64, PIController{Rational{Int64}}, typeof(DiffEqBase.ODE_DEFAULT_NORM), typeof(LinearAlgebra.opnorm), Nothing, CallbackSet{Tuple{}, Tuple{}}, typeof(DiffEqBase.ODE_DEFAULT_ISOUTOFDOMAIN), typeof(DiffEqBase.ODE_DEFAULT_PROG_MESSAGE), typeof(DiffEqBase.ODE_DEFAULT_UNSTABLE_CHECK), DataStructures.BinaryHeap{Float64, DataStructures.FasterForward}, DataStructures.BinaryHeap{Float64, DataStructures.FasterForward}, Nothing, Nothing, Int64, Tuple{}, Tuple{}, Tuple{}}, Vector{Float64}, Float64, Nothing, OrdinaryDiffEq.DefaultInit}, cache::OrdinaryDiffEq.Tsit5Cache{Vector{Float64}, Vector{Float64}, Vector{Float64}, typeof(OrdinaryDiffEq.trivial_limiter!), typeof(OrdinaryDiffEq.trivial_limiter!), Static.False})
    @ OrdinaryDiffEq C:\Users\shiha\.julia\packages\OrdinaryDiffEq\xaB2l\src\perform_step\low_order_rk_perform_step.jl:766
7

Second part of the error message:

 [14] initialize!(integrator::OrdinaryDiffEq.ODEIntegrator{CompositeAlgorithm{Tuple{Tsit5{typeof(OrdinaryDiffEq.trivial_limiter!), typeof(OrdinaryDiffEq.trivial_limiter!), Static.False}, Rosenbrock23{1, false, LinearSolve.DefaultLinearSolver, typeof(OrdinaryDiffEq.DEFAULT_PRECS), Val{:forward}, true, nothing}}, OrdinaryDiffEq.AutoSwitchCache{Tsit5{typeof(OrdinaryDiffEq.trivial_limiter!), typeof(OrdinaryDiffEq.trivial_limiter!), Static.False}, Rosenbrock23{0, false, Nothing, typeof(OrdinaryDiffEq.DEFAULT_PRECS), Val{:forward}, true, nothing}, Rational{Int64}, Int64}}, true, Vector{Float64}, Nothing, Float64, Vector{Float64}, Float64, Float64, Float64, Float64, Vector{Vector{Float64}}, ODESolution{Float64, 2, Vector{Vector{Float64}}, Nothing, Nothing, Vector{Float64}, Vector{Vector{Vector{Float64}}}, ODEProblem{Vector{Float64}, Tuple{Float64, Float64}, true, Vector{Float64}, ODEFunction{true, SciMLBase.AutoSpecialize, FunctionWrappersWrappers.FunctionWrappersWrapper{Tuple{FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Float64}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{Float64}, Float64}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{Float64}, Vector{Float64}, ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{Float64}, ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}}}, false}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing}, Base.Pairs{Symbol, Union{}, Tuple{}, NamedTuple{(), Tuple{}}}, SciMLBase.StandardODEProblem}, CompositeAlgorithm{Tuple{Tsit5{typeof(OrdinaryDiffEq.trivial_limiter!), typeof(OrdinaryDiffEq.trivial_limiter!), Static.False}, Rosenbrock23{1, false, LinearSolve.DefaultLinearSolver, typeof(OrdinaryDiffEq.DEFAULT_PRECS), Val{:forward}, true, nothing}}, OrdinaryDiffEq.AutoSwitchCache{Tsit5{typeof(OrdinaryDiffEq.trivial_limiter!), typeof(OrdinaryDiffEq.trivial_limiter!), Static.False}, Rosenbrock23{0, false, Nothing, typeof(OrdinaryDiffEq.DEFAULT_PRECS), Val{:forward}, true, nothing}, Rational{Int64}, Int64}}, OrdinaryDiffEq.CompositeInterpolationData{ODEFunction{true, SciMLBase.AutoSpecialize, FunctionWrappersWrappers.FunctionWrappersWrapper{Tuple{FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Float64}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{Float64}, Float64}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{Float64}, Vector{Float64}, ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{Float64}, ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}}}, false}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing}, Vector{Vector{Float64}}, Vector{Float64}, Vector{Vector{Vector{Float64}}}, OrdinaryDiffEq.CompositeCache{Tuple{OrdinaryDiffEq.Tsit5Cache{Vector{Float64}, Vector{Float64}, Vector{Float64}, typeof(OrdinaryDiffEq.trivial_limiter!), typeof(OrdinaryDiffEq.trivial_limiter!), Static.False}, OrdinaryDiffEq.Rosenbrock23Cache{Vector{Float64}, Vector{Float64}, Vector{Float64}, Matrix{Float64}, Matrix{Float64}, OrdinaryDiffEq.Rosenbrock23Tableau{Float64}, SciMLBase.TimeGradientWrapper{ODEFunction{true, SciMLBase.AutoSpecialize, FunctionWrappersWrappers.FunctionWrappersWrapper{Tuple{FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Float64}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{Float64}, Float64}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{Float64}, Vector{Float64}, ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{Float64}, ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}}}, false}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing}, Vector{Float64}, Vector{Float64}}, SciMLBase.UJacobianWrapper{ODEFunction{true, SciMLBase.AutoSpecialize, FunctionWrappersWrappers.FunctionWrappersWrapper{Tuple{FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Float64}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{Float64}, Float64}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{Float64}, Vector{Float64}, ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{Float64}, ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}}}, false}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing}, Float64, Vector{Float64}}, LinearSolve.LinearCache{Matrix{Float64}, Vector{Float64}, Vector{Float64}, SciMLBase.NullParameters, LinearSolve.DefaultLinearSolver, LinearSolve.DefaultLinearSolverInit{LinearAlgebra.LU{Float64, Matrix{Float64}, Vector{Int64}}, LinearAlgebra.QRCompactWY{Float64, Matrix{Float64}, Matrix{Float64}}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, LinearAlgebra.LU{Float64, Matrix{Float64}, Vector{Int64}}, Tuple{LinearAlgebra.LU{Float64, Matrix{Float64}, Vector{Int64}}, Vector{Int64}}, Nothing, Nothing, Nothing, LinearAlgebra.SVD{Float64, Float64, Matrix{Float64}, Vector{Float64}}, LinearAlgebra.Cholesky{Float64, Matrix{Float64}}, LinearAlgebra.Cholesky{Float64, Matrix{Float64}}}, LinearSolve.InvPreconditioner{LinearAlgebra.Diagonal{Float64, Vector{Float64}}}, LinearAlgebra.Diagonal{Float64, Vector{Float64}}, Float64, Bool}, FiniteDiff.JacobianCache{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, UnitRange{Int64}, Nothing, Val{:forward}(), Float64}, FiniteDiff.GradientCache{Nothing, Vector{Float64}, Vector{Float64}, Float64, Val{:forward}(), Float64, Val{true}()}, Float64, Rosenbrock23{1, false, LinearSolve.DefaultLinearSolver, typeof(OrdinaryDiffEq.DEFAULT_PRECS), Val{:forward}, true, nothing}, Nothing}}, OrdinaryDiffEq.AutoSwitchCache{Tsit5{typeof(OrdinaryDiffEq.trivial_limiter!), typeof(OrdinaryDiffEq.trivial_limiter!), Static.False}, Rosenbrock23{0, false, Nothing, typeof(OrdinaryDiffEq.DEFAULT_PRECS), Val{:forward}, true, nothing}, Rational{Int64}, Int64}}}, DiffEqBase.Stats, Vector{Int64}}, ODEFunction{true, SciMLBase.AutoSpecialize, FunctionWrappersWrappers.FunctionWrappersWrapper{Tuple{FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Float64}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{Float64}, Float64}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{Float64}, Vector{Float64}, ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{Float64}, ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}}}, false}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing}, OrdinaryDiffEq.CompositeCache{Tuple{OrdinaryDiffEq.Tsit5Cache{Vector{Float64}, Vector{Float64}, Vector{Float64}, typeof(OrdinaryDiffEq.trivial_limiter!), typeof(OrdinaryDiffEq.trivial_limiter!), Static.False}, OrdinaryDiffEq.Rosenbrock23Cache{Vector{Float64}, Vector{Float64}, Vector{Float64}, Matrix{Float64}, Matrix{Float64}, OrdinaryDiffEq.Rosenbrock23Tableau{Float64}, SciMLBase.TimeGradientWrapper{ODEFunction{true, SciMLBase.AutoSpecialize, FunctionWrappersWrappers.FunctionWrappersWrapper{Tuple{FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Float64}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{Float64}, Float64}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{Float64}, Vector{Float64}, ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{Float64}, ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}}}, false}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing}, Vector{Float64}, Vector{Float64}}, SciMLBase.UJacobianWrapper{ODEFunction{true, SciMLBase.AutoSpecialize, FunctionWrappersWrappers.FunctionWrappersWrapper{Tuple{FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Float64}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{Float64}, Float64}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{Float64}, Vector{Float64}, ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{Float64}, ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}}}, false}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing}, Float64, Vector{Float64}}, LinearSolve.LinearCache{Matrix{Float64}, Vector{Float64}, Vector{Float64}, SciMLBase.NullParameters, LinearSolve.DefaultLinearSolver, LinearSolve.DefaultLinearSolverInit{LinearAlgebra.LU{Float64, Matrix{Float64}, Vector{Int64}}, LinearAlgebra.QRCompactWY{Float64, Matrix{Float64}, Matrix{Float64}}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, LinearAlgebra.LU{Float64, Matrix{Float64}, Vector{Int64}}, Tuple{LinearAlgebra.LU{Float64, Matrix{Float64}, Vector{Int64}}, Vector{Int64}}, Nothing, Nothing, Nothing, LinearAlgebra.SVD{Float64, Float64, Matrix{Float64}, Vector{Float64}}, LinearAlgebra.Cholesky{Float64, Matrix{Float64}}, LinearAlgebra.Cholesky{Float64, Matrix{Float64}}}, LinearSolve.InvPreconditioner{LinearAlgebra.Diagonal{Float64, Vector{Float64}}}, LinearAlgebra.Diagonal{Float64, Vector{Float64}}, Float64, Bool}, FiniteDiff.JacobianCache{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, UnitRange{Int64}, Nothing, Val{:forward}(), Float64}, FiniteDiff.GradientCache{Nothing, Vector{Float64}, Vector{Float64}, Float64, Val{:forward}(), Float64, Val{true}()}, Float64, Rosenbrock23{1, false, LinearSolve.DefaultLinearSolver, typeof(OrdinaryDiffEq.DEFAULT_PRECS), Val{:forward}, true, nothing}, Nothing}}, OrdinaryDiffEq.AutoSwitchCache{Tsit5{typeof(OrdinaryDiffEq.trivial_limiter!), typeof(OrdinaryDiffEq.trivial_limiter!), Static.False}, Rosenbrock23{0, false, Nothing, typeof(OrdinaryDiffEq.DEFAULT_PRECS), Val{:forward}, true, nothing}, Rational{Int64}, Int64}}, OrdinaryDiffEq.DEOptions{Float64, Float64, Float64, Float64, PIController{Rational{Int64}}, typeof(DiffEqBase.ODE_DEFAULT_NORM), typeof(LinearAlgebra.opnorm), Nothing, CallbackSet{Tuple{}, Tuple{}}, typeof(DiffEqBase.ODE_DEFAULT_ISOUTOFDOMAIN), typeof(DiffEqBase.ODE_DEFAULT_PROG_MESSAGE), typeof(DiffEqBase.ODE_DEFAULT_UNSTABLE_CHECK), DataStructures.BinaryHeap{Float64, DataStructures.FasterForward}, DataStructures.BinaryHeap{Float64, DataStructures.FasterForward}, Nothing, Nothing, Int64, Tuple{}, Tuple{}, Tuple{}}, Vector{Float64}, Float64, Nothing, OrdinaryDiffEq.DefaultInit}, cache::OrdinaryDiffEq.CompositeCache{Tuple{OrdinaryDiffEq.Tsit5Cache{Vector{Float64}, Vector{Float64}, Vector{Float64}, typeof(OrdinaryDiffEq.trivial_limiter!), typeof(OrdinaryDiffEq.trivial_limiter!), Static.False}, OrdinaryDiffEq.Rosenbrock23Cache{Vector{Float64}, Vector{Float64}, Vector{Float64}, Matrix{Float64}, Matrix{Float64}, OrdinaryDiffEq.Rosenbrock23Tableau{Float64}, SciMLBase.TimeGradientWrapper{ODEFunction{true, SciMLBase.AutoSpecialize, FunctionWrappersWrappers.FunctionWrappersWrapper{Tuple{FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Float64}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{Float64}, Float64}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{Float64}, Vector{Float64}, ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{Float64}, ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}}}, false}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing}, Vector{Float64}, Vector{Float64}}, SciMLBase.UJacobianWrapper{ODEFunction{true, SciMLBase.AutoSpecialize, FunctionWrappersWrappers.FunctionWrappersWrapper{Tuple{FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Float64}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{Float64}, Float64}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{Float64}, Vector{Float64}, ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{Float64}, ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}}}, false}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing}, Float64, Vector{Float64}}, LinearSolve.LinearCache{Matrix{Float64}, Vector{Float64}, Vector{Float64}, SciMLBase.NullParameters, LinearSolve.DefaultLinearSolver, LinearSolve.DefaultLinearSolverInit{LinearAlgebra.LU{Float64, Matrix{Float64}, Vector{Int64}}, LinearAlgebra.QRCompactWY{Float64, Matrix{Float64}, Matrix{Float64}}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, LinearAlgebra.LU{Float64, Matrix{Float64}, Vector{Int64}}, Tuple{LinearAlgebra.LU{Float64, Matrix{Float64}, Vector{Int64}}, Vector{Int64}}, Nothing, Nothing, Nothing, LinearAlgebra.SVD{Float64, Float64, Matrix{Float64}, Vector{Float64}}, LinearAlgebra.Cholesky{Float64, Matrix{Float64}}, LinearAlgebra.Cholesky{Float64, Matrix{Float64}}}, LinearSolve.InvPreconditioner{LinearAlgebra.Diagonal{Float64, Vector{Float64}}}, LinearAlgebra.Diagonal{Float64, Vector{Float64}}, Float64, Bool}, FiniteDiff.JacobianCache{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, UnitRange{Int64}, Nothing, Val{:forward}(), Float64}, FiniteDiff.GradientCache{Nothing, Vector{Float64}, Vector{Float64}, Float64, Val{:forward}(), Float64, Val{true}()}, Float64, Rosenbrock23{1, false, LinearSolve.DefaultLinearSolver, typeof(OrdinaryDiffEq.DEFAULT_PRECS), Val{:forward}, true, nothing}, Nothing}}, OrdinaryDiffEq.AutoSwitchCache{Tsit5{typeof(OrdinaryDiffEq.trivial_limiter!), typeof(OrdinaryDiffEq.trivial_limiter!), Static.False}, Rosenbrock23{0, false, Nothing, typeof(OrdinaryDiffEq.DEFAULT_PRECS), Val{:forward}, true, nothing}, Rational{Int64}, Int64}})
    @ OrdinaryDiffEq C:\Users\shiha\.julia\packages\OrdinaryDiffEq\xaB2l\src\perform_step\composite_perform_step.jl:51
 [15] __init(prob::ODEProblem{Vector{Float64}, Tuple{Float64, Float64}, true, Vector{Float64}, ODEFunction{true, SciMLBase.AutoSpecialize, FunctionWrappersWrappers.FunctionWrappersWrapper{Tuple{FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Float64}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{Float64}, Float64}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{Float64}, Vector{Float64}, ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{Float64}, ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}}}, false}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing}, Base.Pairs{Symbol, Union{}, Tuple{}, NamedTuple{(), Tuple{}}}, SciMLBase.StandardODEProblem}, alg::CompositeAlgorithm{Tuple{Tsit5{typeof(OrdinaryDiffEq.trivial_limiter!), typeof(OrdinaryDiffEq.trivial_limiter!), Static.False}, Rosenbrock23{1, false, LinearSolve.DefaultLinearSolver, typeof(OrdinaryDiffEq.DEFAULT_PRECS), Val{:forward}, true, nothing}}, AutoSwitch{Tsit5{typeof(OrdinaryDiffEq.trivial_limiter!), typeof(OrdinaryDiffEq.trivial_limiter!), Static.False}, Rosenbrock23{0, false, Nothing, typeof(OrdinaryDiffEq.DEFAULT_PRECS), Val{:forward}, true, nothing}, Rational{Int64}, Int64}}, timeseries_init::Tuple{}, ts_init::Tuple{}, ks_init::Tuple{}, recompile::Type{Val{true}}; saveat::Tuple{}, tstops::Tuple{}, d_discontinuities::Tuple{}, save_idxs::Nothing, save_everystep::Bool, save_on::Bool, save_start::Bool, save_end::Nothing, callback::Nothing, dense::Bool, calck::Bool, dt::Float64, dtmin::Nothing, dtmax::Float64, force_dtmin::Bool, adaptive::Bool, gamma::Rational{Int64}, abstol::Nothing, reltol::Nothing, qmin::Rational{Int64}, qmax::Int64, qsteady_min::Int64, qsteady_max::Int64, beta1::Nothing, beta2::Nothing, qoldinit::Rational{Int64}, controller::Nothing, fullnormalize::Bool, failfactor::Int64, maxiters::Int64, internalnorm::typeof(DiffEqBase.ODE_DEFAULT_NORM), internalopnorm::typeof(LinearAlgebra.opnorm), isoutofdomain::typeof(DiffEqBase.ODE_DEFAULT_ISOUTOFDOMAIN), unstable_check::typeof(DiffEqBase.ODE_DEFAULT_UNSTABLE_CHECK), verbose::Bool, timeseries_errors::Bool, dense_errors::Bool, advance_to_tstop::Bool, stop_at_next_tstop::Bool, initialize_save::Bool, progress::Bool, progress_steps::Int64, progress_name::String, progress_message::typeof(DiffEqBase.ODE_DEFAULT_PROG_MESSAGE), userdata::Nothing, allow_extrapolation::Bool, initialize_integrator::Bool, alias_u0::Bool, alias_du0::Bool, initializealg::OrdinaryDiffEq.DefaultInit, kwargs::Base.Pairs{Symbol, Bool, Tuple{Symbol, Symbol}, NamedTuple{(:default_set, :second_time), Tuple{Bool, Bool}}})
    @ OrdinaryDiffEq C:\Users\shiha\.julia\packages\OrdinaryDiffEq\xaB2l\src\solve.jl:499
 [16] __init (repeats 5 times)
    @ C:\Users\shiha\.julia\packages\OrdinaryDiffEq\xaB2l\src\solve.jl:10 [inlined]
 [17] #__solve#648
    @ C:\Users\shiha\.julia\packages\OrdinaryDiffEq\xaB2l\src\solve.jl:5 [inlined]
 [18] __solve(::ODEProblem{Vector{Float64}, Tuple{Float64, Float64}, true, Vector{Float64}, ODEFunction{true, SciMLBase.AutoSpecialize, FunctionWrappersWrappers.FunctionWrappersWrapper{Tuple{FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Float64}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{Float64}, Float64}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{Float64}, Vector{Float64}, ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{Float64}, ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}}}, false}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing}, Base.Pairs{Symbol, Union{}, Tuple{}, NamedTuple{(), Tuple{}}}, SciMLBase.StandardODEProblem}, ::Nothing; default_set::Bool, kwargs::Base.Pairs{Symbol, Bool, Tuple{Symbol}, NamedTuple{(:second_time,), Tuple{Bool}}})
    @ DifferentialEquations C:\Users\shiha\.julia\packages\DifferentialEquations\9GdKy\src\default_solve.jl:14
 [19] __solve
    @ C:\Users\shiha\.julia\packages\DifferentialEquations\9GdKy\src\default_solve.jl:1 [inlined]
 [20] #__solve#49
    @ C:\Users\shiha\.julia\packages\DiffEqBase\tVuJP\src\solve.jl:1247 [inlined]
 [21] __solve
    @ C:\Users\shiha\.julia\packages\DiffEqBase\tVuJP\src\solve.jl:1240 [inlined]
 [22] #solve_call#22
    @ C:\Users\shiha\.julia\packages\DiffEqBase\tVuJP\src\solve.jl:511 [inlined]
 [23] solve_call(::ODEProblem{Vector{Float64}, Tuple{Float64, Float64}, true, Vector{Float64}, ODEFunction{true, SciMLBase.AutoSpecialize, FunctionWrappersWrappers.FunctionWrappersWrapper{Tuple{FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Float64}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{Float64}, Float64}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{Float64}, Vector{Float64}, ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}}, FunctionWrappers.FunctionWrapper{Nothing, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}, Vector{Float64}, ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}}}, false}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing}, Base.Pairs{Symbol, Union{}, Tuple{}, NamedTuple{(), Tuple{}}}, SciMLBase.StandardODEProblem})
    @ DiffEqBase C:\Users\shiha\.julia\packages\DiffEqBase\tVuJP\src\solve.jl:481
 [24] solve_up(::ODEProblem{Vector{Float64}, Tuple{Int64, Int64}, true, Vector{Float64}, ODEFunction{true, SciMLBase.AutoSpecialize, typeof(HM), LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing}, Base.Pairs{Symbol, Union{}, Tuple{}, NamedTuple{(), Tuple{}}}, SciMLBase.StandardODEProblem}, ::Nothing, ::Vector{Float64}, ::Vector{Float64}; kwargs::Base.Pairs{Symbol, Union{}, Tuple{}, NamedTuple{(), Tuple{}}})
    @ DiffEqBase C:\Users\shiha\.julia\packages\DiffEqBase\tVuJP\src\solve.jl:976
 [25] solve_up
    @ C:\Users\shiha\.julia\packages\DiffEqBase\tVuJP\src\solve.jl:945 [inlined]
 [26] solve(::ODEProblem{Vector{Float64}, Tuple{Int64, Int64}, true, Vector{Float64}, ODEFunction{true, SciMLBase.AutoSpecialize, typeof(HM), LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing}, Base.Pairs{Symbol, Union{}, Tuple{}, NamedTuple{(), Tuple{}}}, SciMLBase.StandardODEProblem}; sensealg::Nothing, u0::Nothing, p::Nothing, wrap::Val{true}, kwargs::Base.Pairs{Symbol, Union{}, Tuple{}, NamedTuple{(), Tuple{}}})
    @ DiffEqBase C:\Users\shiha\.julia\packages\DiffEqBase\tVuJP\src\solve.jl:882
 [27] solve(::ODEProblem{Vector{Float64}, Tuple{Int64, Int64}, true, Vector{Float64}, ODEFunction{true, SciMLBase.AutoSpecialize, typeof(HM), LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing}, Base.Pairs{Symbol, Union{}, Tuple{}, NamedTuple{(), Tuple{}}}, SciMLBase.StandardODEProblem})
    @ DiffEqBase C:\Users\shiha\.julia\packages\DiffEqBase\tVuJP\src\solve.jl:872
 [28] top-level scope
    @ In[28]:2
8

It’s saying that it cannot convert your derivatives to Float64, because du is Float64. du is Float64 since u0, your state, was defined as Float64.

julia> u0 = [0, 1/√2, 1/√2]
3-element Vector{Float64}:
 0.0
 0.7071067811865475
 0.7071067811865475

julia> prob = ODEProblem(HM, u0, (0, 100), p0)
ODEProblem with uType Vector{Float64} and tType Int64. In-place: true
timespan: (0, 100)
u0: 3-element Vector{Float64}:
 0.0
 0.7071067811865475
 0.7071067811865475

You instead want your state vector to be complex numbers, so make it complex numbers and it works. Thus the following works:

using DifferentialEquations
function HM(du, u, p, z)
    Δₐ, Δₘ, Δₚ, Δᵣ, α = p 
    du[1] = -im*(Δₐ*u[1] + Δₘ*u[2])
    du[2] = -im*(Δₘ*u[1] + im*α*u[3] + Δₚ*u[2])
    du[3] = -im*(-im*α*u[2] + Δᵣ*u[3])
end
p0 = [1.5e-2, 3e-2, 0, 0, rand(-π:π)]
u0 = [0+0im, 1/√2, 1/√2]
prob = ODEProblem(HM, u0, (0, 100), p0)
Sol = solve(prob)
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