# Using DynamicalSystems: InexactError

**URL:** <https://discourse.julialang.org/t/using-dynamicalsystems-inexacterror/83965>\
**Category:** General Usage\
**Tags:** question\
**Created:** [July 8, 2022, 6:38pm UTC](https://discourse.julialang.org/t/using-dynamicalsystems-inexacterror/83965 "2022-07-08T18:38:23Z")\
**Posts on this page:** 3\
**Page:** 1

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**Author:** ![Griffith\_Mendonca](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/griffith_mendonca/32/37789_2.png) [@Griffith\_Mendonca](https://discourse.julialang.org/u/Griffith_Mendonca)\
**Post date:** [July 8, 2022, 6:38pm UTC](https://discourse.julialang.org/t/using-dynamicalsystems-inexacterror/83965/1 "2022-07-08T18:38:23Z")

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Dear all, I have a question about the meaning of some output messages from the package DynamicalSystems from [https://juliadynamics.github.io/DynamicalSystems.jl/latest/chaos/lyapunovs/](https://juliadynamics.github.io/DynamicalSystems.jl/latest/chaos/lyapunovs/).  
This error message appears when I tried to study a dynamical system with four equations Please find below the very small program and the messages. I would like to obtain the Lyapunov exponent.

Bests regards,

> using DynamicalSystems # also exports relevant StaticArrays names  
> @inline @inbounds function loop(u, p, t)  
> nU = p[1]; nW = p[2];

```
du1=-(-((u[1]*(-nW*(1 + u[1]*u[2]) - 2*nW*u[3]*u[4]))/( 2*(1 + u[1]*u[2] + u[3]*u[4]))) + ( 1im*(u[1]^2*u[2]*(nU + u[3]) - 
u[3]*(1 + u[3]*u[4]) + (-1 + u[1]^2)*(1 + u[1]*u[2] + u[3]*u[4]) +u[1]*(-nU + u[3]*(1 - u[2] + u[3]*u[4]))))/(1 + u[1]*u[2] + u[3]*u[4]))
du2=-((1im*(u[1]*u[2]^2*(nU + u[4]) - u[4]*(1 + u[3]*u[4]) + (-1 + u[2]^2)*(1 + u[1]*u[2] + u[3]*u[4]) + u[2]*(-nU + u[4]*(1 - u[1]+u[3]*u[4]))))/(1+u[1]*u[2]+ 

```

u[3]_u[4]))  
du3=-((nW_u[3]_(1 + u[3]u[4]))/(2(1 + u[1]u[2] + u[3]u[4])) - 1im(1 + u[1] - u[3]^2 + u[3](-u[1] + (nU - nU_u[3]_u[4])/(1 + u[1]u[2] + u[3]u[4]))))  
du4=-((nWu[4](1 + u[3]u[4]))/(2(1 + u[1]u[2] + u[3]u[4])) + 1im(1 + u[2] - u[4]^2 + u[4](-u[2] + (nU - nU_u[3]\*u[4])/(1 + u[1]\*u[2] + u[3]\*u[4]))))  
return SVector{4}(du1, du2, du3, du4)  
end

# Jacobian:

> @inline @inbounds function loop\_jac(u, p, t)  
> nU = p[1]; nW = p[2];  
> J= [(-nW \* (u[1] \* u[2] + 1) - 2 \* nW \* u[3] \* u[4]) / (2 \* u[1] \* u[2] + 2 \* u[3] \* u[4] + 2) - u[1] \* nW \* u[2] / (2 \* u[1] \* u[2] + 2 \* u[3] \* u[4] + 2) - 2 \* u[1] \* (-nW \* (u[1] \* u[2] + 1) - 2 \* nW \* u[3] \* u[4]) / (2 \* u[1] \* u[2] + 2 \* u[3] \* u[4] + 2) ^ 2 \* u[2] + -1im \* (2 \* u[1] \* u[2] \* (nU + u[3]) + 2 \* u[1] \* (u[1] \* u[2] + u[3] \* u[4] + 1) + (u[1] ^ 2 - 1) \* u[2] - nU + u[3] \* (u[3] \* u[4] - u[2] + 1)) / (u[1] \* u[2] + u[3] \* u[4] + 1) + 1im \* (u[1] ^ 2 \* u[2] \* (nU + u[3]) - u[3] \* (u[3] \* u[4] + 1) + (u[1] ^ 2 - 1) \* (u[1] \* u[2] + u[3] \* u[4] + 1) + u[1] \* (-nU + u[3] \* (u[3] \* u[4] - u[2] + 1))) / (u[1] \* u[2] + u[3] \* u[4] + 1) ^ 2 \* u[2] -u[1] ^ 2 \* nW / (2 \* u[1] \* u[2] + 2 \* u[3] \* u[4] + 2) - 2 \* u[1] ^ 2 \* (-nW \* (u[1] \* u[2] + 1) - 2 \* nW \* u[3] \* u[4]) / (2 \* u[1] \* u[2] + 2 \* u[3] \* u[4] + 2) ^ 2 + -1im \* (u[1] ^ 2 \* (nU + u[3]) + (u[1] ^ 2 - 1) \* u[1] - u[1] \* u[3]) / (u[1] \* u[2] + u[3] \* u[4] + 1) + 1im \* (u[1] ^ 2 \* u[2] \* (nU + u[3]) - u[3] \* (u[3] \* u[4] + 1) + (u[1] ^ 2 - 1) \* (u[1] \* u[2] + u[3] \* u[4] + 1) + u[1] \* (-nU + u[3] \* (u[3] \* u[4] - u[2] + 1))) / (u[1] \* u[2] + u[3] \* u[4] + 1) ^ 2 \* u[1] -2 \* u[1] \* nW \* u[4] / (2 \* u[1] \* u[2] + 2 \* u[3] \* u[4] + 2) - 2 \* u[1] \* (-nW \* (u[1] \* u[2] + 1) - 2 \* nW \* u[3] \* u[4]) / (2 \* u[1] \* u[2] + 2 \* u[3] \* u[4] + 2) ^ 2 \* u[4] + -1im \* (u[1] ^ 2 \* u[2] - 2 \* u[3] \* u[4] - 1 + (u[1] ^ 2 - 1) \* u[4] + u[1] \* (2 \* u[3] \* u[4] - u[2] + 1)) / (u[1] \* u[2] + u[3] \* u[4] + 1) + 1im \* (u[1] ^ 2 \* u[2] \* (nU + u[3]) - u[3] \* (u[3] \* u[4] + 1) + (u[1] ^ 2 - 1) \* (u[1] \* u[2] + u[3] \* u[4] + 1) + u[1] \* (-nU + u[3] \* (u[3] \* u[4] - u[2] + 1))) / (u[1] \* u[2] + u[3] \* u[4] + 1) ^ 2 \* u[4] -2 \* u[1] \* nW \* u[3] / (2 \* u[1] \* u[2] + 2 \* u[3] \* u[4] + 2) - 2 \* u[1] \* (-nW \* (u[1] \* u[2] + 1) - 2 \* nW \* u[3] \* u[4]) / (2 \* u[1] \* u[2] + 2 \* u[3] \* u[4] + 2) ^ 2 \* u[3] + -1im \* (-u[3] ^ 2 + (u[1] ^ 2 - 1) \* u[3] + u[1] \* u[3] ^ 2) / (u[1] \* u[2] + u[3] \* u[4] + 1) + 1im \* (u[1] ^ 2 \* u[2] \* (nU + u[3]) - u[3] \* (u[3] \* u[4] + 1) + (u[1] ^ 2 - 1) \* (u[1] \* u[2] + u[3] \* u[4] + 1) + u[1] \* (-nU + u[3] \* (u[3] \* u[4] - u[2] + 1))) / (u[1] \* u[2] + u[3] \* u[4] + 1) ^ 2 \* u[3]; -1im \* (u[2] ^ 2 \* (nU + u[4]) + (u[2] ^ 2 - 1) \* u[2] - u[2] \* u[4]) / (u[1] \* u[2] + u[3] \* u[4] + 1) + 1im \* (u[1] \* u[2] ^ 2 \* (nU + u[4]) - u[4] \* (u[3] \* u[4] + 1) + (u[2] ^ 2 - 1) \* (u[1] \* u[2] + u[3] \* u[4] + 1) + u[2] \* (-nU + u[4] \* (u[3] \* u[4] - u[1] + 1))) / (u[1] \* u[2] + u[3] \* u[4] + 1) ^ 2 \* u[2] -1im \* (2 \* u[1] \* u[2] \* (nU + u[4]) + 2 \* u[2] \* (u[1] \* u[2] + u[3] \* u[4] + 1) + (u[2] ^ 2 - 1) \* u[1] - nU + u[4] \* (u[3] \* u[4] - u[1] + 1)) / (u[1] \* u[2] + u[3] \* u[4] + 1) + 1im \* (u[1] \* u[2] ^ 2 \* (nU + u[4]) - u[4] \* (u[3] \* u[4] + 1) + (u[2] ^ 2 - 1) \* (u[1] \* u[2] + u[3] \* u[4] + 1) + u[2] \* (-nU + u[4] \* (u[3] \* u[4] - u[1] + 1))) / (u[1] \* u[2] + u[3] \* u[4] + 1) ^ 2 \* u[1] -1im \* (-u[4] ^ 2 + (u[2] ^ 2 - 1) \* u[4] + u[2] \* u[4] ^ 2) / (u[1] \* u[2] + u[3] \* u[4] + 1) + 1im \* (u[1] \* u[2] ^ 2 \* (nU + u[4]) - u[4] \* (u[3] \* u[4] + 1) + (u[2] ^ 2 - 1) \* (u[1] \* u[2] + u[3] \* u[4] + 1) + u[2] \* (-nU + u[4] \* (u[3] \* u[4] - u[1] + 1))) / (u[1] \* u[2] + u[3] \* u[4] + 1) ^ 2 \* u[4] -1im \* (u[1] \* u[2] ^ 2 - 2 \* u[3] \* u[4] - 1 + (u[2] ^ 2 - 1) \* u[3] + u[2] \* (2 \* u[3] \* u[4] - u[1] + 1)) / (u[1] \* u[2] + u[3] \* u[4] + 1) + 1im \* (u[1] \* u[2] ^ 2 \* (nU + u[4]) - u[4] \* (u[3] \* u[4] + 1) + (u[2] ^ 2 - 1) \* (u[1] \* u[2] + u[3] \* u[4] + 1) + u[2] \* (-nU + u[4] \* (u[3] \* u[4] - u[1] + 1))) / (u[1] \* u[2] + u[3] \* u[4] + 1) ^ 2 \* u[3]; 2 \* nW \* u[3] \* (u[3] \* u[4] + 1) / (2 \* u[1] \* u[2] + 2 \* u[3] \* u[4] + 2) ^ 2 \* u[2] + 1im \* (1 + u[3] \* (-1 - (-nU \* u[3] \* u[4] + nU) / (u[1] \* u[2] + u[3] \* u[4] + 1) ^ 2 \* u[2])) 2 \* nW \* u[3] \* (u[3] \* u[4] + 1) / (2 \* u[1] \* u[2] + 2 \* u[3] \* u[4] + 2) ^ 2 \* u[1] + -1im \* u[3] \* (-nU \* u[3] \* u[4] + nU) / (u[1] \* u[2] + u[3] \* u[4] + 1) ^ 2 \* u[1] -nW \* (u[3] \* u[4] + 1) / (2 \* u[1] \* u[2] + 2 \* u[3] \* u[4] + 2) - nW \* u[3] \* u[4] / (2 \* u[1] \* u[2] + 2 \* u[3] \* u[4] + 2) + 2 \* nW \* u[3] \* (u[3] \* u[4] + 1) / (2 \* u[1] \* u[2] + 2 \* u[3] \* u[4] + 2) ^ 2 \* u[4] + 1im \* (-2 \* u[3] - u[1] + (-nU \* u[3] \* u[4] + nU) / (u[1] \* u[2] + u[3] \* u[4] + 1) + u[3] \* (-nU \* u[4] / (u[1] \* u[2] + u[3] \* u[4] + 1) - (-nU \* u[3] \* u[4] + nU) / (u[1] \* u[2] + u[3] \* u[4] + 1) ^ 2 \* u[4])) -nW \* u[3] ^ 2 / (2 \* u[1] \* u[2] + 2 \* u[3] \* u[4] + 2) + 2 \* nW \* u[3] ^ 2 \* (u[3] \* u[4] + 1) / (2 \* u[1] \* u[2] + 2 \* u[3] \* u[4] + 2) ^ 2 + 1im \* u[3] \* (-nU \* u[3] / (u[1] \* u[2] + u[3] \* u[4] + 1) - (-nU \* u[3] \* u[4] + nU) / (u[1] \* u[2] + u[3] \* u[4] + 1) ^ 2 \* u[3]); 2 \* nW \* u[4] \* (u[3] \* u[4] + 1) / (2 \* u[1] \* u[2] + 2 \* u[3] \* u[4] + 2) ^ 2 \* u[2] + 1im \* u[4] \* (-nU \* u[3] \* u[4] + nU) / (u[1] \* u[2] + u[3] \* u[4] + 1) ^ 2 \* u[2] 2 \* nW \* u[4] \* (u[3] \* u[4] + 1) / (2 \* u[1] \* u[2] + 2 \* u[3] \* u[4] + 2) ^ 2 \* u[1] + -1im \* (1 + u[4] \* (-1 - (-nU \* u[3] \* u[4] + nU) / (u[1] \* u[2] + u[3] \* u[4] + 1) ^ 2 \* u[1])) -nW \* u[4] ^ 2 / (2 \* u[1] \* u[2] + 2 \* u[3] \* u[4] + 2) + 2 \* nW \* u[4] ^ 2 \* (u[3] \* u[4] + 1) / (2 \* u[1] \* u[2] + 2 \* u[3] \* u[4] + 2) ^ 2 + -1im \* u[4] \* (-nU \* u[4] / (u[1] \* u[2] + u[3] \* u[4] + 1) - (-nU \* u[3] \* u[4] + nU) / (u[1] \* u[2] + u[3] \* u[4] + 1) ^ 2 \* u[4]) -nW \* (u[3] \* u[4] + 1) / (2 \* u[1] \* u[2] + 2 \* u[3] \* u[4] + 2) - nW \* u[3] \* u[4] / (2 \* u[1] \* u[2] + 2 \* u[3] \* u[4] + 2) + 2 \* nW \* u[3] \* (u[3] \* u[4] + 1) / (2 \* u[1] \* u[2] + 2 \* u[3] \* u[4] + 2) ^ 2 \* u[4] + -1im \* (-2 \* u[4] - u[2] + (-nU \* u[3] \* u[4] + nU) / (u[1] \* u[2] + u[3] \* u[4] + 1) + u[4] \* (-nU \* u[3] / (u[1] \* u[2] + u[3] \* u[4] + 1) - (-nU \* u[3] \* u[4] + nU) / (u[1] \* u[2] + u[3] \* u[4] + 1) ^ 2 \* u[3]))]  
> return J  
> end

> ds = ContinuousDynamicalSystem(loop, rand(4), [1, 1, 1], loop\_jac)  
> λ = lyapunov(ds, 100, d0 = 1e-7, upper\_threshold = 1e-4, Ttr = 100)

error message

InexactError: Float64(-0.45320104263522193 + 0.7182458475836356im)

Stacktrace:  
[1] Real  
@ .\complex.jl:44 [inlined]  
[2] convert  
@ .\number.jl:7 [inlined]  
[3] macro expansion  
@ C:\Users\griff.julia\packages\StaticArraysCore\gkLqH\src\StaticArraysCore.jl:79 [inlined]  
[4] convert\_ntuple  
@ C:\Users\griff.julia\packages\StaticArraysCore\gkLqH\src\StaticArraysCore.jl:75 [inlined]  
[5] SVector{4, Float64}(x::NTuple{4, ComplexF64})  
@ StaticArraysCore C:\Users\griff.julia\packages\StaticArraysCore\gkLqH\src\StaticArraysCore.jl:111  
[6] convert  
@ C:\Users\griff.julia\packages\StaticArrays\5bAMi\src\convert.jl:168 [inlined]  
[7] setindex!  
@ .\array.jl:903 [inlined]  
[8] (::DynamicalSystemsBase.var"#30#32"{ContinuousDynamicalSystem{false, SVector{4, Float64}, 4, typeof(loop), Vector{Int64}, typeof(loop\_jac), Matrix{ComplexF64}, false}, Int64})(du::Vector{SVector{4, Float64}}, u::Vector{SVector{4, Float64}}, p::Vector{Int64}, t::Float64)  
@ DynamicalSystemsBase C:\Users\griff.julia\packages\DynamicalSystemsBase\56NGa\src\core\create\_dynamics.jl:83  
[9] ODEFunction  
@ C:\Users\griff.julia\packages\SciMLBase\IJbT7\src\scimlfunctions.jl:1624 [inlined]  
[10] step!  
@ C:\Users\griff.julia\packages\SimpleDiffEq\vIpeG\src\tsit5\atsit5.jl:458 [inlined]  
[11] step!  
@ C:\Users\griff.julia\packages\SimpleDiffEq\vIpeG\src\tsit5\atsit5.jl:592 [inlined]  
[12] step!  
@ C:\Users\griff.julia\packages\SimpleDiffEq\vIpeG\src\tsit5\atsit5.jl:586 [inlined]  
[13] lyapunov(pinteg::SimpleDiffEq.SimpleATsit5Integrator{true, Vector{SVector{4, Float64}}, Float64, Vector{Int64}, SciMLBase.ODEFunction{true, DynamicalSystemsBase.var"#30#32"{ContinuousDynamicalSystem{false, SVector{4, Float64}, 4, typeof(loop), Vector{Int64}, typeof(loop\_jac), Matrix{ComplexF64}, false}, Int64}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT\_OBSERVED), Nothing}, typeof(DynamicalSystemsBase.\_parallelnorm)}, T::Int64, Ttr::Int64, Δt::Int64, d0::Float64, ut::Float64, lt::Float64)  
@ ChaosTools C:\Users\griff.julia\packages\ChaosTools\PHPDF\src\chaosdetection\lyapunovs.jl:355  
[14] lyapunov(ds::ContinuousDynamicalSystem{false, SVector{4, Float64}, 4, typeof(loop), Vector{Int64}, typeof(loop\_jac), Matrix{ComplexF64}, false}, T::Int64; u0::SVector{4, Float64}, Ttr::Int64, d0::Float64, upper\_threshold::Float64, lower\_threshold::Float64, inittest::ChaosTools.var"#217#218"{Int64}, Δt::Int64, diffeq::NamedTuple{(), Tuple{}}, kwargs::Base.Pairs{Symbol, Union{}, Tuple{}, NamedTuple{(), Tuple{}}})  
@ ChaosTools C:\Users\griff.julia\packages\ChaosTools\PHPDF\src\chaosdetection\lyapunovs.jl:347  
[15] top-level scope  
@ In[19]:1  
[16] eval  
@ .\boot.jl:373 [inlined]  
[17] include\_string(mapexpr::typeof(REPL.softscope), mod::Module, code::String, filename::String)  
@ Base .\loading.jl:1196

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<div class="post-metadata">

**Author:** ![ChrisRackauckas](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/chrisrackauckas/32/77_2.png) [@ChrisRackauckas](https://discourse.julialang.org/u/ChrisRackauckas)\
**Post date:** [July 8, 2022, 6:56pm UTC](https://discourse.julialang.org/t/using-dynamicalsystems-inexacterror/83965/2 "2022-07-08T18:56:48Z")

</div>

Your initial condition is real but it looks like you have an `im` in the Jacobian definition at least, so it’s failing to convert the dynamics to complex numbers. You should choose a complex initial condition if you want it to solve in complex numbers.

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<div class="post-metadata">

**Author:** ![Griffith\_Mendonca](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/griffith_mendonca/32/37789_2.png) [@Griffith\_Mendonca](https://discourse.julialang.org/u/Griffith_Mendonca)\
**Post date:** [July 13, 2022, 2:44pm UTC](https://discourse.julialang.org/t/using-dynamicalsystems-inexacterror/83965/3 "2022-07-13T14:44:38Z")

</div>

Dear ChrisRackanuckas thank you so much for your quick response. I will try to replace initial conditions from real to imaginary.

Bests,  
Griffith Rufo.