# Error while using DynamicalSystems.jl to solve for a complex nonlinear system

**URL:** <https://discourse.julialang.org/t/error-while-using-dynamicalsystems-jl-to-solve-for-a-complex-nonlinear-system/96976>\
**Category:** General Usage\
**Tags:** package, modelingtoolkit, nonlinear, differentialequation, dynamical-systems\
**Created:** [April 2, 2023, 5:38pm UTC](https://discourse.julialang.org/t/error-while-using-dynamicalsystems-jl-to-solve-for-a-complex-nonlinear-system/96976 "2023-04-02T17:38:21Z")\
**Posts on this page:** 1\
**Page:** 1

<div class="post-metadata">

**Author:** ![Sh\_166](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/sh_166/32/43965_2.png) [@Sh\_166](https://discourse.julialang.org/u/Sh_166)\
**Post date:** [April 2, 2023, 5:38pm UTC](https://discourse.julialang.org/t/error-while-using-dynamicalsystems-jl-to-solve-for-a-complex-nonlinear-system/96976/1 "2023-04-02T17:38:21Z")

</div>

Hi,

I was having quite a bit of trouble with DynamicalSystems.jl. In particular, I would like to implement an elastic double pendulum (so, 3 degrees of freedoms). The problem with this system is that it is nonlinear, second order and its quite troublesome and tedious to separate the different second order derivatives by hand.

I have solved this system numerically once already, separating the derivatives by using,

> E₁₁, E₂₂, E₃₃ = Symbolics.solve\_for([E₁ ~ 0, E₂ ~ 0, E₃ ~ 0], [d(d(θ)), d(d(ϕ)), d(d(r))])

(The E’s inside the parentheses are the equations of motion I have)

The problem is that this doesn’t seem to work with DynamicalSystems.jl (I am quite new to this package).

My code looks like this:

> using Plots, ModelingToolkit, DifferentialEquations, LaTeXStrings, DynamicalSystems  
> @variables t θ(t) ϕ(t) r(t)  
> @parameters m₁, m₂, l₁, l₂, k, g  
> d = Differential(t)  
> M = m₁+m₂  
> ω₃² = k/m₂  
> μ = m₂/m₁  
> l₀ = g/(ω₃²)  
> l = l₂ + l₀ ##Defining some constants.  
> E₁ = …  
> E₂ = …  
> E₃ = … //These equations contain both the variables and their first-order  
> derivatives  
> E₁₁, E₂₂, E₃₃ = Symbolics.solve\_for([E₁ ~ 0, E₂ ~ 0, E₃ ~ 0], [d(d(θ)), d(d(ϕ)), d(d(r))])  
> function truncated(du, u, p, t)  
> m₁ = p[1] ; m₂ = p[2]; l₁ = p[3]; l₂ = p[4]; k = p[5]; g = p[6]  
> u[1] = θ  
> u[2] = dθ  
> u[3] = ϕ  
> u[4] = dϕ  
> u[5] = r  
> u[6] = dr  
> du[1] = u[2]  
> du[3] = u[4]  
> du[5] = u[6]  
> du[2] = E₁₁  
> du[4] = E₂₂  
> du[6] = E₃₃  
> end

I then define the initial state, the parameters and the dynamical system as:

> u\_0 = [0, 1e-7, 0, 1e-7, 0.05, 0]  
> p\_0 = [0.4, 0.8, 0.786, 0.7, 225.02, 9.8]  
> DS = ContinuousDynamicalSystem(truncated, u\_0, p\_0)

This gives me the following error:

> MethodError: no method matching Float64(::Num)  
> Closest candidates are:  
> (::Type{T})(::Real, ::RoundingMode) where T\<:AbstractFloat at rounding.jl:200  
> (::Type{T})(::T) where T\<:Number at boot.jl:772  
> (::Type{T})(::AbstractChar) where T\<:Union{AbstractChar, Number} at char.jl:50  
> …  
> Stacktrace:  
> [1] convert(#unused#::Type{Float64}, x::Num)  
> @ Base .\number.jl:7  
> [2] setindex!(A::Vector{Float64}, x::Num, i1::Int64)  
> @ Base .\array.jl:966  
> [3] truncated(du::Vector{Float64}, u::Vector{Float64}, p::Vector{Float64}, t::Float64)  
> @ Main .\In[29]:19  
> [4] ODEFunction  
> @ C:\Users\shiha.julia\packages\SciMLBase\nzvrv\src\scimlfunctions.jl:2096 [inlined]  
> [5] initialize!(integrator::OrdinaryDiffEq.ODEIntegrator{Tsit5{typeof(OrdinaryDiffEq.trivial\_limiter!), typeof(OrdinaryDiffEq.trivial\_limiter!), Static.False}, 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, typeof(truncated), 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}, Tsit5{typeof(OrdinaryDiffEq.trivial\_limiter!), typeof(OrdinaryDiffEq.trivial\_limiter!), Static.False}, OrdinaryDiffEq.InterpolationData{ODEFunction{true, SciMLBase.AutoSpecialize, typeof(truncated), 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.Tsit5Cache{Vector{Float64}, Vector{Float64}, Vector{Float64}, OrdinaryDiffEq.Tsit5ConstantCache{Float64, Float64}, typeof(OrdinaryDiffEq.trivial\_limiter!), typeof(OrdinaryDiffEq.trivial\_limiter!), Static.False}}, DiffEqBase.DEStats, Nothing}, ODEFunction{true, SciMLBase.AutoSpecialize, typeof(truncated), LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT\_OBSERVED), Nothing, Nothing}, OrdinaryDiffEq.Tsit5Cache{Vector{Float64}, Vector{Float64}, Vector{Float64}, OrdinaryDiffEq.Tsit5ConstantCache{Float64, Float64}, typeof(OrdinaryDiffEq.trivial\_limiter!), typeof(OrdinaryDiffEq.trivial\_limiter!), Static.False}, OrdinaryDiffEq.DEOptions{Float64, Float64, Float64, Float64, PIController{Rational{Int64}}, typeof(DiffEqBase.ODE\_DEFAULT\_NORM), typeof(LinearAlgebra.opnorm), Bool, 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, Float64, Tuple{}, Tuple{}, Tuple{}}, Vector{Float64}, Float64, Nothing, OrdinaryDiffEq.DefaultInit}, cache::OrdinaryDiffEq.Tsit5Cache{Vector{Float64}, Vector{Float64}, Vector{Float64}, OrdinaryDiffEq.Tsit5ConstantCache{Float64, Float64}, typeof(OrdinaryDiffEq.trivial\_limiter!), typeof(OrdinaryDiffEq.trivial\_limiter!), Static.False})  
> @ OrdinaryDiffEq C:\Users\shiha.julia\packages\OrdinaryDiffEq\GxEzW\src\perform\_step\low\_order\_rk\_perform\_step.jl:736  
> [6] \_\_init(prob::ODEProblem{Vector{Float64}, Tuple{Float64, Float64}, true, Vector{Float64}, ODEFunction{true, SciMLBase.AutoSpecialize, typeof(truncated), 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::Tsit5{typeof(OrdinaryDiffEq.trivial\_limiter!), typeof(OrdinaryDiffEq.trivial\_limiter!), Static.False}, 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::Bool, callback::Nothing, dense::Bool, calck::Bool, dt::Float64, dtmin::Nothing, dtmax::Float64, force\_dtmin::Bool, adaptive::Bool, gamma::Rational{Int64}, abstol::Float64, reltol::Float64, 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::Float64, 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, Union{}, Tuple{}, NamedTuple{(), Tuple{}}})  
> @ OrdinaryDiffEq C:\Users\shiha.julia\packages\OrdinaryDiffEq\GxEzW\src\solve.jl:493  
> [7] CoupledODEs(prob::ODEProblem{Vector{Float64}, Tuple{Float64, Float64}, true, Vector{Float64}, ODEFunction{true, SciMLBase.AutoSpecialize, typeof(truncated), 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}, diffeq::NamedTuple{(:alg, :abstol, :reltol), Tuple{Tsit5{typeof(OrdinaryDiffEq.trivial\_limiter!), typeof(OrdinaryDiffEq.trivial\_limiter!), Static.False}, Float64, Float64}}; special\_kwargs::Base.Pairs{Symbol, Union{}, Tuple{}, NamedTuple{(), Tuple{}}})  
> @ DynamicalSystemsBase C:\Users\shiha.julia\packages\DynamicalSystemsBase\byMNf\src\core\_systems\continuous\_time\_ode.jl:94  
> [8] CoupledODEs(prob::ODEProblem{Vector{Float64}, Tuple{Float64, Float64}, true, Vector{Float64}, ODEFunction{true, SciMLBase.AutoSpecialize, typeof(truncated), 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}, diffeq::NamedTuple{(:alg, :abstol, :reltol), Tuple{Tsit5{typeof(OrdinaryDiffEq.trivial\_limiter!), typeof(OrdinaryDiffEq.trivial\_limiter!), Static.False}, Float64, Float64}})  
> @ DynamicalSystemsBase C:\Users\shiha.julia\packages\DynamicalSystemsBase\byMNf\src\core\_systems\continuous\_time\_ode.jl:89  
> [9] CoupledODEs(f::Function, u0::Vector{Float64}, p::Vector{Float64}; t0::Int64, diffeq::NamedTuple{(:alg, :abstol, :reltol), Tuple{Tsit5{typeof(OrdinaryDiffEq.trivial\_limiter!), typeof(OrdinaryDiffEq.trivial\_limiter!), Static.False}, Float64, Float64}})  
> @ DynamicalSystemsBase C:\Users\shiha.julia\packages\DynamicalSystemsBase\byMNf\src\core\_systems\continuous\_time\_ode.jl:85  
> [10] CoupledODEs(f::Function, u0::Vector{Float64}, p::Vector{Float64})  
> @ DynamicalSystemsBase C:\Users\shiha.julia\packages\DynamicalSystemsBase\byMNf\src\core\_systems\continuous\_time\_ode.jl:80  
> [11] top-level scope  
> @ In[31]:1  
> [12] eval  
> @ .\boot.jl:368 [inlined]  
> [13] include\_string(mapexpr::typeof(REPL.softscope), mod::Module, code::String, filename::String)  
> @ Base .\loading.jl:1428

I expect I am making an elementary error since I am very new to this package. I have mixed up the syntax from quite a few different packages here since I am not too sure how to proceed without separating the second order derivatives by hand (I am using Plots, ModelingToolkit, DifferentialEquations, LaTeXStrings, DynamicalSystems; apart from the latter, the rest I have used earlier to numerically solve this very system).  
Thanks in advance!