Add solution key for Henon-Heles

[SebastianM-C]

I added the solution key for problem 5. Let me know if the approach is ok.
I was wondering if I should have used ModeingTookit.jl for the system definition.

For the initial condition generation I used a simple rejection sampling like in @Datseris example here.

I also think there was a typo in the mathematical description of the the SecondOrderODEProblem.

Right now there is a problem with the initial conditions and with threading this crashes julia

signal (11): Segmentation fault
in expression starting at /home/sebastian/Documents/julia/DiffEqTutorials.jl/tutorials/exercises/02-workshop_solutions.jmd:263
┌ Warning: dt <= dtmin. Aborting. There is either an error in your model specification or the true solution is unstable.

Probably the numbers have to be tweaked a bit. I will investigate this further.

I was also thinking of doing the initial condition generation a bit more general (so that there is no need to hardcode arbitrary numbers) with IntervalConstraintProgramming.jl, but for a exercise marked for beginners, I think it's a bit too much.

[SebastianM-C]

I rebased on master and fixed the problem in Part 3 and added Part 4.
@ChrisRackauckas I'm not sure what can I do about the plots. For this system the most used visualization is the Poincare map, but here I have full solutions.
Computing the Poincare section via a discrete callback is on my todo list, but I don't think it fits here.
Also one could do some kind of post solution interpolation, but again I'm not exactly sure if it would be something too complicated.

At Part 4 I get an error when trying to evaluate the solve statement

InvalidIRError: compiling gpu_kernel(Cassette.Context{nametype(Ctx),Nothing,Nothing,getfield(GPUifyLoops, Symbol("##PassType#363")),Nothing,Cassette.DisableHooks}, typeof(DiffEqGPU.gpu_kernel), ODEFunction{true,typeof(henon),LinearAlgebra.UniformScaling{Bool},Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing}, CUDAnative.CuDeviceArray{Float64,2,CUDAnative.AS.Global}, CUDAnative.CuDeviceArray{Float64,2,CUDAnative.AS.Global}, CUDAnative.CuDeviceArray{Nothing,2,CUDAnative.AS.Global}, Float64) resulted in invalid LLVM IR
Reason: unsupported call to the Julia runtime (call to jl_f_tuple)
Stacktrace:
 [1] overdub at /mnt/storage/.julia/packages/Cassette/IwsFs/src/overdub.jl:537
 [2] throw_boundserror at subarray.jl:43
 [3] checkbounds at abstractarray.jl:449
 [4] copyto! at multidimensional.jl:874
 [5] copyto! at broadcast.jl:838
 [6] copyto! at broadcast.jl:797
 [7] materialize! at broadcast.jl:756
 [8] henon at C:\Users\sebastian\Documents\julia\DiffEqTutorials.jl\tutorials\exercises\02-workshop_solutions.jmd:197
 [9] ODEFunction at /mnt/storage/.julia/packages/DiffEqBase/u5mMn/src/diffeqfunction.jl:193
 [10] gpu_kernel at /mnt/storage/.julia/packages/DiffEqGPU/QB1WC/src/DiffEqGPU.jl:6
 [11] overdub at /mnt/storage/.julia/packages/Cassette/IwsFs/src/overdub.jl:0
Reason: unsupported call to the Julia runtime (call to jl_f_getfield)
Stacktrace:
 [1] overdub at /mnt/storage/.julia/packages/Cassette/IwsFs/src/overdub.jl:537
 [2] throw_boundserror at subarray.jl:43
 [3] checkbounds at abstractarray.jl:449
 [4] copyto! at multidimensional.jl:874
 [5] copyto! at broadcast.jl:838
 [6] copyto! at broadcast.jl:797
 [7] materialize! at broadcast.jl:756
 [8] henon at C:\Users\sebastian\Documents\julia\DiffEqTutorials.jl\tutorials\exercises\02-workshop_solutions.jmd:197
 [9] ODEFunction at /mnt/storage/.julia/packages/DiffEqBase/u5mMn/src/diffeqfunction.jl:193
 [10] gpu_kernel at /mnt/storage/.julia/packages/DiffEqGPU/QB1WC/src/DiffEqGPU.jl:6
 [11] overdub at /mnt/storage/.julia/packages/Cassette/IwsFs/src/overdub.jl:0
Reason: unsupported call to the Julia runtime (call to jl_f_tuple)
Stacktrace:
 [1] overdub at /mnt/storage/.julia/packages/Cassette/IwsFs/src/overdub.jl:537
 [2] throw_boundserror at subarray.jl:43
 [3] checkbounds at abstractarray.jl:449
 [4] getindex at subarray.jl:260
 [5] henon at C:\Users\sebastian\Documents\julia\DiffEqTutorials.jl\tutorials\exercises\02-workshop_solutions.jmd:191
 [6] ODEFunction at /mnt/storage/.julia/packages/DiffEqBase/u5mMn/src/diffeqfunction.jl:193
 [7] gpu_kernel at /mnt/storage/.julia/packages/DiffEqGPU/QB1WC/src/DiffEqGPU.jl:6
 [8] overdub at /mnt/storage/.julia/packages/Cassette/IwsFs/src/overdub.jl:0
Reason: unsupported call to the Julia runtime (call to jl_f_getfield)
Stacktrace:
 [1] overdub at /mnt/storage/.julia/packages/Cassette/IwsFs/src/overdub.jl:537
 [2] throw_boundserror at subarray.jl:43
 [3] checkbounds at abstractarray.jl:449
 [4] getindex at subarray.jl:260
 [5] henon at C:\Users\sebastian\Documents\julia\DiffEqTutorials.jl\tutorials\exercises\02-workshop_solutions.jmd:191
 [6] ODEFunction at /mnt/storage/.julia/packages/DiffEqBase/u5mMn/src/diffeqfunction.jl:193
 [7] gpu_kernel at /mnt/storage/.julia/packages/DiffEqGPU/QB1WC/src/DiffEqGPU.jl:6
 [8] overdub at /mnt/storage/.julia/packages/Cassette/IwsFs/src/overdub.jl:0
Reason: unsupported call through a literal pointer (call to jl_alloc_array_1d)
Stacktrace:
 [1] Type at boot.jl:402
 [2] Type at boot.jl:411
 [3] Type at boot.jl:419
 [4] similar at abstractarray.jl:618
 [5] similar at abstractarray.jl:617
 [6] vect at array.jl:130
 [7] henon at C:\Users\sebastian\Documents\julia\DiffEqTutorials.jl\tutorials\exercises\02-workshop_solutions.jmd:197
 [8] ODEFunction at /mnt/storage/.julia/packages/DiffEqBase/u5mMn/src/diffeqfunction.jl:193
 [9] gpu_kernel at /mnt/storage/.julia/packages/DiffEqGPU/QB1WC/src/DiffEqGPU.jl:6
 [10] overdub at /mnt/storage/.julia/packages/Cassette/IwsFs/src/overdub.jl:0
Reason: unsupported call through a literal pointer (call to jl_object_id)
Stacktrace:
....

[SebastianM-C]

I also tried the following for the GPU part

function henon_gpu(dz,z,p,t)
  @inbounds begin
    dz[1] = -z[3]*(1 + 2z[4])
    dz[2] = -z[4]-(z[3]^2 - z[4]^2)
    dz[3] = z[1]
    dz[4] = z[2]
  end
  return nothing
end

z0 = generate_ics(0.125, 5)
prob_gpu = ODEProblem(henon_gpu, u₀, (0., 10.))
ensprob = EnsembleProblem(prob_gpu, prob_func=prob_func)

but the error is still present.

[SebastianM-C]

The problem was due to the input not being Float32, as mentioned by Chris on slack.
Everything should work now.

[ChrisRackauckas]

Thanks!