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Hénon-Heiles Energy Conservation

In this notebook we will study the energy conservation properties of several high-order methods for the Hénon-Heiles system. We will se how the energy error behaves at very tight tolerances and how different techniques, such as using symplectic solvers or manifold projections, benchmark against each other. The Hamiltonian for this system is given by:

\[\mathcal{H}=\frac{1}{2}(p_1^2 + p_2^2) + \frac{1}{2}\left(q_1^2 + q_2^2 + 2q_1^2 q_2 - \frac{2}{3}q_2^3\right)\]

We will also compare the in place approach with the out of place approach by using Arrays (for the in place version) and StaticArrays (for out of place versions). In order to separate these two, we will use iip for the in-place names and oop for out of place ones.

using OrdinaryDiffEq, Plots, DiffEqCallbacks
using SciMLBenchmarks
using TaylorIntegration, LinearAlgebra, StaticArrays
gr(fmt=:png)
default(fmt=:png)

T(p) = 1//2 * norm(p)^2
V(q) = 1//2 * (q[1]^2 + q[2]^2 + 2q[1]^2 * q[2]- 2//3 * q[2]^3)
H(p,q, params) = T(p) + V(q)

function iip_dq(dq,p,q,params,t)
    dq[1] = p[1]
    dq[2] = p[2]
end

function iip_dp(dp,p,q,params,t)
    dp[1] = -q[1] * (1 + 2q[2])
    dp[2] = -q[2] - (q[1]^2 - q[2]^2)
end

const iip_q0 = [0.1, 0.]
const iip_p0 = [0., 0.5]

function oop_dq(p, q, params, t)
    p
end

function oop_dp(p, q, params, t)
    dp1 = -q[1] * (1 + 2q[2])
    dp2 = -q[2] - (q[1]^2 - q[2]^2)
    @SVector [dp1, dp2]
end

const oop_q0 = @SVector [0.1, 0.]
const oop_p0 = @SVector [0., 0.5]

function hamilton(du,u,p,t)
    dq, q = @views u[3:4], du[3:4]
    dp, p = @views u[1:2], du[1:2]

    dp[1] = -q[1] * (1 + 2q[2])
    dp[2] = -q[2] - (q[1]^2 - q[2]^2)
    dq .= p

    return nothing
end

function g(resid, u, p)
    resid[1] = H([u[1],u[2]], [u[3],u[4]], nothing) - E
    resid[2:4] .= 0
end

const cb = ManifoldProjection(g, nlopts=Dict(:ftol=>1e-13))

const E = H(iip_p0, iip_q0, nothing)
0.13

For the comparison we will use the following function

energy_err(sol) = map(i->H([sol[1,i], sol[2,i]], [sol[3,i], sol[4,i]], nothing)-E, 1:length(sol.u))
abs_energy_err(sol) = [abs.(H([sol[1,j], sol[2,j]], [sol[3,j], sol[4,j]], nothing) - E) for j=1:length(sol.u)]

function compare(mode=:inplace, all=true, plt=nothing; tmax=1e2)
    if mode == :inplace
        prob = DynamicalODEProblem(iip_dp, iip_dq, iip_p0, iip_q0, (0., tmax))
    else
        prob = DynamicalODEProblem(oop_dp, oop_dq, oop_p0, oop_q0, (0., tmax))
    end
    prob_linear = ODEProblem(hamilton, vcat(iip_p0, iip_q0), (0., tmax))

    GC.gc()
    (mode == :inplace && all) && @time sol1 = solve(prob, Vern9(), callback=cb, abstol=1e-14, reltol=1e-14)
    GC.gc()
    @time sol2 = solve(prob, KahanLi8(), dt=1e-2, maxiters=1e10)
    GC.gc()
    @time sol3 = solve(prob, SofSpa10(), dt=1e-2, maxiters=1e8)
    GC.gc()
    @time sol4 = solve(prob, Vern9(), abstol=1e-14, reltol=1e-14)
    GC.gc()
    @time sol5 = solve(prob, DPRKN12(), abstol=1e-14, reltol=1e-14)
    GC.gc()
    (mode == :inplace && all) && @time sol6 = solve(prob_linear, TaylorMethod(50), abstol=1e-20)

    (mode == :inplace && all) && println("Vern9 + ManifoldProjection max energy error:\t"*
        "$(maximum(abs_energy_err(sol1)))\tin\t$(length(sol1.u))\tsteps.")
    println("KahanLi8 max energy error:\t\t\t$(maximum(abs_energy_err(sol2)))\tin\t$(length(sol2.u))\tsteps.")
    println("SofSpa10 max energy error:\t\t\t$(maximum(abs_energy_err(sol3)))\tin\t$(length(sol3.u))\tsteps.")
    println("Vern9 max energy error:\t\t\t\t$(maximum(abs_energy_err(sol4)))\tin\t$(length(sol4.u))\tsteps.")
    println("DPRKN12 max energy error:\t\t\t$(maximum(abs_energy_err(sol5)))\tin\t$(length(sol5.u))\tsteps.")
    (mode == :inplace && all) && println("TaylorMethod max energy error:\t\t\t$(maximum(abs_energy_err(sol6)))"*
        "\tin\t$(length(sol6.u))\tsteps.")

    if plt === nothing
        plt = plot(xlabel="t", ylabel="Energy error")
    end
    (mode == :inplace && all) && plot!(sol1.t, energy_err(sol1), label="Vern9 + ManifoldProjection")
    plot!(sol2.t, energy_err(sol2), label="KahanLi8", ls=mode==:inplace ? :solid : :dash)
    plot!(sol3.t, energy_err(sol3), label="SofSpa10", ls=mode==:inplace ? :solid : :dash)
    plot!(sol4.t, energy_err(sol4), label="Vern9", ls=mode==:inplace ? :solid : :dash)
    plot!(sol5.t, energy_err(sol5), label="DPRKN12", ls=mode==:inplace ? :solid : :dash)
    (mode == :inplace && all) && plot!(sol6.t, energy_err(sol6), label="TaylorMethod")

    return plt
end
compare (generic function with 4 methods)

The mode argument choses between the in place approach and the out of place one. The all parameter is used to compare only the integrators that support both the in place and the out of place versions (we reffer here only to the 6 high order methods chosen bellow). The plt argument can be used to overlay the results over a previous plot and the tmax keyword determines the simulation time.

Note:

  1. The Vern9 method is used with ODEProblem because of performance issues with ArrayPartition indexing which manifest for DynamicalODEProblem.
  2. The NLsolve call used by ManifoldProjection was modified to use ftol=1e-13 in order to obtain a very low energy error.

Here are the results of the comparisons between the in place methods:

compare(tmax=1e2)
86.677919 seconds (158.88 M allocations: 9.349 GiB, 5.14% gc time, 99.99% 
compilation time)
  2.716626 seconds (5.31 M allocations: 334.646 MiB, 3.38% gc time, 99.72% 
compilation time)
  2.254694 seconds (2.93 M allocations: 187.289 MiB, 1.25% gc time, 99.54% 
compilation time)
 44.576013 seconds (51.73 M allocations: 3.128 GiB, 3.06% gc time, 99.99% c
ompilation time)
  3.806314 seconds (3.49 M allocations: 219.307 MiB, 1.46% gc time, 99.97% 
compilation time)
  1.821381 seconds (2.28 M allocations: 154.086 MiB, 1.86% gc time, 99.94% 
compilation time)
Vern9 + ManifoldProjection max energy error:	3.885780586188048e-16	in	1881	
steps.
KahanLi8 max energy error:			4.9404924595819466e-15	in	10001	steps.
SofSpa10 max energy error:			5.440092820663267e-15	in	10001	steps.
Vern9 max energy error:				3.885780586188048e-16	in	941	steps.
DPRKN12 max energy error:			1.942890293094024e-16	in	385	steps.
TaylorMethod max energy error:			0.0	in	2	steps.

compare(tmax=1e3)
0.079207 seconds (1.03 M allocations: 69.172 MiB)
  0.070062 seconds (700.10 k allocations: 61.899 MiB)
  0.108297 seconds (700.10 k allocations: 61.901 MiB)
  0.034577 seconds (326.72 k allocations: 20.258 MiB)
  0.005689 seconds (72.14 k allocations: 3.121 MiB)
  0.000714 seconds (2.02 k allocations: 759.844 KiB)
Vern9 + ManifoldProjection max energy error:	4.9960036108132044e-15	in	1865
9	steps.
KahanLi8 max energy error:			1.815214645262131e-14	in	100002	steps.
SofSpa10 max energy error:			2.8033131371785203e-14	in	100002	steps.
Vern9 max energy error:				4.9960036108132044e-15	in	9330	steps.
DPRKN12 max energy error:			1.27675647831893e-15	in	3787	steps.
TaylorMethod max energy error:			0.0	in	2	steps.

compare(tmax=1e4)
1.248434 seconds (10.25 M allocations: 685.919 MiB, 29.03% gc time)
  0.926814 seconds (7.00 M allocations: 579.841 MiB, 12.48% gc time)
  1.381545 seconds (7.00 M allocations: 579.843 MiB, 19.20% gc time)
  0.337701 seconds (3.26 M allocations: 203.181 MiB)
  0.054402 seconds (718.64 k allocations: 32.470 MiB)
  0.000698 seconds (2.02 k allocations: 759.844 KiB)
Vern9 + ManifoldProjection max energy error:	4.440892098500626e-14	in	18643
1	steps.
KahanLi8 max energy error:			3.161360062620133e-14	in	1000001	steps.
SofSpa10 max energy error:			1.136590821460004e-13	in	1000001	steps.
Vern9 max energy error:				4.440892098500626e-14	in	93216	steps.
DPRKN12 max energy error:			8.104628079763643e-15	in	37813	steps.
TaylorMethod max energy error:			0.0	in	2	steps.

compare(tmax=5e4)
6.573684 seconds (51.45 M allocations: 3.340 GiB, 29.33% gc time, 1.73% c
ompilation time)
  5.580664 seconds (35.00 M allocations: 2.831 GiB, 26.87% gc time)
  7.366414 seconds (35.00 M allocations: 2.831 GiB, 24.72% gc time)
  1.670743 seconds (16.31 M allocations: 1001.902 MiB)
  0.273859 seconds (3.59 M allocations: 155.672 MiB)
  0.001116 seconds (2.02 k allocations: 759.844 KiB)
Vern9 + ManifoldProjection max energy error:	1.0000333894311098e-13	in	9320
93	steps.
KahanLi8 max energy error:			1.2331802246023926e-13	in	5000001	steps.
SofSpa10 max energy error:			1.5035195310986182e-13	in	5000001	steps.
Vern9 max energy error:				2.108035968007016e-13	in	466047	steps.
DPRKN12 max energy error:			3.6387559632089506e-14	in	189040	steps.
TaylorMethod max energy error:			0.0	in	2	steps.

We can see that as the simulation time increases, the energy error increases. For this particular example the energy error for all the methods is comparable. For relatively short simulation times, if a highly accurate solution is required, the symplectic method is not recommended as its energy error fluctuations are larger than for other methods. An other thing to notice is the fact that the two versions of Vern9 behave identically, as expected, until the energy error set by ftol is reached.

We will now compare the in place with the out of place versions. In the plots bellow we will use a dashed line for the out of place versions.

function in_vs_out(;all=false, tmax=1e2)
    println("In place versions:")
    plt = compare(:inplace, all, tmax=tmax)
    println("\nOut of place versions:")
    plt = compare(:oop, false, plt; tmax=tmax)
end
in_vs_out (generic function with 1 method)

First, here is a summary of all the available methods for tmax = 1e3:

in_vs_out(all=true, tmax=1e3)
In place versions:
  0.100256 seconds (1.03 M allocations: 69.172 MiB)
  0.069883 seconds (700.10 k allocations: 61.899 MiB)
  0.108527 seconds (700.10 k allocations: 61.901 MiB)
  0.034573 seconds (326.72 k allocations: 20.258 MiB)
  0.005688 seconds (72.14 k allocations: 3.121 MiB)
  0.000707 seconds (2.02 k allocations: 759.844 KiB)
Vern9 + ManifoldProjection max energy error:	4.9960036108132044e-15	in	1865
9	steps.
KahanLi8 max energy error:			1.815214645262131e-14	in	100002	steps.
SofSpa10 max energy error:			2.8033131371785203e-14	in	100002	steps.
Vern9 max energy error:				4.9960036108132044e-15	in	9330	steps.
DPRKN12 max energy error:			1.27675647831893e-15	in	3787	steps.
TaylorMethod max energy error:			0.0	in	2	steps.

Out of place versions:
  1.513064 seconds (2.69 M allocations: 193.834 MiB, 4.12% gc time, 95.92% 
compilation time)
  0.704514 seconds (1.07 M allocations: 83.387 MiB, 94.24% compilation time
)
  1.441128 seconds (2.90 M allocations: 155.806 MiB, 2.36% gc time, 99.68% 
compilation time)
  0.988159 seconds (1.08 M allocations: 68.799 MiB, 99.82% compilation time
)
KahanLi8 max energy error:			1.815214645262131e-14	in	100002	steps.
SofSpa10 max energy error:			2.8033131371785203e-14	in	100002	steps.
Vern9 max energy error:				4.496403249731884e-15	in	9330	steps.
DPRKN12 max energy error:			1.27675647831893e-15	in	3787	steps.

Now we will compare the in place and the out of place versions, but only for the integrators that are compatible with StaticArrays

in_vs_out(tmax=1e2)
In place versions:
  0.007299 seconds (70.09 k allocations: 5.806 MiB)
  0.009681 seconds (70.09 k allocations: 5.808 MiB)
  0.002990 seconds (33.10 k allocations: 2.076 MiB)
  0.000616 seconds (7.49 k allocations: 352.203 KiB)
KahanLi8 max energy error:			4.9404924595819466e-15	in	10001	steps.
SofSpa10 max energy error:			5.440092820663267e-15	in	10001	steps.
Vern9 max energy error:				3.885780586188048e-16	in	941	steps.
DPRKN12 max energy error:			1.942890293094024e-16	in	385	steps.

Out of place versions:
  0.002734 seconds (10.02 k allocations: 1.682 MiB)
  0.004039 seconds (10.02 k allocations: 1.682 MiB)
  0.000514 seconds (971 allocations: 500.562 KiB)
  0.000281 seconds (412 allocations: 99.344 KiB)
KahanLi8 max energy error:			4.9404924595819466e-15	in	10001	steps.
SofSpa10 max energy error:			5.440092820663267e-15	in	10001	steps.
Vern9 max energy error:				3.3306690738754696e-16	in	941	steps.
DPRKN12 max energy error:			1.942890293094024e-16	in	385	steps.

in_vs_out(tmax=1e3)
In place versions:
  0.066382 seconds (700.10 k allocations: 61.899 MiB)
  0.102688 seconds (700.10 k allocations: 61.901 MiB)
  0.033129 seconds (326.72 k allocations: 20.258 MiB)
  0.005666 seconds (72.14 k allocations: 3.121 MiB)
KahanLi8 max energy error:			1.815214645262131e-14	in	100002	steps.
SofSpa10 max energy error:			2.8033131371785203e-14	in	100002	steps.
Vern9 max energy error:				4.9960036108132044e-15	in	9330	steps.
DPRKN12 max energy error:			1.27675647831893e-15	in	3787	steps.

Out of place versions:
  0.029030 seconds (100.03 k allocations: 21.886 MiB)
  0.039693 seconds (100.03 k allocations: 21.887 MiB)
  0.004676 seconds (9.37 k allocations: 4.464 MiB)
  0.001817 seconds (3.82 k allocations: 803.531 KiB)
KahanLi8 max energy error:			1.815214645262131e-14	in	100002	steps.
SofSpa10 max energy error:			2.8033131371785203e-14	in	100002	steps.
Vern9 max energy error:				4.496403249731884e-15	in	9330	steps.
DPRKN12 max energy error:			1.27675647831893e-15	in	3787	steps.

in_vs_out(tmax=1e4)
In place versions:
  1.053828 seconds (7.00 M allocations: 579.841 MiB, 24.32% gc time)
  1.298132 seconds (7.00 M allocations: 579.843 MiB, 14.45% gc time)
  0.341107 seconds (3.26 M allocations: 203.181 MiB)
  0.053934 seconds (718.64 k allocations: 32.470 MiB)
KahanLi8 max energy error:			3.161360062620133e-14	in	1000001	steps.
SofSpa10 max energy error:			1.136590821460004e-13	in	1000001	steps.
Vern9 max energy error:				4.440892098500626e-14	in	93216	steps.
DPRKN12 max energy error:			8.104628079763643e-15	in	37813	steps.

Out of place versions:
  0.315389 seconds (1.00 M allocations: 167.850 MiB, 6.65% gc time)
  0.438038 seconds (1.00 M allocations: 167.850 MiB, 8.03% gc time)
  0.049011 seconds (93.26 k allocations: 45.375 MiB)
  0.016480 seconds (37.85 k allocations: 10.021 MiB)
KahanLi8 max energy error:			3.161360062620133e-14	in	1000001	steps.
SofSpa10 max energy error:			1.136590821460004e-13	in	1000001	steps.
Vern9 max energy error:				4.4797499043625066e-14	in	93215	steps.
DPRKN12 max energy error:			8.104628079763643e-15	in	37813	steps.

in_vs_out(tmax=5e4)
In place versions:
  7.513751 seconds (35.00 M allocations: 2.831 GiB, 45.90% gc time)
  6.613498 seconds (35.00 M allocations: 2.831 GiB, 16.13% gc time)
  1.670954 seconds (16.31 M allocations: 1001.902 MiB)
  0.272723 seconds (3.59 M allocations: 155.672 MiB)
KahanLi8 max energy error:			1.2331802246023926e-13	in	5000001	steps.
SofSpa10 max energy error:			1.5035195310986182e-13	in	5000001	steps.
Vern9 max energy error:				2.108035968007016e-13	in	466047	steps.
DPRKN12 max energy error:			3.6387559632089506e-14	in	189040	steps.

Out of place versions:
  1.527744 seconds (5.00 M allocations: 839.236 MiB, 3.89% gc time)
  2.091271 seconds (5.00 M allocations: 839.237 MiB, 5.05% gc time)
  0.231359 seconds (466.10 k allocations: 208.068 MiB)
  0.079695 seconds (189.09 k allocations: 39.705 MiB)
KahanLi8 max energy error:			1.2331802246023926e-13	in	5000001	steps.
SofSpa10 max energy error:			1.5035195310986182e-13	in	5000001	steps.
Vern9 max energy error:				2.246258734572848e-13	in	466047	steps.
DPRKN12 max energy error:			3.6387559632089506e-14	in	189040	steps.

As we see from the above comparisons, the StaticArray versions are significantly faster and use less memory. The speedup provided for the out of place version is more proeminent at larger values for tmax. We can see again that if the simulation time is increased, the energy error of the symplectic methods is less noticeable compared to the rest of the methods.

The benchmarks were performed on a machine with

Appendix

These benchmarks are a part of the SciMLBenchmarks.jl repository, found at: https://github.com/SciML/SciMLBenchmarks.jl. For more information on high-performance scientific machine learning, check out the SciML Open Source Software Organization https://sciml.ai.

To locally run this benchmark, do the following commands:

using SciMLBenchmarks
SciMLBenchmarks.weave_file("benchmarks/DynamicalODE","Henon-Heiles_energy_conservation_benchmark.jmd")

Computer Information:

Julia Version 1.9.4
Commit 8e5136fa297 (2023-11-14 08:46 UTC)
Build Info:
  Official https://julialang.org/ release
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: 128 × AMD EPYC 7502 32-Core Processor
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-14.0.6 (ORCJIT, znver2)
  Threads: 128 on 128 virtual cores
Environment:
  JULIA_CPU_THREADS = 128
  JULIA_DEPOT_PATH = /cache/julia-buildkite-plugin/depots/5b300254-1738-4989-ae0a-f4d2d937f953
  JULIA_IMAGE_THREADS = 1

Package Information:

Status `/cache/build/exclusive-amdci1-0/julialang/scimlbenchmarks-dot-jl/benchmarks/DynamicalODE/Project.toml`
⌃ [459566f4] DiffEqCallbacks v2.27.0
⌃ [055956cb] DiffEqPhysics v3.11.0
  [b305315f] Elliptic v1.0.1
⌃ [1dea7af3] OrdinaryDiffEq v6.53.4
⌃ [65888b18] ParameterizedFunctions v5.15.0
⌃ [91a5bcdd] Plots v1.38.17
⌃ [d330b81b] PyPlot v2.11.1
  [31c91b34] SciMLBenchmarks v0.1.3
⌃ [90137ffa] StaticArrays v1.6.2
⌃ [92b13dbe] TaylorIntegration v0.14.2
  [37e2e46d] LinearAlgebra
  [de0858da] Printf
  [10745b16] Statistics v1.9.0
Info Packages marked with ⌃ have new versions available and may be upgradable.
Warning The project dependencies or compat requirements have changed since the manifest was last resolved. It is recommended to `Pkg.resolve()` or consider `Pkg.update()` if necessary.

And the full manifest:

Status `/cache/build/exclusive-amdci1-0/julialang/scimlbenchmarks-dot-jl/benchmarks/DynamicalODE/Manifest.toml`
⌅ [47edcb42] ADTypes v0.1.6
⌅ [c3fe647b] AbstractAlgebra v0.31.0
  [1520ce14] AbstractTrees v0.4.4
⌃ [79e6a3ab] Adapt v3.6.2
  [ec485272] ArnoldiMethod v0.2.0
⌃ [4fba245c] ArrayInterface v7.4.11
  [30b0a656] ArrayInterfaceCore v0.1.29
⌃ [6e4b80f9] BenchmarkTools v1.3.2
⌃ [e2ed5e7c] Bijections v0.1.4
⌃ [d1d4a3ce] BitFlags v0.1.7
  [62783981] BitTwiddlingConvenienceFunctions v0.1.5
⌃ [2a0fbf3d] CPUSummary v0.2.3
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  [49dc2e85] Calculus v0.5.1
⌃ [d360d2e6] ChainRulesCore v1.16.0
  [fb6a15b2] CloseOpenIntervals v0.1.12
⌃ [944b1d66] CodecZlib v0.7.2
⌃ [35d6a980] ColorSchemes v3.22.0
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  [5ae59095] Colors v0.12.10
  [861a8166] Combinatorics v1.0.2
  [a80b9123] CommonMark v0.8.12
  [38540f10] CommonSolve v0.2.4
  [bbf7d656] CommonSubexpressions v0.3.0
⌃ [34da2185] Compat v4.8.0
  [b152e2b5] CompositeTypes v0.1.3
⌃ [f0e56b4a] ConcurrentUtilities v2.2.1
⌃ [8f4d0f93] Conda v1.9.1
⌃ [187b0558] ConstructionBase v1.5.3
  [d38c429a] Contour v0.6.2
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  [a8cc5b0e] Crayons v4.1.1
  [9a962f9c] DataAPI v1.15.0
⌃ [864edb3b] DataStructures v0.18.14
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  [8bb1440f] DelimitedFiles v1.9.1
⌃ [2b5f629d] DiffEqBase v6.127.0
⌃ [459566f4] DiffEqCallbacks v2.27.0
⌃ [055956cb] DiffEqPhysics v3.11.0
  [163ba53b] DiffResults v1.1.0
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⌃ [b4f34e82] Distances v0.10.9
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⌃ [d5909c97] GroupsCore v0.4.0
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⌃ [1019f520] JLFzf v0.1.5
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⌃ [7e49a35a] RuntimeGeneratedFunctions v0.5.11
⌃ [fdea26ae] SIMD v3.4.5
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⌃ [476501e8] SLEEFPirates v0.6.39
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⌃ [e9a6253c] SciMLNLSolve v0.1.8
⌃ [c0aeaf25] SciMLOperators v0.3.6
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⌃ [82ae8749] StatsAPI v1.6.0
⌃ [2913bbd2] StatsBase v0.34.0
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⌃ [0c5d862f] Symbolics v5.5.1
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⌃ [92b13dbe] TaylorIntegration v0.14.2
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⌃ [410a4b4d] Tricks v0.1.7
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⌃ [5c2747f8] URIs v1.4.2
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⌃ [f8c6e375] Git_jll v2.36.1+2
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⌃ [aacddb02] JpegTurbo_jll v2.1.91+0
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⌅ [e9f186c6] Libffi_jll v3.2.2+1
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⌃ [94ce4f54] Libiconv_jll v1.16.1+2
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⌃ [a2964d1f] Wayland_jll v1.21.0+0
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⌃ [02c8fc9c] XML2_jll v2.10.3+0
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⌃ [ffd25f8a] XZ_jll v5.4.3+1
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  [9fa8497b] Future
  [b77e0a4c] InteractiveUtils
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  [37e2e46d] LinearAlgebra
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Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`
Warning The project dependencies or compat requirements have changed since the manifest was last resolved. It is recommended to `Pkg.resolve()` or consider `Pkg.update()` if necessary.