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 see 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 OrdinaryDiffEqRKN, OrdinaryDiffEqSymplecticRK
using OrdinaryDiffEqTaylorSeries
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, 0.0]
const iip_p0 = [0.0, 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, 0.0]
const oop_p0 = @SVector [0.0, 0.5]

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

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

    return nothing
end

let u = vcat(iip_p0, iip_q0), du = fill(NaN, 4)
    u_before = copy(u)
    hamilton(du, u, nothing, 0.0)
    @assert u == u_before "hamilton must not mutate its input state"
    @assert du ≈ [-0.1, -0.01, 0.0, 0.5] "hamilton returned an incorrect derivative"
end

function hamilton_taylor!(du, u, p, t)
    du[1] = -u[3] * (1 + 2u[4])
    du[2] = -u[4] - (u[3]^2 - u[4]^2)
    du[3] = u[1]
    du[4] = u[2]
    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

function g_jacobian(J, u, p)
    J[1, 1] = u[1]
    J[1, 2] = u[2]
    J[1, 3] = u[3]
    J[1, 4] = u[4]
    J[2:4, :] .= 0
end

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

const E = H(iip_p0, iip_q0, nothing)

0.13

For the comparison we will use the following function

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))
end
function abs_energy_err(sol)
    [abs.(H([sol[1, j], sol[2, j]], [sol[3, j], sol[4, j]], nothing) - E)
     for j in 1:length(sol.u)]
end

function compare(mode = :inplace, all = true, plt = nothing; tmax = 1e2)
    if mode == :inplace
        prob = DynamicalODEProblem(iip_dp, iip_dq, iip_p0, iip_q0, (0.0, tmax))
    else
        prob = DynamicalODEProblem(oop_dp, oop_dq, oop_p0, oop_q0, (0.0, tmax))
    end
    prob_linear = ODEProblem(hamilton, vcat(iip_p0, iip_q0), (0.0, tmax))
    prob_taylor = ODEProblem{true, SciMLBase.FullSpecialize}(
        hamilton_taylor!, vcat(iip_p0, iip_q0), (0.0, tmax))

    # Cap saved points so energy-error plots stay CI-friendly. Default
    # save_everystep+dense at tmax=5e4 stores ~5e6 states per symplectic
    # solve and Plots.jl of multi-million-point series is what pinned the
    # self-hosted runner for multi-day runs (see CI run 30654781033).
    nsave = clamp(Int(round(tmax)) + 1, 101, 1001)
    saveat = range(0.0, tmax; length = nsave)
    common = (; dense = false, saveat)

    GC.gc()
    (mode == :inplace && all) &&
        @time sol1 = solve(prob, Vern9(), callback = cb, abstol = 1e-14, reltol = 1e-14;
            common...)
    GC.gc()
    @time sol2 = solve(prob, KahanLi8(), dt = 1e-2, maxiters = 1e7; common...)
    GC.gc()
    @time sol3 = solve(prob, SofSpa10(), dt = 1e-2, maxiters = 1e7; common...)
    GC.gc()
    @time sol4 = solve(prob, Vern9(), abstol = 1e-14, reltol = 1e-14; common...)
    GC.gc()
    @time sol5 = solve(prob, DPRKN12(), abstol = 1e-14, reltol = 1e-14; common...)
    GC.gc()
    (mode == :inplace && all) &&
        @time sol6 = solve(prob_linear, TaylorMethod(50), abstol = 1e-20; common...)
    GC.gc()
    (mode == :inplace && all) &&
        @time sol7 = solve(prob_taylor, ExplicitTaylor(order = Val(8)),
            abstol = 1e-14, reltol = 1e-14; common...)

    (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.")
    (mode == :inplace && all) &&
        println("ExplicitTaylor max energy error:\t\t\t$(maximum(abs_energy_err(sol7)))\tin\t$(length(sol7.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")
    (mode == :inplace && all) && plot!(sol7.t, energy_err(sol7), label = "ExplicitTaylor")

    return plt
end

compare (generic function with 4 methods)

The mode argument chooses 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 refer here only to the 6 high order methods chosen below). 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)

98.124480 seconds (182.98 M allocations: 9.096 GiB, 5.32% gc time, 99.98% 
compilation time: <1% of which was recompilation)
  4.278598 seconds (5.41 M allocations: 292.028 MiB, 1.15% gc time, 99.84% 
compilation time)
  2.997120 seconds (3.04 M allocations: 171.177 MiB, 99.62% compilation tim
e)
 31.716866 seconds (68.67 M allocations: 3.511 GiB, 4.19% gc time, 99.98% c
ompilation time)
 10.035292 seconds (14.05 M allocations: 748.541 MiB, 2.25% gc time, 99.95%
 compilation time)
  6.094978 seconds (7.58 M allocations: 535.031 MiB, 3.08% gc time, 97.70% 
compilation time)
 44.734015 seconds (27.58 M allocations: 1.410 GiB, 1.05% gc time, 99.88% c
ompilation time: 9% of which was recompilation)
Vern9 + ManifoldProjection max energy error:	1.582067810090848e-15	in	1040	
steps.
KahanLi8 max energy error:			4.718447854656915e-15	in	101	steps.
SofSpa10 max energy error:			5.2735593669694936e-15	in	101	steps.
Vern9 max energy error:				1.582067810090848e-15	in	101	steps.
DPRKN12 max energy error:			9.990739409687643e-6	in	101	steps.
TaylorMethod max energy error:			1.942890293094024e-16	in	101	steps.
ExplicitTaylor max energy error:			1.1546319456101628e-14	in	101	steps.

compare(tmax = 1e3)

0.078428 seconds (412.06 k allocations: 28.735 MiB)
  0.059519 seconds (4.11 k allocations: 217.305 KiB)
  0.105577 seconds (4.11 k allocations: 219.664 KiB)
  0.053770 seconds (85.49 k allocations: 2.605 MiB)
  0.007243 seconds (20.33 k allocations: 549.156 KiB)
  0.712693 seconds (1.96 M allocations: 438.041 MiB, 21.89% gc time)
  0.049140 seconds (7.57 k allocations: 414.750 KiB, 72.77% compilation tim
e)
Vern9 + ManifoldProjection max energy error:	5.245803791353865e-15	in	10330
	steps.
KahanLi8 max energy error:			1.8096635301390052e-14	in	1001	steps.
SofSpa10 max energy error:			2.7533531010703882e-14	in	1001	steps.
Vern9 max energy error:				5.245803791353865e-15	in	1001	steps.
DPRKN12 max energy error:			1.0635715396917211e-5	in	1001	steps.
TaylorMethod max energy error:			3.885780586188048e-16	in	1001	steps.
ExplicitTaylor max energy error:			8.149037000748649e-14	in	1001	steps.

# Long-horizon comparison without Taylor / ExplicitTaylor / ManifoldProjection
# (those are the expensive paths; energy-trend story is carried by the symplectic
# and high-order RK methods already).
compare(:inplace, false; tmax = 1e4)

0.596428 seconds (4.11 k allocations: 217.305 KiB)
  0.961624 seconds (4.11 k allocations: 219.664 KiB)
  0.341150 seconds (421.03 k allocations: 9.004 MiB)
  0.058994 seconds (156.44 k allocations: 3.132 MiB)
KahanLi8 max energy error:			3.1002977962657496e-14	in	1001	steps.
SofSpa10 max energy error:			1.1304845948245656e-13	in	1001	steps.
Vern9 max energy error:				4.421463195569686e-14	in	1001	steps.
DPRKN12 max energy error:			1.0780853129166434e-5	in	1001	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 = 1e2 (kept short so Taylor / ExplicitTaylor stay in the CI budget):

in_vs_out(all = true, tmax = 1e2)

In place versions:
  0.010696 seconds (41.64 k allocations: 2.923 MiB)
  0.006036 seconds (503 allocations: 28.750 KiB)
  0.010656 seconds (503 allocations: 31.109 KiB)
  0.005489 seconds (8.72 k allocations: 276.125 KiB)
  0.001203 seconds (3.11 k allocations: 94.820 KiB)
  0.037175 seconds (196.73 k allocations: 43.936 MiB)
  0.001481 seconds (301 allocations: 20.422 KiB)
Vern9 + ManifoldProjection max energy error:	1.582067810090848e-15	in	1040	
steps.
KahanLi8 max energy error:			4.718447854656915e-15	in	101	steps.
SofSpa10 max energy error:			5.2735593669694936e-15	in	101	steps.
Vern9 max energy error:				1.582067810090848e-15	in	101	steps.
DPRKN12 max energy error:			9.990739409687643e-6	in	101	steps.
TaylorMethod max energy error:			1.942890293094024e-16	in	101	steps.
ExplicitTaylor max energy error:			1.1546319456101628e-14	in	101	steps.

Out of place versions:
  3.538339 seconds (3.75 M allocations: 202.249 MiB, 99.91% compilation tim
e)
  1.649109 seconds (1.68 M allocations: 96.980 MiB, 99.70% compilation time
)
  3.224454 seconds (4.40 M allocations: 217.349 MiB, 99.98% compilation tim
e)
  2.662170 seconds (2.66 M allocations: 138.900 MiB, 99.97% compilation tim
e)
KahanLi8 max energy error:			4.718447854656915e-15	in	101	steps.
SofSpa10 max energy error:			5.2735593669694936e-15	in	101	steps.
Vern9 max energy error:				1.7208456881689926e-15	in	101	steps.
DPRKN12 max energy error:			9.0510035500313e-6	in	101	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:
  0.006112 seconds (503 allocations: 28.750 KiB)
  0.010486 seconds (503 allocations: 31.109 KiB)
  0.005401 seconds (8.72 k allocations: 276.125 KiB)
  0.001254 seconds (3.11 k allocations: 94.820 KiB)
KahanLi8 max energy error:			4.718447854656915e-15	in	101	steps.
SofSpa10 max energy error:			5.2735593669694936e-15	in	101	steps.
Vern9 max energy error:				1.582067810090848e-15	in	101	steps.
DPRKN12 max energy error:			9.990739409687643e-6	in	101	steps.

Out of place versions:
  0.002855 seconds (35 allocations: 10.055 KiB)
  0.006240 seconds (35 allocations: 11.039 KiB)
  0.000564 seconds (34 allocations: 10.742 KiB)
  0.000685 seconds (1.77 k allocations: 169.531 KiB)
KahanLi8 max energy error:			4.718447854656915e-15	in	101	steps.
SofSpa10 max energy error:			5.2735593669694936e-15	in	101	steps.
Vern9 max energy error:				1.7208456881689926e-15	in	101	steps.
DPRKN12 max energy error:			9.0510035500313e-6	in	101	steps.

in_vs_out(tmax = 1e3)

In place versions:
  0.059546 seconds (4.11 k allocations: 217.305 KiB)
  0.105802 seconds (4.11 k allocations: 219.664 KiB)
  0.056449 seconds (85.49 k allocations: 2.605 MiB)
  0.009148 seconds (20.33 k allocations: 549.156 KiB)
KahanLi8 max energy error:			1.8096635301390052e-14	in	1001	steps.
SofSpa10 max energy error:			2.7533531010703882e-14	in	1001	steps.
Vern9 max energy error:				5.245803791353865e-15	in	1001	steps.
DPRKN12 max energy error:			1.0635715396917211e-5	in	1001	steps.

Out of place versions:
  0.021970 seconds (41 allocations: 72.148 KiB)
  0.038007 seconds (41 allocations: 73.133 KiB)
  0.004040 seconds (40 allocations: 72.836 KiB)
  0.003646 seconds (8.58 k allocations: 1.317 MiB)
KahanLi8 max energy error:			1.8096635301390052e-14	in	1001	steps.
SofSpa10 max energy error:			2.7533531010703882e-14	in	1001	steps.
Vern9 max energy error:				5.800915303666443e-15	in	1001	steps.
DPRKN12 max energy error:			1.1218924397565777e-5	in	1001	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 prominent 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 1.11.9
Commit 53a02c0720c (2026-02-06 00:27 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
  LLVM: libLLVM-16.0.6 (ORCJIT, znver2)
Threads: 128 default, 0 interactive, 64 GC (on 128 virtual cores)
Environment:
  JULIA_DEPOT_PATH = /home/crackauc/github-runners/amdci8-1/.julia
  JULIA_NUM_THREADS = auto

Package Information:

Status `~/github-runners/amdci8-1/_work/SciMLBenchmarks.jl/SciMLBenchmarks.jl/benchmarks/DynamicalODE/Project.toml`
  [459566f4] DiffEqCallbacks v4.19.4
  [055956cb] DiffEqPhysics v3.22.0
  [b305315f] Elliptic v1.0.1
  [1dea7af3] OrdinaryDiffEq v7.8.1
  [af6ede74] OrdinaryDiffEqRKN v2.2.0
  [fa646aed] OrdinaryDiffEqSymplecticRK v2.2.2
  [9c7f1690] OrdinaryDiffEqTaylorSeries v2.3.0
  [65888b18] ParameterizedFunctions v5.27.0
  [91a5bcdd] Plots v1.41.7
  [31c91b34] SciMLBenchmarks v0.2.1
  [90137ffa] StaticArrays v1.9.20
  [10745b16] Statistics v1.11.5
  [92b13dbe] TaylorIntegration v0.18.16
  [37e2e46d] LinearAlgebra v1.11.0
  [de0858da] Printf v1.11.0

And the full manifest:

Status `~/github-runners/amdci8-1/_work/SciMLBenchmarks.jl/SciMLBenchmarks.jl/benchmarks/DynamicalODE/Manifest.toml`
  [47edcb42] ADTypes v1.24.0
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  [79e6a3ab] Adapt v4.7.0
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⌃ [4fba245c] ArrayInterface v7.30.1
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⌅ [861a8166] Combinatorics v1.0.2
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⌅ [53c48c17] FixedPointNumbers v0.8.6
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  [28b8d3ca] GR v0.73.27
  [86223c79] Graphs v1.15.0
  [076d061b] HashArrayMappedTries v0.2.0
⌅ [eafb193a] Highlights v0.5.3
  [3263718b] ImplicitDiscreteSolve v2.3.0
  [d25df0c9] Inflate v0.1.5
  [18e54dd8] IntegerMathUtils v0.1.4
  [8197267c] IntervalSets v0.7.14
  [3587e190] InverseFunctions v0.1.17
  [92d709cd] IrrationalConstants v0.2.6
  [82899510] IteratorInterfaceExtensions v1.0.0
  [033835bb] JLD2 v0.6.6
  [1019f520] JLFzf v0.1.11
  [692b3bcd] JLLWrappers v1.8.0
⌅ [682c06a0] JSON v0.21.4
  [ae98c720] Jieko v0.2.1
⌃ [ccbc3e58] JumpProcesses v9.32.3
  [ba0b0d4f] Krylov v0.10.10
  [2faa5264] LHLFactorization v2.2.2
  [b964fa9f] LaTeXStrings v1.4.1
  [23fbe1c1] Latexify v0.16.12
  [87fe0de2] LineSearch v0.1.18
⌃ [7ed4a6bd] LinearSolve v5.17.3
  [2ab3a3ac] LogExpFunctions v1.0.1
  [e6f89c97] LoggingExtras v1.2.0
  [1914dd2f] MacroTools v0.5.16
  [bb5d69b7] MaybeInplace v0.1.8
  [442fdcdd] Measures v0.3.3
  [e1d29d7a] Missings v1.2.0
  [961ee093] ModelingToolkit v11.43.1
⌃ [7771a370] ModelingToolkitBase v1.71.2
  [6bb917b9] ModelingToolkitTearing v1.20.6
⌅ [2e0e35c7] Moshi v0.3.9
  [46d2c3a1] MuladdMacro v0.2.7
  [102ac46a] MultivariatePolynomials v0.5.19
  [ffc61752] Mustache v1.0.21
  [d8a4904e] MutableArithmetics v1.8.0
  [77ba4419] NaNMath v1.1.4
⌃ [8913a72c] NonlinearSolve v4.30.0
⌃ [be0214bd] NonlinearSolveBase v2.49.5
⌃ [5959db7a] NonlinearSolveFirstOrder v2.6.1
  [9a2c21bd] NonlinearSolveQuasiNewton v1.15.3
  [26075421] NonlinearSolveSpectralMethods v1.8.3
  [6fe1bfb0] OffsetArrays v1.17.0
  [bac558e1] OrderedCollections v2.0.1
  [1dea7af3] OrdinaryDiffEq v7.8.1
⌃ [6ad6398a] OrdinaryDiffEqBDF v2.4.9
⌃ [bbf590c4] OrdinaryDiffEqCore v4.17.2
  [50262376] OrdinaryDiffEqDefault v2.6.2
  [4302a76b] OrdinaryDiffEqDifferentiation v3.12.0
  [127b3ac7] OrdinaryDiffEqNonlinearSolve v2.9.8
  [af6ede74] OrdinaryDiffEqRKN v2.2.0
  [43230ef6] OrdinaryDiffEqRosenbrock v2.7.3
  [b4bd8bb3] OrdinaryDiffEqRosenbrockTableaus v2.4.2
⌃ [2d112036] OrdinaryDiffEqSDIRK v2.9.4
  [fa646aed] OrdinaryDiffEqSymplecticRK v2.2.2
  [9c7f1690] OrdinaryDiffEqTaylorSeries v2.3.0
  [b1df2697] OrdinaryDiffEqTsit5 v2.1.4
  [79d7bb75] OrdinaryDiffEqVerner v2.4.1
  [65888b18] ParameterizedFunctions v5.27.0
  [d96e819e] Parameters v0.13.1
⌅ [69de0a69] Parsers v2.8.8
  [ccf2f8ad] PlotThemes v3.3.0
  [995b91a9] PlotUtils v1.4.4
  [91a5bcdd] Plots v1.41.7
  [e409e4f3] PoissonRandom v0.4.13
  [d236fae5] PreallocationTools v1.7.1
⌅ [aea7be01] PrecompileTools v1.2.1
  [21216c6a] Preferences v1.6.0
  [08abe8d2] PrettyTables v3.4.8
  [27ebfcd6] Primes v0.5.7
  [43287f4e] PtrArrays v1.4.0
  [0c0d3e7f] PureKLU v1.5.0
  [988b38a3] ReadOnlyArrays v0.2.0
  [795d4caa] ReadOnlyDicts v1.0.1
  [c1ae055f] RealDot v0.1.0
  [3cdcf5f2] RecipesBase v1.3.4
  [01d81517] RecipesPipeline v0.6.12
  [731186ca] RecursiveArrayTools v4.5.1
  [189a3867] Reexport v1.2.2
  [05181044] RelocatableFolders v1.0.1
  [ae029012] Requires v1.3.1
  [9fe22ead] RespecializeParams v1.3.0
  [7e49a35a] RuntimeGeneratedFunctions v0.5.26
  [9dfe8606] SCCNonlinearSolve v1.15.3
  [0bca4576] SciMLBase v3.54.0
  [31c91b34] SciMLBenchmarks v0.2.1
  [19f34311] SciMLJacobianOperators v0.1.19
  [a6db7da4] SciMLLogging v2.1.0
⌃ [c0aeaf25] SciMLOperators v1.30.0
  [431bcebd] SciMLPublic v1.3.0
  [53ae85a6] SciMLStructures v1.10.5
  [7e506255] ScopedValues v1.6.2
  [6c6a2e73] Scratch v1.3.0
  [efcf1570] Setfield v1.1.2
  [992d4aef] Showoff v1.1.1
  [727e6d20] SimpleNonlinearSolve v2.14.5
  [699a6c99] SimpleTraits v0.9.6
  [a2af1166] SortingAlgorithms v1.2.3
  [a57abbd0] SparseColumnPivotedQR v2.1.8
  [dc90abb0] SparseInverseSubset v0.1.3
  [0a514795] SparseMatrixColorings v0.4.28
  [276daf66] SpecialFunctions v2.9.0
  [860ef19b] StableRNGs v1.0.4
  [0c0c59c1] StarAlgebras v0.3.0
  [64909d44] StateSelection v1.11.1
  [90137ffa] StaticArrays v1.9.20
  [1e83bf80] StaticArraysCore v1.4.4
  [10745b16] Statistics v1.11.5
  [82ae8749] StatsAPI v1.8.0
  [2913bbd2] StatsBase v0.34.13
  [69024149] StringEncodings v0.3.7
⌅ [892a3eda] StringManipulation v0.5.0
  [09ab397b] StructArrays v0.7.3
  [2efcf032] SymbolicIndexingInterface v0.3.55
  [19f23fe9] SymbolicLimits v1.2.1
  [d1185830] SymbolicUtils v4.46.6
  [0c5d862f] Symbolics v7.39.2
  [3783bdb8] TableTraits v1.0.1
  [bd369af6] Tables v1.14.0
  [ed4db957] TaskLocalValues v0.1.3
  [b36ab563] TaylorDiff v0.3.5
  [92b13dbe] TaylorIntegration v0.18.16
  [6aa5eb33] TaylorSeries v0.22.4
  [62fd8b95] TensorCore v0.1.1
  [8ea1fca8] TermInterface v2.0.0
  [a759f4b9] TimerOutputs v1.2.1
  [781d530d] TruncatedStacktraces v1.4.0
  [3a884ed6] UnPack v1.0.2
  [1cfade01] UnicodeFun v0.4.1
  [41fe7b60] Unzip v0.2.0
  [d30d5f5c] WeakCacheSets v0.1.0
  [44d3d7a6] Weave v0.10.12
  [ddb6d928] YAML v0.4.16
  [6e34b625] Bzip2_jll v1.0.9+0
  [83423d85] Cairo_jll v1.18.7+0
  [ee1fde0b] Dbus_jll v1.16.2+0
  [2702e6a9] EpollShim_jll v0.0.20230411+1
  [2e619515] Expat_jll v2.8.4+0
⌅ [b22a6f82] FFMPEG_jll v8.1.2+0
  [a3f928ae] Fontconfig_jll v2.17.1+0
  [d7e528f0] FreeType2_jll v2.14.3+1
  [559328eb] FriBidi_jll v1.0.17+0
  [0656b61e] GLFW_jll v3.5.1+0
  [d2c73de3] GR_jll v0.73.27+0
⌅ [b0724c58] GettextRuntime_jll v0.22.4+0
  [61579ee1] Ghostscript_jll v9.55.1+0
  [7746bdde] Glib_jll v2.88.3+0
  [3b182d85] Graphite2_jll v1.3.16+0
  [2e76f6c2] HarfBuzz_jll v100.14004.0+0
  [1d5cc7b8] IntelOpenMP_jll v2025.2.0+0
  [aacddb02] JpegTurbo_jll v3.2.0+1
  [c1c5ebd0] LAME_jll v3.100.3+0
  [88015f11] LERC_jll v4.2.0+0
  [1d63c593] LLVMOpenMP_jll v23.1.1+0
⌅ [e9f186c6] Libffi_jll v3.4.7+0
  [7e76a0d4] Libglvnd_jll v1.7.1+1
  [94ce4f54] Libiconv_jll v1.18.0+0
  [4b2f31a3] Libmount_jll v2.42.0+0
  [89763e89] Libtiff_jll v4.7.3+0
  [38a345b3] Libuuid_jll v2.42.0+0
  [856f044c] MKL_jll v2025.2.0+0
  [e7412a2a] Ogg_jll v1.3.6+0
  [458c3c95] OpenSSL_jll v3.5.8+0
  [efe28fd5] OpenSpecFun_jll v0.5.6+0
  [91d4177d] Opus_jll v1.6.1+0
  [36c8627f] Pango_jll v1.58.2+0
  [30392449] Pixman_jll v0.46.4+0
  [c0090381] Qt6Base_jll v6.10.2+2
  [629bc702] Qt6Declarative_jll v6.10.2+2
  [ce943373] Qt6ShaderTools_jll v6.10.2+1
  [6de9746b] Qt6Svg_jll v6.10.2+0
  [e99dba38] Qt6Wayland_jll v6.10.2+1
  [a44049a8] Vulkan_Loader_jll v1.3.243+0
  [a2964d1f] Wayland_jll v1.24.0+0
  [ffd25f8a] XZ_jll v5.8.4+0
  [f67eecfb] Xorg_libICE_jll v1.1.2+0
  [c834827a] Xorg_libSM_jll v1.2.6+0
  [4f6342f7] Xorg_libX11_jll v1.8.13+0
  [0c0b7dd1] Xorg_libXau_jll v1.0.13+0
  [935fb764] Xorg_libXcursor_jll v1.2.4+0
  [a3789734] Xorg_libXdmcp_jll v1.1.6+0
  [1082639a] Xorg_libXext_jll v1.3.8+0
  [d091e8ba] Xorg_libXfixes_jll v6.0.2+0
  [a51aa0fd] Xorg_libXi_jll v1.8.4+0
  [d1454406] Xorg_libXinerama_jll v1.1.7+0
  [ec84b674] Xorg_libXrandr_jll v1.5.6+0
  [ea2f1a96] Xorg_libXrender_jll v0.9.12+0
  [a65dc6b1] Xorg_libpciaccess_jll v0.19.0+0
  [c7cfdc94] Xorg_libxcb_jll v1.17.1+0
  [cc61e674] Xorg_libxkbfile_jll v1.2.0+0
  [e920d4aa] Xorg_xcb_util_cursor_jll v0.1.6+0
  [12413925] Xorg_xcb_util_image_jll v0.4.1+0
  [2def613f] Xorg_xcb_util_jll v0.4.1+0
  [975044d2] Xorg_xcb_util_keysyms_jll v0.4.1+0
  [0d47668e] Xorg_xcb_util_renderutil_jll v0.3.10+0
  [c22f9ab0] Xorg_xcb_util_wm_jll v0.4.2+0
  [35661453] Xorg_xkbcomp_jll v1.4.7+0
  [33bec58e] Xorg_xkeyboard_config_jll v2.47.0+2
  [c5fb5394] Xorg_xtrans_jll v1.6.0+0
  [3161d3a3] Zstd_jll v1.5.7+1
  [35ca27e7] eudev_jll v3.2.14+0
⌅ [214eeab7] fzf_jll v0.61.1+0
  [a4ae2306] libaom_jll v3.14.1+0
  [0ac62f75] libass_jll v0.17.5+0
  [1183f4f0] libdecor_jll v0.2.2+0
  [8e53e030] libdrm_jll v2.4.134+0
  [2db6ffa8] libevdev_jll v1.13.4+0
  [f638f0a6] libfdk_aac_jll v2.0.4+0
  [36db933b] libinput_jll v1.28.1+0
  [b53b4c65] libpng_jll v1.6.58+0
  [9a156e7d] libva_jll v2.23.0+0
  [f27f6e37] libvorbis_jll v1.3.8+0
  [009596ad] mtdev_jll v1.1.7+0
  [1317d2d5] oneTBB_jll v2022.3.0+0
⌅ [1270edf5] x264_jll v10164.0.1+0
  [dfaa095f] x265_jll v4.1.0+0
  [d8fb68d0] xkbcommon_jll v1.13.0+0
  [0dad84c5] ArgTools v1.1.2
  [56f22d72] Artifacts v1.11.0
  [2a0f44e3] Base64 v1.11.0
  [ade2ca70] Dates v1.11.0
  [8ba89e20] Distributed v1.11.0
  [f43a241f] Downloads v1.6.0
  [7b1f6079] FileWatching v1.11.0
  [9fa8497b] Future v1.11.0
  [b77e0a4c] InteractiveUtils v1.11.0
  [4af54fe1] LazyArtifacts v1.11.0
  [b27032c2] LibCURL v0.6.4
  [76f85450] LibGit2 v1.11.0
  [8f399da3] Libdl v1.11.0
  [37e2e46d] LinearAlgebra v1.11.0
  [56ddb016] Logging v1.11.0
  [d6f4376e] Markdown v1.11.0
  [a63ad114] Mmap v1.11.0
  [ca575930] NetworkOptions v1.2.0
  [44cfe95a] Pkg v1.11.0
  [de0858da] Printf v1.11.0
  [3fa0cd96] REPL v1.11.0
  [9a3f8284] Random v1.11.0
  [ea8e919c] SHA v0.7.0
  [9e88b42a] Serialization v1.11.0
  [6462fe0b] Sockets v1.11.0
  [2f01184e] SparseArrays v1.11.0
  [f489334b] StyledStrings v1.11.0
  [4607b0f0] SuiteSparse
  [fa267f1f] TOML v1.0.3
  [a4e569a6] Tar v1.10.0
  [8dfed614] Test v1.11.0
  [cf7118a7] UUIDs v1.11.0
  [4ec0a83e] Unicode v1.11.0
  [e66e0078] CompilerSupportLibraries_jll v1.1.1+0
  [deac9b47] LibCURL_jll v8.6.0+0
  [e37daf67] LibGit2_jll v1.7.2+0
  [29816b5a] LibSSH2_jll v1.11.0+1
  [c8ffd9c3] MbedTLS_jll v2.28.6+0
  [14a3606d] MozillaCACerts_jll v2023.12.12
  [4536629a] OpenBLAS_jll v0.3.27+1
  [05823500] OpenLibm_jll v0.8.5+0
  [efcefdf7] PCRE2_jll v10.42.0+1
  [bea87d4a] SuiteSparse_jll v7.7.0+0
  [83775a58] Zlib_jll v1.2.13+1
  [8e850b90] libblastrampoline_jll v5.11.0+0
  [8e850ede] nghttp2_jll v1.59.0+0
  [3f19e933] p7zip_jll v17.4.0+2
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`