Extension to the SciML ecosystem

The SciML ecosystem provides a rich set of tools to solve (non)-linear equations, differential equations and inverse problems. We provide an interface (in the form of Package extensions) to export the the derived harmonic equations computed with harmonic balance method or krylov-bogoliubov method to the SciML ecosystem.

ModeligToolkit.jl

The ModelingToolkit.jl (MTK) package provides a symbolic framework for defining and simplifying mathematical models. Through, MTK SciML provides a symbolic interface for their ecosystem

SciMLBase.ODEProblem Method

ODEProblem(
    eom::Union{DifferentialEquation, HarmonicEquation},
    u0,
    tspan::Tuple,
    p::AbstractDict;
    in_place,
    kwargs...
) -> Any

Creates and ModelingToolkit.ODEProblem from a DifferentialEquation or HarmonicEquation.

Example

using ModelingToolkit, StaticArrays

@variables α ω ω0 F γ t x(t)
diff_eq = DifferentialEquation(
    d(x, t, 2) + ω0^2 * x + α * x^3 + γ * d(x, t) ~ F * cos(ω * t), x
)
add_harmonic!(diff_eq, x, ω) #
harmonic_eq = get_harmonic_equations(diff_eq)

# in place (most performant for large systems)
ODEProblem(harmonic_eq, [1.0, 0.0], (0, 100), param)

# out of place (most performant for small systems with StaticArrays)
ODEProblem(
    harmonic_eq, [1.0, 0.0], (0, 100), param;
    in_place=false, u0_constructor=x -> SVector(x...)
)

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ModelingToolkit.ODESystem Type

ODESystem(eom::HarmonicEquation) -> Any

Creates and ModelingToolkit.ODESystem from a HarmonicEquation.

Example

using ModelingToolkit

@variables α ω ω0 F γ t x(t)
diff_eq = DifferentialEquation(
    d(x, t, 2) + ω0^2 * x + α * x^3 + γ * d(x, t) ~ F * cos(ω * t), x
)
add_harmonic!(diff_eq, x, ω) #
harmonic_eq = get_harmonic_equations(diff_eq)

sys = ODESystem(harmonic_eq)
param = (α => 1.0, ω0 => 1.1, F => 0.01, γ => 0.01, ω => 1.1)
ODEProblem(sys, [1.0, 0.0], (0, 100), param)

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ODESystem(diff_eq::DifferentialEquation) -> Any

Creates and ModelingToolkit.ODESystem from a DifferentialEquation.

Example

using ModelingToolkit

@variables α ω ω0 F γ t x(t)
diff_eq = DifferentialEquation(
    d(x, t, 2) + ω0^2 * x + α * x^3 + γ * d(x, t) ~ F * cos(ω * t), x
)
sys = ODESystem(diff_eq)

param = (α => 1.0, ω0 => 1.1, F => 0.01, γ => 0.01, ω => 1.1)

ODEProblem(sys, [1.0, 0.0], (0, 100), param)

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SciMLBase.SteadyStateProblem Type

SteadyStateProblem(
    eom::HarmonicEquation,
    u0,
    p::AbstractDict;
    in_place,
    kwargs...
) -> Any

Creates and ModelingToolkit.SteadyStateProblem from a HarmonicEquation.

Example

using ModelingToolkit, StaticArrays

@variables α ω ω0 F γ t x(t)
diff_eq = DifferentialEquation(
    d(x, t, 2) + ω0^2 * x + α * x^3 + γ * d(x, t) ~ F * cos(ω * t), x
)
add_harmonic!(diff_eq, x, ω) #
harmonic_eq = get_harmonic_equations(diff_eq)


SteadyStateProblem(harmonic_eq, [1.0, 0.0], param)

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SciMLBase.NonlinearProblem Type

NonlinearProblem(
    eom::HarmonicEquation,
    u0,
    p::AbstractDict;
    in_place,
    kwargs...
) -> Any

Creates and ModelingToolkit.NonlinearProblem from a HarmonicEquation.

Example

using ModelingToolkit, StaticArrays

@variables α ω ω0 F γ t x(t)
diff_eq = DifferentialEquation(
    d(x, t, 2) + ω0^2 * x + α * x^3 + γ * d(x, t) ~ F * cos(ω * t), x
)
add_harmonic!(diff_eq, x, ω) #
harmonic_eq = get_harmonic_equations(diff_eq)


NonlinearProblem(harmonic_eq, [1.0, 0.0], param)

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