Time evolution

Generally, solving the ODE of oscillatory systems in time requires numerically tracking the oscillations. This is a computationally expensive process; however, using the harmonic ansatz removes the oscillatory time-dependence. Simulating instead the harmonic variables of a HarmonicEquation is vastly more efficient - a steady state of the system appears as a fixed point in multidimensional space rather than an oscillatory function.

The extension TimeEvolution is used to interface HarmonicEquation with the solvers contained in OrdinaryDiffEq.jl. Time-dependent parameter sweeps are defined using the object AdiabaticSweep. To use the TimeEvolution extension, one must first load the OrdinaryDiffEq.jl package.

SciMLBase.ODEProblem Method

ODEProblem(
    eom::HarmonicEquation,
    fixed_parameters;
    sweep,
    u0,
    timespan,
    perturb_initial,
    kwargs...
)

Creates an ODEProblem object used by OrdinaryDiffEqTsit5.jl from the equations in eom to simulate time-evolution within timespan. fixed_parameters must be a dictionary mapping parameters+variables to numbers (possible to use a solution index, e.g. solutions[x][y] for branch y of solution x). If u0 is specified, it is used as an initial condition; otherwise the values from fixed_parameters are used.

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HarmonicSteadyState.AdiabaticSweep Type

Represents a sweep of one or more parameters of a HarmonicEquation. During a sweep, the selected parameters vary linearly over some timespan and are constant elsewhere.

Sweeps of different variables can be combined using +.

Fields

• functions::Dict{Num, Function}: Maps each swept parameter to a function.

Examples

# create a sweep of parameter a from 0 to 1 over time 0 -> 100
julia> @variables a,b;
julia> sweep = AdiabaticSweep(a => [0., 1.], (0, 100));
julia> sweep[a](50)
0.5
julia> sweep[a](200)
1.0

# do the same, varying two parameters simultaneously
julia> sweep = AdiabaticSweep([a => [0.,1.], b => [0., 1.]], (0,100))

Successive sweeps can be combined,

sweep1 = AdiabaticSweep(ω => [0.95, 1.0], (0, 2e4))
sweep2 = AdiabaticSweep(λ => [0.05, 0.01], (2e4, 4e4))
sweep = sweep1 + sweep2

multiple parameters can be swept simultaneously,

sweep = AdiabaticSweep([ω => [0.95;1.0], λ => [5e-2;1e-2]], (0, 2e4))

and custom sweep functions may be used.

ωfunc(t) = cos(t)
sweep = AdiabaticSweep(ω => ωfunc)

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In addition, one can use the steady_state_sweep function from SteadyStateDiffEqExt to perform a parameter sweep over the steady states of a system. For this one has to load SteadyStateDiffEq.jl.

HarmonicSteadyState.steady_state_sweep Function

steady_state_sweep(prob::SteadyStateProblem, alg::DynamicSS; varied::Pair, kwargs...)

Sweeps through a range of parameter values using a dynamic steady state solver DynamicSS of the SteadyStateDiffEq.jl package. Given a steady state problem and a parameter to vary, computes the steady state solution for each value in the sweep range. The solutions are returned as a vector where each element corresponds to the steady state found at that parameter value.

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steady_state_sweep(prob_np::NonlinearProblem, prob_ss::SteadyStateProblem,
                  alg_np, alg_ss::DynamicSS; varied::Pair, kwargs...)

Performs a parameter sweep by combining nonlinear root alg_np and steady state solvers alg_ss. For each parameter value, it first attempts a direct nonlinear root solver and checks its stability. If the solution is unstable or not found, it switches to a dynamic steady state solver. This hybrid approach is much faster then only using a steady state solver.

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Plotting

RecipesBase.plot Method

plot(soln::ODESolution, f::String, harm_eq::HarmonicEquation; kwargs...)

Plot a function f of a time-dependent solution soln of harm_eq.

As a function of time

plot(soln::ODESolution, f::String, harm_eq::HarmonicEquation; kwargs...)

f is parsed by Symbolics.jl

parametric plots

plot(soln::ODESolution, f::Vector{String}, harm_eq::HarmonicEquation; kwargs...)

Parametric plot of f[1] against f[2]

Also callable as plot!

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Miscellaneous

Using a time-dependent simulation can verify solution stability in cases where the Jacobian is too expensive to compute.

HarmonicSteadyState.is_stable Function

is_stable(
    steady_solution::OrderedCollections.OrderedDict,
    eom::HarmonicEquation;
    timespan,
    tol,
    perturb_initial
)

Numerically investigate the stability of a solution soln of eom within timespan. The initial condition is displaced by perturb_initial.

Return true the solution evolves within tol of the initial value (interpreted as stable).

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is_stable(
    soln::OrderedCollections.OrderedDict,
    res::HarmonicSteadyState.Result;
    kwargs...
) -> Any

Returns true if the solution soln of the Result res is stable. Stable solutions are real and have all Jacobian eigenvalues Re(λ) <= 0. im_tol : an absolute threshold to distinguish real/complex numbers. rel_tol: Re(λ) considered <=0 if real.(λ) < rel_tol*abs(λmax)

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