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BifurcationKit Extension (take 2)
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# [Bifurcation Diagrams](@id bifurcation_diagrams) | ||
Bifurcation diagrams describes how, for a dynamic system, the quantity and quality of its steady states changes with a parameter's value. These can be computed through the [BifurcationKit.jl](https://github.com/bifurcationkit/BifurcationKit.jl) package. ModelingToolkit provides a simple interface for creating BifurcationKit compatible `BifurcationProblem`s from `NonlinearSystem`s and `ODESystem`s. All teh features provided by BifurcationKit can then be applied to these systems. This tutorial provides a brief introduction for these features, with BifurcationKit.jl providing [a more extensive documentation](https://bifurcationkit.github.io/BifurcationKitDocs.jl/stable/). | ||
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### Creating a `BifurcationProblem` | ||
Let us first consider a simple `NonlinearSystem`: | ||
```@example Bif1 | ||
using ModelingToolkit | ||
@variables t x(t) y(t) | ||
@parameters μ α | ||
eqs = [0 ~ μ*x - x^3 + α*y, | ||
0 ~ -y] | ||
@named nsys = NonlinearSystem(eqs, [x, y], [μ, α]) | ||
``` | ||
we wish to compute a bifurcation diagram for this system as we vary the parameter `μ`. For this, we need to provide the following information: | ||
1. The system for which we wish to compute the bifurcation diagram (`nsys`). | ||
2. The parameter which we wish to vary (`μ`). | ||
3. The parameter set for which we want to compute the bifurcation diagram. | ||
4. An initial guess of the state of the system for which there is a steady state at our provided parameter value. | ||
5. The variable which value we wish to plot in the bifurcation diagram (this argument is optional, if not provided, BifurcationKit default plot functions are used). | ||
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We declare this additional information: | ||
```@example Bif1 | ||
bif_par = μ | ||
p_start = [μ => -1.0, α => 1.0] | ||
u0_guess = [x => 1.0, y => 1.0] | ||
plot_var = x; | ||
``` | ||
For the initial state guess (`u0_guess`), typically any value can be provided, however, read BifurcatioKit's documentation for more details. | ||
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We can now create our `BifurcationProblem`, which can be provided as input to BifurcationKit's various functions. | ||
```@example Bif1 | ||
using BifurcationKit | ||
bprob = BifurcationProblem(nsys, u0_guess, p_start, bif_par; plot_var=plot_var, jac=false) | ||
``` | ||
Here, the `jac` argument (by default set to `true`) sets whenever to provide BifurcationKit with a Jacobian or not. | ||
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### Computing a bifurcation diagram | ||
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Let us consider the `BifurcationProblem` from the last section. If we wish to compute the corresponding bifurcation diagram we must first declare various settings used by BifurcationKit to compute the diagram. These are stored in a `ContinuationPar` structure (which also contain a `NewtonPar` structure). | ||
```@example Bif1 | ||
p_span = (-4.0, 6.0) | ||
opt_newton = NewtonPar(tol = 1e-9, max_iterations = 20) | ||
opts_br = ContinuationPar(dsmin = 0.001, dsmax = 0.05, ds = 0.01, | ||
max_steps = 100, nev = 2, newton_options = opt_newton, | ||
p_min = p_span[1], p_max = p_span[2], | ||
detect_bifurcation = 3, n_inversion = 4, tol_bisection_eigenvalue = 1e-8, dsmin_bisection = 1e-9); | ||
``` | ||
Here, `p_span` sets the interval over which we wish to compute the diagram. | ||
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Next, we can use this as input to our bifurcation diagram, and then plot it. | ||
```@example Bif1 | ||
bf = bifurcationdiagram(bprob, PALC(), 2, (args...) -> opts_br; bothside=true) | ||
``` | ||
Here, the value `2` sets how sub-branches of the diagram that BifurcationKit should compute. Generally, for bifurcation diagrams, it is recommended to use the `bothside=true` argument. | ||
```@example Bif1 | ||
using Plots | ||
plot(bf; putspecialptlegend=false, markersize=2, plotfold=false, xguide="μ", yguide = "x") | ||
``` | ||
Here, the system exhibits a pitchfork bifurcation at *μ=0.0*. | ||
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### Using `ODESystem` inputs | ||
It is also possible to use `ODESystem`s (rather than `NonlinearSystem`s) as input to `BifurcationProblem`. Here follows a brief such example. | ||
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```@example Bif2 | ||
using BifurcationKit, ModelingToolkit, Plots | ||
@variables t x(t) y(t) | ||
@parameters μ | ||
D = Differential(t) | ||
eqs = [D(x) ~ μ*x - y - x*(x^2+y^2), | ||
D(y) ~ x + μ*y - y*(x^2+y^2)] | ||
@named osys = ODESystem(eqs, t) | ||
bif_par = μ | ||
plot_var = x | ||
p_start = [μ => 1.0] | ||
u0_guess = [x => 0.0, y=> 0.0] | ||
bprob = BifurcationProblem(osys, u0_guess, p_start, bif_par; plot_var=plot_var, jac=false) | ||
p_span = (-3.0, 3.0) | ||
opt_newton = NewtonPar(tol = 1e-9, max_iterations = 20) | ||
opts_br = ContinuationPar(dsmin = 0.001, dsmax = 0.05, ds = 0.01, | ||
max_steps = 100, nev = 2, newton_options = opt_newton, | ||
p_max = p_span[2], p_min = p_span[1], | ||
detect_bifurcation = 3, n_inversion = 4, tol_bisection_eigenvalue = 1e-8, dsmin_bisection = 1e-9) | ||
bf = bifurcationdiagram(bprob, PALC(), 2, (args...) -> opts_br; bothside=true) | ||
using Plots | ||
plot(bf; putspecialptlegend=false, markersize=2, plotfold=false, xguide="μ", yguide = "x") | ||
``` | ||
Here, the value of `x` in the steady state does not change, however, at `μ=0` a Hopf bifurcation occur and the steady state turn unstable. |
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module MTKBifurcationKitExt | ||
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### Preparations ### | ||
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# Imports | ||
using ModelingToolkit, Setfield | ||
import BifurcationKit | ||
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### Creates BifurcationProblem Overloads ### | ||
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# When input is a NonlinearSystem. | ||
function BifurcationKit.BifurcationProblem(nsys::NonlinearSystem, u0_bif, ps, bif_par, args...; plot_var=nothing, record_from_solution=BifurcationKit.record_sol_default, jac=true, kwargs...) | ||
# Creates F and J functions. | ||
ofun = NonlinearFunction(nsys; jac=jac) | ||
F = ofun.f | ||
J = jac ? ofun.jac : nothing | ||
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# Computes bifurcation parameter and plot var indexes. | ||
bif_idx = findfirst(isequal(bif_par), parameters(nsys)) | ||
if !isnothing(plot_var) | ||
plot_idx = findfirst(isequal(plot_var), states(nsys)) | ||
record_from_solution = (x, p) -> x[plot_idx] | ||
end | ||
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# Converts the input state guess. | ||
u0_bif = ModelingToolkit.varmap_to_vars(u0_bif, states(nsys)) | ||
ps = ModelingToolkit.varmap_to_vars(ps, parameters(nsys)) | ||
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return BifurcationKit.BifurcationProblem(F, u0_bif, ps, (@lens _[bif_idx]), args...; record_from_solution = record_from_solution, J = J, kwargs...) | ||
end | ||
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# When input is a ODESystem. | ||
function BifurcationKit.BifurcationProblem(osys::ODESystem, args...; kwargs...) | ||
nsys = NonlinearSystem([0 ~ eq.rhs for eq in equations(osys)], states(osys), parameters(osys); name=osys.name) | ||
return BifurcationKit.BifurcationProblem(nsys, args...; kwargs...) | ||
end | ||
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end # module |
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using BifurcationKit, ModelingToolkit, Test | ||
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# Checks pitchfork diagram and that there are the correct number of branches (a main one and two children) | ||
let | ||
@variables t x(t) y(t) | ||
@parameters μ α | ||
eqs = [0 ~ μ*x - x^3 + α*y, | ||
0 ~ -y] | ||
@named nsys = NonlinearSystem(eqs, [x, y], [μ, α]) | ||
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bif_par = μ | ||
p_start = [μ => -1.0, α => 1.0] | ||
u0_guess = [x => 1.0, y => 1.0] | ||
plot_var = x; | ||
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using BifurcationKit | ||
bprob = BifurcationProblem(nsys, u0_guess, p_start, bif_par; plot_var=plot_var, jac=false) | ||
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p_span = (-4.0, 6.0) | ||
opt_newton = NewtonPar(tol = 1e-9, max_iterations = 20) | ||
opts_br = ContinuationPar(dsmin = 0.001, dsmax = 0.05, ds = 0.01, | ||
max_steps = 100, nev = 2, newton_options = opt_newton, | ||
p_min = p_span[1], p_max = p_span[2], | ||
detect_bifurcation = 3, n_inversion = 4, tol_bisection_eigenvalue = 1e-8, dsmin_bisection = 1e-9); | ||
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bf = bifurcationdiagram(bprob, PALC(), 2, (args...) -> opts_br; bothside=true) | ||
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@test length(bf.child) == 2 | ||
end |
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