|
| 1 | +# [Analysing model steady state properties with DynamicalSystems.jl](@id dynamical_systems) |
| 2 | +The [DynamicalSystems.jl package](https://github.com/JuliaDynamics/DynamicalSystems.jl) implements a wide range of methods for analysing dynamical systems. This includes both continuous-time systems (i.e. ODEs) and discrete-times ones (difference equations, however, these are not relevant to chemical reaction network modelling). Here we give two examples of how DynamicalSystems.jl can be used, with the package's [documentation describing many more features](https://juliadynamics.github.io/DynamicalSystemsDocs.jl/dynamicalsystems/dev/tutorial/). Finally, it should also be noted that DynamicalSystems.jl contain several tools for [analysing data measured from dynamical systems](https://juliadynamics.github.io/DynamicalSystemsDocs.jl/dynamicalsystems/dev/contents/#Exported-submodules). |
| 3 | + |
| 4 | +## [Finding basins of attraction](@id dynamical_systems_basins_of_attraction) |
| 5 | +Given enough time, an ODE will eventually reach a so-called [*attractor*](https://en.wikipedia.org/wiki/Attractor). For chemical reaction networks (CRNs), this will typically be either a *steady state* or a *limit cycle*. Since ODEs are deterministic, which attractor a simulation will reach is uniquely determined by the initial condition (assuming parameter values are fixed). Conversely, each attractor is associated with a set of initial conditions such that model simulations originating in these will tend to that attractor. These sets are called *basins of attraction*. Here, phase space (the space of all possible states of the system) can be divided into a number of basins of attraction equal to the number of attractors. |
| 6 | + |
| 7 | +DynamicalSystems.jl provides a simple interface for finding an ODE's basins of attraction across any given subspace of phase space. In this example we will use the bistable [Wilhelm model](https://bmcsystbiol.biomedcentral.com/articles/10.1186/1752-0509-3-90) (which steady states we have previous [computed using homotopy continuation](@ref homotopy_continuation_basic_example)). As a first step, we create an `ODEProblem` corresponding to the model which basins of attraction we wish to compute. For this application, `u0` and `tspan` is unused, and their values are of little importance (the only exception is than `tspan`, for implementation reason, must provide a not too small interval, we recommend minimum `(0.0, 1.0)`). |
| 8 | +```@example dynamical_systems_basins |
| 9 | +using Catalyst |
| 10 | +wilhelm_model = @reaction_network begin |
| 11 | + k1, Y --> 2X |
| 12 | + k2, 2X --> X + Y |
| 13 | + k3, X + Y --> Y |
| 14 | + k4, X --> 0 |
| 15 | +end |
| 16 | +
|
| 17 | +u0 = [:X => 0.0, :Y => .0] |
| 18 | +tspan = (0.0, 10.0) |
| 19 | +ps = [:k1 => 8.0, :k2 => 2.0, :k3 => 1.0, :k4 => 1.5] |
| 20 | +oprob = ODEProblem(wilhelm_model, u0, tspan, ps) |
| 21 | +nothing # hide |
| 22 | +``` |
| 23 | +Next, for any application of DynamicalSystems.jl, our `ODEProblem` must be converted into a so-called `CoupledODEs` structure. This is done by combining the ODE with the solver (and potential solver options) with which we wish to simulate it (just like when it is simulated using `solve`). Here, we will simply designate the `Tsit5` numeric solver (but provide no other options). |
| 24 | +```@example dynamical_systems_basins |
| 25 | +using DynamicalSystems, OrdinaryDiffEq |
| 26 | +ds = CoupledODEs(oprob, (alg = Tsit5(),)) |
| 27 | +``` |
| 28 | +We can now compute the basins of attraction. This is done by first creating a grid that designates which subspace of phase-space we wish to investigate (here, the corresponding basin of attraction is found for every point on the grid). Next, we create a `AttractorsViaRecurrences` struct, that maps initial conditions to attractors, and then use that as input to the `basins_of_attraction` function. |
| 29 | +```@example dynamical_systems_basins |
| 30 | +# We provide one grid of values for each species. These are then bundled into a tuple. |
| 31 | +x_grid = 0.0:0.03:6.0 |
| 32 | +y_grid = 0.0:0.03:9.0 |
| 33 | +grid = (x_grid, y_grid) |
| 34 | +avr = AttractorsViaRecurrences(ds, grid) |
| 35 | +basins, attractors = basins_of_attraction(avr, grid; show_progress = false) |
| 36 | +attractors |
| 37 | +``` |
| 38 | +Here, `attractors` is a dictionary that maps attractor labels (integers) to attractors. In this case we have two fixed points, one at $(0.0,0.0)$ and one at $(4.5,6.0)$. Next, `basins` is a matrix of equal size to `grid`, where each value is an integer describing to which attractor's basin that state belongs. |
| 39 | + |
| 40 | +DynamicalSystems.jl also provides a simple interface for plotting the resulting basins. This uses [Makie.jl](https://docs.makie.org/stable/), an alternative plotting package to [Plots.jl](https://github.com/JuliaPlots/Plots.jl) (which is typically the preferred plotting package within the context of Catalyst). Generally, Makie is good at creating animations or interactive graphics (however, it is also a [popular competitor to Plots.jl for general-purpose plotting](https://juliapackagecomparisons.github.io/pages/plotting/)). |
| 41 | +```@example dynamical_systems_basins |
| 42 | +using CairoMakie |
| 43 | +heatmap_basins_attractors(grid, basins, attractors) |
| 44 | +``` |
| 45 | +Here, in addition to the basins of attraction, the system's three steady states are marked (the one at the intersection of the two basins is unstable). |
| 46 | + |
| 47 | +!!! warning |
| 48 | + Both Makie and Plots.jl exports a function called `plot`. Hence, if both these packages are imported into the same session, calls to `plot` must be prepended with the package one wishes to use (e.g. `Plots.plot(sol)`). |
| 49 | + |
| 50 | +More information on how to compute basins of attractions for ODEs using DynamicalSystems.jl can be found [here](https://juliadynamics.github.io/DynamicalSystemsDocs.jl/attractors/stable/basins/). |
| 51 | + |
| 52 | +## [Computing Lyapunov exponents](@id dynamical_systems_lyapunov_exponents) |
| 53 | +[*Lyapunov exponents*](https://en.wikipedia.org/wiki/Lyapunov_exponent) are scalar values that can be computed for any attractor of an ODE. For an ODE with $n$ variables, for each state, a total of $n$ Lyapunov exponents can be computed (and these are collectively called the *Lyapunov spectrum*). Positive Lyapunov exponents indicate that trajectories initially infinitesimally close diverge from each other. Conversely, negative Lyapunov exponents suggests that trajectories converge to each others. |
| 54 | + |
| 55 | +While Lyapunov exponents can be used for other purposes, they are primarily used to characterise [*chaotic behaviours*](https://en.wikipedia.org/wiki/Chaos_theory) (where small changes in initial conditions has large effect on the resulting trajectories). Generally, an ODE exhibit chaotic behaviour if its attractor(s) have *at least one* positive Lyapunov exponent. Practically, Lyapunov exponents can be computed using DynamicalSystems.jl's `lyapunovspectrum` function. Here we will use it to investigate two models, one which exhibits chaos and one which do not. |
| 56 | + |
| 57 | +First, let us consider the [Willamowski–Rössler model](@ref ref), which is known to exhibit chaotic behaviour. |
| 58 | +```@example dynamical_systems_lyapunov |
| 59 | +using Catalyst |
| 60 | +wr_model = @reaction_network begin |
| 61 | + k1, 2X --> 3X |
| 62 | + k2, X --> 2X |
| 63 | + k3, Z + 2X --> 2Z |
| 64 | + k4, Y + X --> 2Y |
| 65 | + k5, Y --> ∅ |
| 66 | + k6, 2Z --> ∅ |
| 67 | + k7, Z --> ∅ |
| 68 | +end |
| 69 | +``` |
| 70 | +We can simulate the model, noting that its behaviour seems chaotic. |
| 71 | +```@example dynamical_systems_lyapunov |
| 72 | +using OrdinaryDiffEq, Plots |
| 73 | +
|
| 74 | +u0 = [:X => 1.5, :Y => 1.5, :Z => 1.5] |
| 75 | +tspan = (0.0, 100.0) |
| 76 | +p = [:k1 => 2.1, :k2 => 0.7, :k3 => 2.9, :k4 => 1.1, :k5 => 1.0, :k6 => 0.5, :k7 => 2.7] |
| 77 | +
|
| 78 | +oprob = ODEProblem(wr_model, u0, tspan, p) |
| 79 | +sol = solve(oprob, Rodas5P()) |
| 80 | +plot(sol; idxs=(:X, :Y, :Z)) |
| 81 | +``` |
| 82 | +Next, like when we [computed basins of attraction](@ref dynamical_systems_basins_of_attraction), we create a `CoupledODEs` corresponding to the model and state for which we wish to compute our Lyapunov spectrum. Lke previously, `tspan` must provide some small interval (at least `(0.0, 1.0)` is recommended), but else have no impact on the computed Lyapunov spectrum. |
| 83 | +```@example dynamical_systems_lyapunov |
| 84 | +using DynamicalSystems |
| 85 | +ds = CoupledODEs(oprob, (alg = Rodas5P(autodiff = false),)) |
| 86 | +nothing # hide |
| 87 | +``` |
| 88 | +Here, the `autodiff = false` argument is required when Lyapunov spectrums are computed. We can now provide our `CoupledODEs` (`ds`) to `lyapunovspectrum` to compute the lyapunov spectrum. This function requires a second argument (here set to `100`). Generally setting this to a higher value will increase accuracy, but also increase runtime (since `lyapunovspectrum` is fast for most systems, setting this to a large value is recommended). |
| 89 | +```@example dynamical_systems_lyapunov |
| 90 | +lyapunovspectrum(ds, 100) |
| 91 | +``` |
| 92 | +Here, the largest exponent is positive, suggesting that the model is chaotic (or, more accurately, it has at least one chaotic attractor, to which is approached from the initial condition $(1.5,1.5,1.5)). |
| 93 | + |
| 94 | +Next, we consider the [Brusselator] model. First we simulate the model for two similar initial conditions, confirming that they converge to the same limit cycle: |
| 95 | +```@example dynamical_systems_lyapunov |
| 96 | +brusselator = @reaction_network begin |
| 97 | + A, ∅ --> X |
| 98 | + 1, 2X + Y --> 3X |
| 99 | + B, X --> Y |
| 100 | + 1, X --> ∅ |
| 101 | +end |
| 102 | +
|
| 103 | +u0_1 = [:X => 1.0, :Y => 1.0] |
| 104 | +u0_2 = [:X => 1.2, :Y => 1.0] |
| 105 | +tspan = (0., 25.) |
| 106 | +ps = [:A => 1.0, :B => 4.0] |
| 107 | +
|
| 108 | +oprob1 = ODEProblem(brusselator, u0_1, tspan, ps) |
| 109 | +oprob2 = ODEProblem(brusselator, u0_2, tspan, ps) |
| 110 | +osol1 = solve(oprob1, Tsit5()) |
| 111 | +osol2 = solve(oprob2, Tsit5()) |
| 112 | +plot(osol1; idxs = (:X, :Y)) |
| 113 | +plot!(osol2; idxs = (:X, :Y)) |
| 114 | +``` |
| 115 | +Next, we compute the Lyapunov spectrum at one of the initial conditions: |
| 116 | +```@example dynamical_systems_lyapunov |
| 117 | +ds = CoupledODEs(oprob1, (alg = Rodas5P(autodiff = false),)) |
| 118 | +lyapunovspectrum(ds, 100) |
| 119 | +``` |
| 120 | +Here, all Lyapunov exponents are negative, confirming that the brusselator is non-chaotic. |
| 121 | + |
| 122 | +More details on how to compute Lyapunov exponents using DynamicalSystems.jl can be found [here](https://juliadynamics.github.io/ChaosTools.jl/stable/lyapunovs/). A full overview of tools for analysing chaotic behaviours (using the "ChaosTools.jl subpackage) can be found [here](https://juliadynamics.github.io/ChaosTools.jl/stable/). |
| 123 | + |
| 124 | + |
| 125 | +--- |
| 126 | +## [Citations](@id dynamical_systems_citations) |
| 127 | +If you use this functionality in your research, [in addition to Catalyst](@ref catalyst_citation), please cite the following paper to support the author of the DynamicalSystems.jl package: |
| 128 | +``` |
| 129 | +@article{DynamicalSystems.jl-2018, |
| 130 | + doi = {10.21105/joss.00598}, |
| 131 | + url = {https://doi.org/10.21105/joss.00598}, |
| 132 | + year = {2018}, |
| 133 | + month = {mar}, |
| 134 | + volume = {3}, |
| 135 | + number = {23}, |
| 136 | + pages = {598}, |
| 137 | + author = {George Datseris}, |
| 138 | + title = {DynamicalSystems.jl: A Julia software library for chaos and nonlinear dynamics}, |
| 139 | + journal = {Journal of Open Source Software} |
| 140 | +} |
| 141 | +``` |
| 142 | + |
| 143 | + |
| 144 | +--- |
| 145 | +## Learning more |
| 146 | + |
| 147 | +If you want to learn more about analysing dynamical systems, including chaotic behaviour, you can have a look at the textbook [Nonlinear Dynamics](https://link.springer.com/book/10.1007/978-3-030-91032-7). It utilizes DynamicalSystems.jl and provides a concise, hands-on approach to learning nonlinear dynamics and analysing dynamical systems [^1]. |
| 148 | + |
| 149 | + |
| 150 | +--- |
| 151 | +## References |
| 152 | +[^1]: [G. Datseris, U. Parlitz, *Nonlinear dynamics: A concise introduction interlaced with code*, Springer (2022).](https://link.springer.com/book/10.1007/978-3-030-91032-7) |
| 153 | +[^2]: [S. H. Strogatz, *Nonlinear Dynamics and Chaos*, Westview Press (1994).](http://users.uoa.gr/~pjioannou/nonlin/Strogatz,%20S.%20H.%20-%20Nonlinear%20Dynamics%20And%20Chaos.pdf) |
| 154 | +[^3]: [A. M. Lyapunov, *The general problem of the stability of motion*, International Journal of Control (1992).](https://www.tandfonline.com/doi/abs/10.1080/00207179208934253) |
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