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Author: TorkelE

docs/src/catalyst_applications/dynamical_systems.md

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 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, these are, however, 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).
 
 ## [Finding basins of attraction](@id dynamical_systems_basins_of_attraction)
-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 (almost always) 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. 
+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. 
 
-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 a `ODEProblem` corresponding to the model which basins of attraction we wish to compute. For this application, `u0` is unused, and its value is irrelevant. The value of `tspan` is not very important, however, it must provide long enough a time range (else basins will not be successfully found). 
+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 a `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 to small interval, we recommend minimum `(0.0, 1.0)`). 
 ```@example dynamical_systems_basins
 using Catalyst
 wilhelm_model = @reaction_network begin
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 sol = solve(oprob, Rodas5P())
 plot(sol; idxs=(:X, :Y, :Z))
 ```
-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 (here, the `tspan` of the used `ODEProblem` have no impact on the computed Lyapunov spectrum).
+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.
 ```@example dynamical_systems_lyapunov
 using DynamicalSystems
 ds = CoupledODEs(oprob, (alg = Rodas5P(autodiff=false),))
@@ -139,9 +139,13 @@ If you use this functionality in your research, [in addition to Catalyst](@ref c
   journal = {Journal of Open Source Software}
 }
 ```
+
+
+---
 ## Learning more
 
-If you want to learn more about analyzing dynamical systems, including chaotic behavior, 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 analyzing dynamical systems.
+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].
+
 
 ---
 ## References