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Update docs/src/catalyst_applications/dynamical_systems.md
Co-authored-by: George Datseris <[email protected]>
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docs/src/catalyst_applications/dynamical_systems.md

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@@ -51,7 +51,7 @@ Here, in addition to the basins of attraction, the system's three steady states
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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/).
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## [Computing Lyapunov exponents](@id dynamical_systems_lyapunov_exponents)
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[*Lyapunov exponents*](https://en.wikipedia.org/wiki/Lyapunov_exponent) are scalar values that can be computed for any state 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*). Large Lyapunov exponents indicates that trajectories infinitesimally close to the given state diverge from each other. Conversely, small Lyapunov exponents suggests that trajectories converge to each others.
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[*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.
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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 behaviours in a point in phase space if that point 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.
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