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Update steady_state_stability_computation.md
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isaacsas committed Jul 23, 2024
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After system steady states have been found using [HomotopyContinuation.jl](@ref homotopy_continuation), [NonlinearSolve.jl](@ref steady_state_solving), or other means, their stability can be computed using Catalyst's `steady_state_stability` function. Systems with conservation laws will automatically have these removed, permitting stability computation on systems with singular Jacobian.

!!! warning
Catalyst currently computes steady state stabilities using the naive approach of checking whether a system's largest eigenvalue real part is negative. While more advanced stability computation methods exist (and would be a welcome addition to Catalyst), there is no direct plans to implement these. Furthermore, Catalyst uses a tolerance `tol = 10*sqrt(eps())` to determine whether a computed eigenvalue is far away enough from 0 to be reliably used. This threshold can be changed through the `tol` keyword argument.
Catalyst currently computes steady state stabilities using the naive approach of checking whether a system's largest eigenvalue real part is negative. Furthermore, Catalyst uses a tolerance `tol = 10*sqrt(eps())` to determine whether a computed eigenvalue is far away enough from 0 to be reliably considered non-zero. This threshold can be changed through the `tol` keyword argument.

## [Basic examples](@id steady_state_stability_basics)
Let us consider the following basic example:
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```

!!! warning
For systems with [conservation laws](@ref homotopy_continuation_conservation_laws), `steady_state_jac` must be supplied a `u0` vector (indicating species concentrations for conservation law computation). This is required to eliminate the conserved quantities, preventing a singular Jacobian. These are supplied using the `u0` optional argument.
For systems with [conservation laws](@ref homotopy_continuation_conservation_laws), `steady_state_jac` must be supplied a `u0` vector (indicating species concentrations for conservation law computation). This is required to eliminate the conserved quantities, preventing a singular Jacobian. These are supplied using the `u0` optional argument.

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