# Computing the Lyapunov exponent of the Hénon map with DynamicalSystems.jl

## Issue

I am trying to compute the Lyapunov exponent of the Hénon map with `b = 0.3` and `a = 1.8` but the `lyapunov` function from `DynamicalSystems.jl` seems to not converge. Is there something I am doing wrong or is this expected?

## Minimal working example

This does not converge:

``````using DynamicalSystems

henon = Systems.henon(a = 1.75)
λ = lyapunov(henon, 10000, d0 = 1e-7, upper_threshold = 1e-4, Ttr = 100)
``````

This converges very quickly:

``````henon = Systems.henon(a = 1.)
λ = lyapunov(henon, 10000, d0 = 1e-7, upper_threshold = 1e-4, Ttr = 100)
``````

Hi,

1. What does “converge” mean?
2. `upper_threshold = 1e-4`. Lyapunov exponents are in theory defined for infinitesimal perturbations. You want `Δt` and `threshold` to be such that perturbation growth is limited to linearized dynamics. Is this true here?
3. What is the trajectory of the dynamical system doing for these parameters?

I see:

``````julia> tr = trajectory(henon, 100; Ttr = 1000)
2-dimensional Dataset{Float64} with 101 points
-Inf  -Inf
-Inf  -Inf
-Inf  -Inf
-Inf  -Inf
-Inf  -Inf
-Inf  -Inf
-Inf  -Inf
-Inf  -Inf
-Inf  -Inf
-Inf  -Inf
⋮
-Inf  -Inf
-Inf  -Inf
-Inf  -Inf
-Inf  -Inf
-Inf  -Inf
-Inf  -Inf
-Inf  -Inf
-Inf  -Inf
-Inf  -Inf
``````

the system goes to infinity for `a=1.75`

1 Like

Thank you for the quick reply! You are indeed correct, I did not realise the trajectory was diverging for `a = 1.75`.