# Implementation of Bennetin's algorithm for Lyapunov exponents

**URL:** <https://discourse.julialang.org/t/implementation-of-bennetins-algorithm-for-lyapunov-exponents/97207>\
**Category:** General Usage\
**Tags:** nonlinear, differentialequation, dynamical-systems\
**Created:** [April 7, 2023, 8:41am UTC](https://discourse.julialang.org/t/implementation-of-bennetins-algorithm-for-lyapunov-exponents/97207 "2023-04-07T08:41:58Z")\
**Posts on this page:** 2\
**Page:** 1

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**Author:** ![Sh\_166](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/sh_166/32/43965_2.png) [@Sh\_166](https://discourse.julialang.org/u/Sh_166)\
**Post date:** [April 7, 2023, 8:41am UTC](https://discourse.julialang.org/t/implementation-of-bennetins-algorithm-for-lyapunov-exponents/97207/1 "2023-04-07T08:41:58Z")

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I want to calculate the Lyapunov indices for a system - problem is that I can’t get DynamicalSystems.jl to work (described [here](https://discourse.julialang.org/t/error-while-using-dynamicalsystems-jl-to-solve-for-a-complex-nonlinear-system/96976)).

Therefore, I decided to do it a much more crude way by writing out the steps myself. Now, if I understand it right, the algorithm will compare the systems (with initial distance d\_0) at a certain time. We note the distance in the 2 trajectories here as d\_1, move the perturbed system back to d\_0 distance from the evolved unperturbed trajectory, run the evolution for the next interval, repeat.

In order to do this, I decided to solve my original system for some tspan. Now, getting d\_1 is easy enough - solve the same system with different initial conditions and subtract the 2 timeseries. So, for d\_1, lets say I evolve the system for time, t\_j. I am not sure what to do to get d\_2, but I thought of the following way - solve the same system again but this time I use,

```julia
u_0 = [x = sol[1, j] + d_0 ]
tspan = (t_j, t_k)

```

So, I choose the initial value of my variable this time as the value of the unperturbed variable at t\_j plus the deviation, d\_0. Now, I evolve this system from t\_j to some other time t\_k. Repeat the process.

I have not yet tried this (I will as soon as I get the chance), but I would like to know what the people here, who are familiar with Lyapunov exponents and so on, think of the validity and the viability of the process.

Thanks in advance!!

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<div class="post-metadata">

**Author:** ![empet](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/empet/32/221303_2.png) [@empet](https://discourse.julialang.org/u/empet)\
**Post date:** [April 9, 2023, 5:32pm UTC](https://discourse.julialang.org/t/implementation-of-bennetins-algorithm-for-lyapunov-exponents/97207/2 "2023-04-09T17:32:56Z")

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Instead of reimplementing the well working Bennetin algorithm, from ChaosTools, better post the natural form of your system (not as you tried to define it [here](https://discourse.julialang.org/t/error-while-using-dynamicalsystems-jl-to-solve-for-a-complex-nonlinear-system/96976)) to get help in redefining it as a right `ds`. Julia discourse renders a LaTeX text, hence you can write it down with all details.