# Weird behavior of lyapynovspectrum from ChaosTools.jl

**URL:** <https://discourse.julialang.org/t/weird-behavior-of-lyapynovspectrum-from-chaostools-jl/124136>\
**Category:** General Usage\
**Tags:** dynamical-systems, dynamicalsystemsjl\
**Created:** [December 23, 2024, 7:45pm UTC](https://discourse.julialang.org/t/weird-behavior-of-lyapynovspectrum-from-chaostools-jl/124136 "2024-12-23T19:45:04Z")\
**Posts on this page:** 4\
**Page:** 1

<div class="post-metadata">

**Author:** ![EvgenijGr](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/evgenijgr/32/32211_2.png) [@EvgenijGr](https://discourse.julialang.org/u/EvgenijGr)\
**Post date:** [December 23, 2024, 7:45pm UTC](https://discourse.julialang.org/t/weird-behavior-of-lyapynovspectrum-from-chaostools-jl/124136/1 "2024-12-23T19:45:04Z")

</div>

Hello!

I’ve encountered a weird behavior of `lyapunovspectrum` when I was investigating my nonlinear system. A MWE can be found here:

```julia
using ChaosTools, DifferentialEquations, CairoMakie

function couple(x, y, vThr, k, hOffset)
    return (vThr - x)/(1 + exp(k*(hOffset-y)))
end

@inbounds function odesys(u::T, p, t=0.0) where T 
    u1, v1, w1, u2, v2, w2 = u
    α, β, d, vThr, k, hOffset = p 
    du1 = u1*(1 - u1 - α*v1 - β*w1) + d * couple(u1, u2, vThr, k, hOffset)
    dv1 = v1*(1 - v1 - α*w1 - β*u1) + d * couple(v1, v2, vThr, k, hOffset)
    dw1 = w1*(1 - w1 - α*u1 - β*v1) + d * couple(w1, w2, vThr, k, hOffset)
    du2 = u2*(1 - u2 - α*w2 - β*v2) + d * couple(u2, u1, vThr, k, hOffset)
    dv2 = v2*(1 - v2 - α*u2 - β*w2) + d * couple(v2, v1, vThr, k, hOffset)
    dw2 = w2*(1 - w2 - α*v2 - β*u2) + d * couple(w2, w1, vThr, k, hOffset)
    return T([du1, dv1, dw1, du2, dv2, dw2])
end 

# initial point 
u1 = 0.08020870490181783
v1 = 0.042614670405671906
w1 = 0.7652877851548388
u2 = 0.7652877851548388
v2 = 0.042614670405671906
w2 = 0.08020870490181783
ic = [u1, v1, w1, u2, v2, w2]

# param values
alpha = 0.2451
beta = 2.324
d = 0.3606476945244957
W = 0.1
k = 50.0
h = 0.5
pars = [alpha, beta, d, W, k, h]

# integrator parameters 
alg = RK4()
# alg = DP8()
abstol = 1e-9
reltol = 1e-9
dtmax = 5e-1
dtmin = 1e-20
diffeq = (dtmax = dtmax, 
            dtmin = dtmin, 
            alg = alg, 
            abstol = abstol, 
            reltol = reltol, 
            )

# doing DS stuff and looking for Lyapunov exponents 
dsys = CoupledODEs(odesys, ic, pars; diffeq)
tanDsys = TangentDynamicalSystem(dsys)
lyaps = lyapunovspectrum(tanDsys, 100_000; Ttr=3000.)
@show lyaps 
lle = lyapunov(dsys, 100_000.)
@show lle

# plot projections 
totalTime = 6_000
X, t = trajectory(dsys, totalTime)
u1, v1, w1, u2 , v2 , w2 = columns(X)
fig1 = lines(u1, w1)
save("fig1.png", fig1)

```

While the projection of a found attractor looks like a simple limit cycle, there is a weird discrepancy between the calculations of the Largest Lyapunov Exponent.

 ![fig1](https://global.discourse-cdn.com/julialang/original/3X/e/1/e11ca84172b509dbe0e1817d2ec8b6d8e70ac4ac.png)

The output on my computer is the following:

```julia
lyaps = [0.001347582572478865, 0.0013624061655748648, 8.583893939022226e-6, -0.1916855026931509, -0.8523570829071934, -1.276863571175826]
lle = 3.3207502234233297e-6

```

The `lle` variable (LLE computed by Benettin algorithm from `lyapunov` function) is close to zero and corresponds to a limit cycle, while the top Lyapunov exponent of spectrum vector `lyaps` is noticeably larger than zero. Increasing the time of LE computations does not seem to change that behaviour. Is there any caveat that I might be overlooking?

---

<div class="post-metadata">

**Author:** ![Datseris](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/datseris/32/13406_2.png) [@Datseris](https://discourse.julialang.org/u/Datseris)\
**Post date:** [December 23, 2024, 8:03pm UTC](https://discourse.julialang.org/t/weird-behavior-of-lyapynovspectrum-from-chaostools-jl/124136/2 "2024-12-23T20:03:20Z")

</div>

increase computation time 10x or 100x and see if they converge first!

---

<div class="post-metadata">

**Author:** ![Datseris](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/datseris/32/13406_2.png) [@Datseris](https://discourse.julialang.org/u/Datseris)\
**Post date:** [December 23, 2024, 8:04pm UTC](https://discourse.julialang.org/t/weird-behavior-of-lyapynovspectrum-from-chaostools-jl/124136/3 "2024-12-23T20:04:31Z")

</div>

> [@EvgenijGr](#):
>
> `0.001347582572478865`

To me this seems pretty close to zero. Remember, there is no such thing as an absolute value in dynamical systems. The inverse of this number is a timescale. What is the typical timescale of your system? What is the period of the limit cycle? if `1/λ` is millios of times larger you can safely consider it zero.

---

<div class="post-metadata">

**Author:** ![EvgenijGr](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/evgenijgr/32/32211_2.png) [@EvgenijGr](https://discourse.julialang.org/u/EvgenijGr)\
**Post date:** [December 23, 2024, 9:18pm UTC](https://discourse.julialang.org/t/weird-behavior-of-lyapynovspectrum-from-chaostools-jl/124136/4 "2024-12-23T21:18:12Z")

</div>

Thank you for your answer! I’ve checked the spectrum after the 10kk of iterations (it’s 100x more than in the original code), the result is essentially the same:

```julia
lyaps = [0.0013397005481988005, 0.0013397729161342767, -2.130190046311042e-8, -0.1916860048254532, -0.8523688935218153, -1.276849452732373]

```

The period of this limit cycle is roughly 20 units of time.