# Max Lyapunov Exponent of a Double Pendulum [closed]

Using Euler's method I got this graph. I used separation between angles $10^{-10}$, $\Delta t$ of integration 0.0001s and max time 100s. The initial angles are the same ($\theta_1=\theta_2$).

I expected the values to be a lot higher at high initial energy positions. Have I made any mistakes?

• What is being plotted here? – Kyle Kanos Jan 19 '18 at 12:27
• Specifically, what's on the x-axis? – probably_someone Jan 19 '18 at 19:22
• What kind of double pendulum is this? One rigid pendulum hanging from the end of another? Real pendulum, or point masses with zero mass rods? – Sean E. Lake Jan 19 '18 at 19:24
• Equivalently, what is the Lagrangian or Hamiltonian you're using to get your equations of motion? – Sean E. Lake Jan 19 '18 at 19:29
• @SeanE.Lake I obtained equations of motion using Lagrangian, I was using rigid bodies, constant rod length, point mass of bob and rods without mass, no friction etc. so idealised mathematical model. – Mateusz Belka Jan 20 '18 at 8:09

First, remember that, for an angle, $\pi\approx 6.28\approx 0$, i.e., it's a periodic variable, so your plot does show higher values for higher angles.
Main references for you could be the papers A numerical analysis of chaos in the double pendulum (e-print), Double pendulum: An experiment in chaos (e-print), and Chaos in a double pendulum (e-print). Notice that the last two give $\lambda_+\in [3, 8]$, so your current values might indeed be too low.
• I will try increasing runtime of simulation. I made this post because I am scared that calling the behaviour chaotic with such low lyapunov exponent might be seen as a critical mistake. P.S. I meant the lyapunov exponent at higher initial energy positions. I know that after $\pi$ the values are mirrored. – Mateusz Belka Jan 19 '18 at 18:46