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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? test

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    $\begingroup$ What is being plotted here? $\endgroup$ – Kyle Kanos Jan 19 '18 at 12:27
  • $\begingroup$ Specifically, what's on the x-axis? $\endgroup$ – probably_someone Jan 19 '18 at 19:22
  • $\begingroup$ 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? $\endgroup$ – Sean E. Lake Jan 19 '18 at 19:24
  • $\begingroup$ Equivalently, what is the Lagrangian or Hamiltonian you're using to get your equations of motion? $\endgroup$ – Sean E. Lake Jan 19 '18 at 19:29
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    $\begingroup$ @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. $\endgroup$ – Mateusz Belka Jan 20 '18 at 8:09
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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.

As for the values themselves, you should simulate for longer times, but, without further details (such as initial velocities and parameter values) they seem like they could be going in the right direction. You can check the questions Why is my Lyapunov exponent similar for single and double pendulum?, and Do the Lyapunov exponents depend on integrals of motion for common mistakes to avoid and tips on the calculation. And, for comparison, a plot similar to yours can be found here, and it's probably worth it checking this.

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.

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  • $\begingroup$ 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. $\endgroup$ – Mateusz Belka Jan 19 '18 at 18:46
  • $\begingroup$ @MateuszBelka I took a second look and your values do look too low. I added new relevant references to the answer. $\endgroup$ – stafusa Jan 19 '18 at 20:05

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