computational physics - Max Lyapunov Exponent of a Double Pendulum

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?

Answer

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.