Practice: Lorenz System

Module 1: Low-Dimensional Dynamics

This is ungraded practice on the optional Lorenz material. You now have the full workflow for a first chaos experiment: define the Lorenz vector field, integrate the trajectory numerically, visualize the attractor in three dimensions, and compare nearby initial conditions (Lorenz 1963; Strogatz 2024).

You can follow the guide here:

Lorenz Attractor Sensitivity and Chaos

Goals

Your work should show that you can:

  • implement and simulate the Lorenz system correctly;
  • visualize the attractor and the divergence of nearby trajectories;
  • explain why the system is deterministic but hard to predict over long times.

Tasks

  • Implement lorenz() and simulate_lorenz() using forward Euler time stepping.
  • Plot one three-dimensional trajectory for the standard parameter choice \(s=10\), \(r=28\), \(b=8/3\).
  • Run two simulations with slightly different initial conditions and compare them over the same time window.
  • Plot the pointwise separation between the two trajectories as a function of time.
  • Compare at least two values of r and describe how the qualitative behavior changes.

Short discussion

Write a short paragraph that answers:

  • For how long do the nearby trajectories remain visually close?
  • Does the system appear random, periodic, or something in between? Explain.
  • What role does the parameter r seem to play in the shape or complexity of the trajectory?
Optional extension
  • Replace Euler’s method by a higher-order integrator and compare the trajectories.
  • Estimate the growth rate of the separation during the early part of the simulation.
  • Plot only one coordinate, for example \(x(t)\), and compare the two runs in time.
  • Explore other initial conditions on the same attractor.

Tips

  • Start with a small dt, such as 0.01, before you vary parameters.
  • Use the same dt and number of steps when you compare two trajectories.
  • Plot the separation on a logarithmic vertical axis if the early growth is hard to see.
  • Keep the parameter comparison focused. Two or three values are enough if you interpret them well.
Tip

Reference code is available in amlab/extra/lorenz_attractor.py.

References

Lorenz, Edward N. 1963. “Deterministic Nonperiodic Flow.” Journal of the Atmospheric Sciences 20 (2): 130–41. https://doi.org/10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2.
Strogatz, Steven H. 2024. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. Chapman; Hall/CRC.