Lorenz Attractor Quick Start

Set up parameters, run your first simulation, and compare 3D phase space with projected 2D phase portraits for the Lorenz system.

Steps

1. Open the 3D attractor workflow in the toolbox.
2. Choose Lorenz from the system selector.
3. Keep the classical preset values sigma = 10, rho = 28, beta = 8/3, and initial state (1, 1, 1).
4. Use dt = 0.01, total time around 40, and Runge-Kutta 4.
5. Run the simulation, then rotate the 3D canvas to check that both Lorenz lobes are visible.
6. Switch to 2D phase portraits and time series to compare projections and coordinate traces before exporting.

Theory

The Lorenz system is a system of three ordinary differential equations first studied by Edward Lorenz in 1963 as a simplified model for atmospheric convection. In the classical chaotic parameter regime, the trajectory traces a butterfly-like shape in three-dimensional phase space.

The axes are the state variables x, y and z. The curve is not a physical path through ordinary space; it is the history of the system state. The two lobes show that the orbit repeatedly visits two regions, but the switching time is irregular.

Read the 3D plot together with the time series and the 2D projections. The 3D plot shows geometry, x(t), y(t), and z(t) show temporal evolution, and the 2D portraits reveal how pairs of variables fold and return.

How Parameters Change This Figure

• rho: changing rho can move Lorenz from stable equilibria into lobe switching and chaotic motion.
• Initial condition: nearby starts can follow similar paths briefly and then separate in the chaotic regime.
• Step size: too large a dt can cause numerical divergence or a false-looking trajectory.
• Total time: too short shows mostly transient; longer windows reveal the attractor geometry.
• Projection: x-y, x-z, and y-z projections emphasize different folds, so inspect more than one view.