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.

Difficulty Level Beginner

Question What does a trajectory look like?

Objective

Run a reproducible classical Lorenz simulation and connect its 3D geometry with two-dimensional projections and coordinate time series.

Workspace

GUI tab

Atractor 3D, followed by Retratos 2D and Series temporales

Procedure

Select Lorenz and retain the classical parameter values sigma=10, rho=28, beta=8/3 with initial state (1,1,1).
Choose RK4, dt=0.01, and a duration long enough to move beyond the visible startup segment.
Generate the 3D trajectory and rotate it to inspect both lobes and the folded geometry.
Open Retratos 2D and Series temporales with the same simulation inputs, or reuse the compatible last trajectory when that option is available.
Repeat with a smaller step or longer time before using the output in a research record.

Result

Expected result

A bounded butterfly-shaped 3D trajectory with repeated visits to both Lorenz lobes.
Consistent x-y, x-z, and y-z projections plus time traces derived from the same numerical setup.

Interpretation

The curve is a path through state space rather than physical space; the lobes and switching describe the computed evolution of x, y, and z.

The projections reveal different folds but can hide depth, while the time series reveal switching order without displaying the complete geometry.

Export

Export and reproducibility

Save each required figure only after labels, line density, color, and projection are legible.
Record the system, ordered parameters, initial state, method, dt, duration, transient treatment, plotted coordinates, and whether a prior compatible trajectory was reused.

Applications

Introducing phase space, deterministic sensitivity, transients, and projection effects.
Checking numerical settings before method comparison, spectral analysis, or Lyapunov estimation.
Producing reproducible reference figures for dynamical-systems teaching and research protocols.

Limits

One finite trajectory cannot describe every initial condition or establish a basin of attraction.
A familiar butterfly shape can be distorted or created by unsuitable step size, duration, or transient handling.
The visualization alone does not prove chaos, global attraction, uniqueness, or hiddenness.

Lorenz attractor with phase-space and time-series panels generated by the toolbox — Expected output from the Lorenz template: 3D trajectory on the left and state time traces on the right.

Lorenz 3D trajectory with 2D phase portraits projected onto coordinate planes — 3D reading aid: the x-y, x-z, and y-z portraits are projected onto the coordinate planes around the 3D Lorenz trajectory.

Separate Lorenz 2D phase portraits — Separate 2D portraits: the same trajectory is inspected through x-y, x-z, and y-z projections.

Steps

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