The Lorenz attractor is a strange attractor that arises from a simplified mathematical model of atmospheric convection, discovered by Edward Lorenz in 1963. It is one of the most famous examples of chaos theory, showing how small changes in initial conditions can lead to vastly different outcomes (the butterfly effect 🦋).

Lorenz System Equations

The Lorenz attractor is described by three coupled differential equations:

Where:

• x, y, z are system variables (related to convection in a fluid).
• 𝜎 (Prandtl number), 𝜌 (Rayleigh number), and 𝛽 (a geometric factor) are parameters that control system behavior.
• Common values: 𝜎=10 , 𝜌=28, β=8/3 produce chaotic motion.

The butterfly shape of the Lorenz attractor will remain the same, even if you change the starting point. However, the specific trajectory within the attractor will be different.

Why?

1. Bounded Structure: The Lorenz system has an inherent mathematical structure that forces all trajectories to stay within the butterfly-shaped region.
2. Sensitive Dependence: Small changes in the starting point lead to wildly different paths within the attractor but do not change its overall shape.
3. Self-Similar Patterns: Although individual trajectories look unpredictable, they always stay within the attractor’s framework, forming a repeating yet chaotic pattern.

What If You Change the Parameters?

• If you modify 𝜎, ρ, β then the shape can change!
• For example, decreasing 𝜌 may cause the system to stabilize at a fixed point instead of forming a chaotic butterfly.