Lorenz system

The Lorenz system is a system of ordinary differential equations first studied by Edward Lorenz. It is notable for having chaotic solutions for certain parameter values and initial conditions. In particular, the Lorenz attractor is a set of chaotic solutions of the Lorenz system which, when plotted, resemble a butterfly or figure eight.

In 1963, Edward Lorenz developed a simplified mathematical model for atmospheric convection. The model is a system of three ordinary differential equations now known as the Lorenz equations:

Here ${\displaystyle x}$, ${\displaystyle y}$, and ${\displaystyle z}$ make up the system state, ${\displaystyle t}$ is time, and ${\displaystyle \sigma }$, ${\displaystyle \rho }$, ${\displaystyle \beta }$ are the system parameters. The Lorenz equations also arise in simplified models for lasers,dynamos,thermosyphons, brushless DC motors,electric circuits,chemical reactions and forward osmosis.

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