Lorenz attractor
The Lorenz attractor is a geometric figure that arises from plotting the solutions of a system of nonlinear differential equations developed by meteorologist Edward Lorenz in 1963. This system was originally created as a simplified model of atmospheric convection, but it later became one of the most iconic examples of deterministic chaos in dynamical systems theory.
Lorenz system
The Lorenz system is defined by a set of three nonlinear ordinary differential equations that describe the time evolution of three variables: temperature, air velocity, and the pressure difference between two points. These equations are:
$$\frac{dx}{dt} = \sigma \left( y - x \right)$$
$$\frac{dy}{dt} = x \left( \rho - z \right) - y$$
$$\frac{dz}{dt} = xy - \beta z$$
where:
- \(x\) represents the rate of convection in the fluid system. It measures how fast warm fluid rises and cold fluid sinks.
- \(y\) represents the horizontal temperature difference between rising and falling air (or fluid) currents.
- \(z\) represents the vertical temperature deviation from a linear temperature profile (i.e., how much the actual temperature differs from a smooth gradient).
- \(\sigma\) (sigma) known as the Prandtl number, it represents the ratio between the fluid's viscosity and its thermal diffusivity. In the context of atmospheric convection, it reflects how quickly momentum diffuses compared to heat.
- \(\rho\) (rho) this is a scaled version of the Rayleigh number, which measures the intensity of the thermal forcing in the system. A higher \(\rho\) means stronger heating from below, which promotes convection. It controls the system's tendency to become chaotic.
- \(\beta\) (beta) this parameter is related to the geometric aspect ratio of the convection cell. In the simplified Lorenz model, it often takes the value \(\beta = \frac{8}{3}\), which is derived from physical assumptions about the flow.
Typical values used by Lorenz:
- \(\sigma = 10\)
- \(\rho = 28\)
- \(\beta = \frac{8}{3}\)
Characteristics of the Lorenz Attractor
- Chaotic but deterministic: Although the system is deterministic (no randomness in the equations), small differences in initial conditions lead to drastically different behaviors, a phenomenon known as sensitivity to initial conditions.
- Butterfly or wing shape: Visually, the attractor resembles a butterfly or a pair of wings, which has become a symbol of the butterfly effect.
- Strange attractor: It is called "strange" because its trajectories neither converge nor diverge completely, but loop infinitely around two regions without repeating or escaping a bounded space.
Importance
- It was one of the first concrete examples of a chaotic system, helping to establish chaos theory as a formal field in mathematics and physics.
- It is used to illustrate how simple systems can produce unpredictable and complex behavior.
- It has applications not only in meteorology but also in engineering, economics, biology, and other fields that study dynamical systems.
Simulation
Visualization
If the system is plotted in 3D with axes \(x\), \(y\), and \(z\), the trajectories form a double-loop shape that never intersects itself, nor does it repeat, creating a fractal structure.
Instructions
Adjust the system values by moving the sliders for \(\sigma\), \(\rho\), and \(\beta\). Use the left or right mouse button to rotate the 3D plot, and scroll with the mouse wheel to zoom in or out. Pause the animation with the pause button, and restart it using the looped arrows button.