chaos

Chaos refers to a behavior of dynamical systems characterized by high sensitivity to initial conditions, leading to long-term unpredictability despite deterministic rules. It appears in nonlinear systems across physics, mathematics, biology, economics, and weather.

In chaos theory, systems are deterministic (governed by fixed equations) yet exhibit unpredictable evolution due to nonlinear interactions and complex feedback. A hallmark is exponential divergence of nearby trajectories, quantified by positive Lyapunov exponents, and the presence of strange attractors in phase space.

Classic models include the Lorenz system, the logistic map at certain parameters, and the double pendulum. The Lorenz attractor popularized the idea of deterministic chaos, showing a fractal structure in phase space. Chaotic behavior can occur in dissipative systems, where volume in phase space contracts.

The concept contrasts with randomness, illustrating that apparent randomness can arise from deterministic processes. Applications include meteorology, fluid dynamics, population dynamics, secure communications, and engineering. The study of chaos has influenced mathematics, physics, and philosophy, emphasizing limits of long-term prediction and the richness of nonlinear dynamics.