Chaos theory

Chaos theory studies deterministic nonlinear dynamical systems whose long-term behavior can be highly sensitive to small differences in initial conditions, limiting predictability.

Lyapunov exponent: A quantity measuring the average exponential rate at which nearby trajectories in a dynamical system separate or converge. A positive largest exponent quantifies sensitive dependence on initial conditions.
Lorenz system: A three-dimensional system of differential equations introduced as a simplified model of atmospheric convection. Its bounded, aperiodic trajectories became a landmark example of deterministic chaos.
Henri Poincaré: A French mathematician whose work on differential equations and celestial mechanics helped found modern dynamical systems theory. His analysis of the three-body problem exposed intricate, nonperiodic dynamics.
Stochastic process: A collection of random variables indexed by time or space, used to model probabilistic evolution. Randomness differs from deterministic chaos, even when observed patterns look similar.
Chaos control: Methods for stabilizing or directing chaotic behavior through small, targeted interventions. Sensitive dynamics can make small control inputs unusually effective.
Strange attractor: A bounded invariant set with fractal structure toward which nearby trajectories evolve in a chaotic system. It captures the long-term geometry of many chaotic trajectories.
Logistic map: The discrete equation xₙ₊₁ = r xₙ(1 − xₙ), used to model population growth and nonlinear dynamics. Changing its parameter produces period-doubling cascades and chaotic regimes.
Edward Lorenz: An American meteorologist who discovered sensitive dependence in a simplified model of atmospheric convection. His 1963 analysis made deterministic chaos visible in a concrete model.
Linear system: A system whose outputs depend linearly on its inputs and whose governing equations satisfy superposition. Linear systems lack the nonlinear feedback central to standard routes into chaos.
Data assimilation: The process of combining observations with a model to estimate a system’s evolving state. It helps initialize forecasts when errors in state estimates can rapidly grow.

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