A deterministic dynamical system can be unpredictable in practice even when its equations contain no randomness. Chaos refers to bounded deterministic dynamics with strong sensitivity to initial conditions together with sufficiently rich long-term behavior.

If two nearby trajectories begin a distance $\delta_0$ apart, their separation often grows approximately as $$\delta(t)\approx \delta_0e^{\lambda t}$$ over a range of times. The rate $\lambda$ is a Lyapunov exponent. A positive largest Lyapunov exponent means nearby states separate exponentially, so finite uncertainty in the initial state limits long-term prediction.

If $\lambda=0.5,\mathrm{s^{-1}}$, an initial uncertainty of $10^{-6}$ grows to about $$10^{-6}e^{0.5t}.$$ Reaching order $10^{-1}$ requires $$t\approx\frac{\ln(10^5)}{0.5}\approx23,\mathrm{s}.$$ Increasing measurement precision postpones this prediction horizon only logarithmically.

A classic discrete example is the logistic map $$x_{n+1}=rx_n(1-x_n).$$ As $r$ is increased, fixed points lose stability through period-doubling bifurcations and chaotic parameter ranges appear. The map is exactly deterministic, yet nearby initial values eventually produce very different sequences.

Chaos should not be confused with randomness or numerical error. Numerical errors can be amplified by chaotic dynamics, but the sensitivity belongs to the underlying system. Nor does chaos imply that nothing can be predicted: invariant measures, frequencies, bounds and statistical properties can remain highly reproducible even when individual long trajectories cannot.