Nonlinear Dynamics & Chaos
Sensitive dependence on initial conditions, strange attractors, and the Lorenz system.
Dissipative nonlinear systems can exhibit deterministic chaos: aperiodic, bounded trajectories with exponential sensitivity to initial conditions. The hallmark is a positive Lyapunov exponent
$$|\delta(t)| \sim |\delta_0| e^{\lambda t}, \qquad \lambda > 0.$$Two nearby trajectories diverge exponentially fast, so long-term prediction is impossible even though the dynamics are fully deterministic — Lorenz's "butterfly effect."
Classical examples:
- Logistic map $x_{n+1} = r x_n (1 - x_n)$: period-doubling cascade ends in chaos at $r_\infty \approx 3.5699$. Feigenbaum's constant $\delta \approx 4.6692$ is universal.
- Lorenz system: $\dot x = \sigma(y-x), \dot y = x(\rho-z)-y, \dot z = xy - \beta z$. At $(\sigma, \rho, \beta) = (10, 28, 8/3)$ the trajectory winds endlessly around two unstable foci — the Lorenz attractor, of fractal dimension ~ 2.06.
Chaos is everywhere: weather, double pendulum, three-body problem, fluid turbulence. The strange attractor is a fractal set with zero phase-space volume but positive measure on its surface.