~vaibhav

The Lorenz Equations and One-Dimensional Maps

Stub. The Lorenz equations as the canonical example of deterministic chaos in a 3D flow — the chaotic waterwheel, basic properties, the strange attractor, and the Lorenz map (a 1D return map extracted from the 3D flow) — which motivates studying one-dimensional maps in their own right: the logistic map's numerics and analysis, periodic windows, the Liapunov exponent as a quantitative measure of chaos, universality, and renormalization.

· 2 min read · 250 words

Corresponds to Strogatz, Chapters 9–10. The Lorenz equations are where the book crosses into chaos proper: a 3D autonomous flow (motivated by a physical chaotic waterwheel) simple enough to write down, yet with trajectories that never settle to a fixed point or periodic orbit and never repeat, instead wandering forever on a strange attractor. The Lorenz map — plot each local maximum of one coordinate against the next — collapses the 3D flow onto an (almost) 1D map, and this is the bridge to the chapter’s second half: one-dimensional maps xn+1=f(xn)x_{n+1}=f(x_n) studied directly, via the logistic map as the running example. Covers fixed points and cobwebbing as the discrete analogue of the phase line, the period-doubling route to chaos, periodic windows embedded within chaotic parameter ranges, the Liapunov exponent (quantifying sensitive dependence on initial conditions — the defining feature of chaos), and the universality/renormalization results explaining why unrelated chaotic systems share the same period-doubling constants.

Outline (TODO — flesh out each)

  • A chaotic waterwheel
  • Simple properties of the Lorenz equations
  • Chaos on a strange attractor
  • The Lorenz map
  • Exploring parameter space
  • Using chaos to send secret messages
  • Fixed points and cobwebs
  • The logistic map — numerics
  • The logistic map — analysis
  • Periodic windows
  • The Liapunov exponent
  • Universality and experiments
  • Renormalization

← All notes

Mentioned in