= Solution
A useful operational definition of <chaos> for a deterministic flow is bounded, nonperiodic long-time dynamics on an <invariant set> with <sensitive dependence on initial conditions>: arbitrarily close starting states can separate by a fixed macroscopic amount. Boundedness distinguishes recurrent irregular dynamics from simple exponential escape; aperiodicity excludes a <limit cycle>; sensitivity distinguishes chaotic unpredictability from a regular quasiperiodic signal. These are properties of the asymptotic dynamics, not a short irregular transient. One precise topological convention, <Devaney chaos> on a <compact> <invariant set>, instead specifies <topological transitivity>, dense <periodic orbits>, and sensitive dependence. Transitivity expresses indecomposability of the recurrent motion, while dense <periodic orbits> mean every region contains unstable recurrent structure. The convention should be stated because these formulations are not identical in complete generality.
An <attractor> is an <invariant set> approached by trajectories from a neighbourhood or positive-measure basin, according to the adopted <attractor> convention. Calling it strange refers to irregular, typically <fractal geometry> rather than a smooth point, cycle or <torus>. \b[Strangeness alone does not force <chaos>.] <Strange nonchaotic attractors> provide counterexamples: geometrically irregular attracting sets can lack positive exponential separation. Primary examples appear in https://doi.org/10.1016/0167-2789(84)90282-3[Strange attractors that are not chaotic, Physica D 13 (1984), pages 261–268]{ref}. This geometric convention does not assert that every such example occurs in the autonomous three-variable Lorenz family.
Two complementary numerical diagnostics are to estimate the largest <Lyapunov exponent> and to construct a return map. For the first, integrate the <variational equation> $\dot\eta=DF(x(t))\eta$ alongside the Lorenz trajectory, repeatedly normalize $\eta$, and average the logarithms of the accumulated stretch factors after discarding a transient. A positive converged exponent suggests sensitive exponential separation; the full spectrum should include a zero flow exponent and have sum $-(\sigma+1+b)$, providing checks. For the second, record transverse crossings of a Poincaré section or successive maxima of a state variable, then look for the stretching and folding of an aperiodic return set rather than one point or a finite cycle. Nearby-orbit separation or a broadband <power spectrum> can add evidence, but plots alone do not prove <chaos>. Repeat with smaller integration step, longer runs and several <initial conditions> to distinguish numerical error, long transients and coexistence of <attractors> from persistent dynamics.
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