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 (this is really handy when, for instance, the same system can show several different classes of behavior for different initial conditions, and keeps the phase diagram from becoming too crowded).

Contrast this to the butterfly-hurricane case from above, when trajectories that started very close together diverged over time; the small difference in initial conditions was magnified over time in one case, but not in the other. This is what it means for a system to behave chaotically: small differences in initial condition are magnified into larger differences as the system evolves, so trajectories that start very close together in state space need not stay close together.

Lorenz (1963) discusses a system of equations first articulated by Saltzman (1962) to describe the convective transfer of some quantity (e.g. average kinetic energy) across regions of a fluid:

In this system of equations, $$x$$, $$y$$, and $$z$$ represent the modeled system’s position in a three-dimensional state space represents the intensity of convective motion, while $$\sigma$$, $$\rho$$, and $$\beta$$ are parameterizations representing how strongly (and in what way) changes in each of the

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