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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)^[1].

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:

{\tfrac {dx}{dt}}=\sigma (y-x)

5(e)

{\tfrac {dy}{dt}}=x\;(\rho -z)

5(f)

{\tfrac {dz}{dt}}=xy-\beta z

5(g)

In this system of equations, $x$ , $y$ , and $z$ represent the modeled system’s position in a three-dimensional state space^[2] 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

↑ Indeed, even our pendulum is like this! There is another possible qualitatively identical class of trajectories that’s not shown in Figure 1. Think about what would happen if we start things not by dropping the pendulum, but by giving it a big push. If we add in enough initial energy, the angular velocity will be high enough that, rather than coming to rest at the apex of its swing toward the other side and dropping back down, the pendulum will continue on and spin over the top, something most schoolchildren have tried to do on playground swings. Depending on the initial push given, this over-the-top spin may happen only once, or it may happen several times. Eventually though, the behavior of the pendulum will decay back down into the class of trajectories depicted here, an event known as a phase change.
↑ Precisely what this means, of course, depends on the system being modeled. In Lorenz’s original discussion, $x$ represents the intensity of convective energy transfer, $y$ represents the relative temperature of flows moving in opposite directions, and $z$ represents the the degree to which (and how) the vertical temperature profile of the fluid diverges from a smooth, linear flow.

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[1] Indeed, even our pendulum is like this! There is another possible qualitatively identical class of trajectories that’s not shown in Figure 1. Think about what would happen if we start things not by dropping the pendulum, but by giving it a big push. If we add in enough initial energy, the angular velocity will be high enough that, rather than coming to rest at the apex of its swing toward the other side and dropping back down, the pendulum will continue on and spin over the top, something most schoolchildren have tried to do on playground swings. Depending on the initial push given, this over-the-top spin may happen only once, or it may happen several times. Eventually though, the behavior of the pendulum will decay back down into the class of trajectories depicted here, an event known as a phase change.

[2] Precisely what this means, of course, depends on the system being modeled. In Lorenz’s original discussion, $x$ represents the intensity of convective energy transfer, $y$ represents the relative temperature of flows moving in opposite directions, and $z$ represents the the degree to which (and how) the vertical temperature profile of the fluid diverges from a smooth, linear flow.

[1]

[2]