values of three variables – each of which is a function of time – constrained by three equations $$\label{eq:lorenz}
  \begin{cases}
    \frac{dx}{dt} = a(y-x),\\
    \frac{dy}{dt} = x(b-z) - y,\\
    \frac{dz}{dt} = xy - cz,\\
  \end{cases}$$ where $a,b,c$ are constants. The values of $x$ and $y$ determine the future values of $x$, but all three are implicated in determining $y$ and $z$. The solutions exhibit intricate behavior because the values of $z$ constrain the values of $x$ even though there is not a direct causal relationship.

![Dependency relations among state variables in the Lorenz system (left), and among variables and their derivatives (right)[]{data-label="fig:lorenz1"}](lorenz1){width="3in"}

One way to gain an understanding of the behavior of solutions to is to build a visual representation of the causal relationships between the state variables. The left frame of Figure \[fig:lorenz1\] shows one such representation, where an arrow from one variable $x \to y$ to another indicates that $x$ partially determines future values of $y$.

This representation isn’t entirely true to the way the equations in are written, because the equations also involve derivatives of the state variables. If we include derivatives of state variables as new state variables in their own right, then we obtain a rather 