75.

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		2.3.8— The Shadowing Theorem

		The coordinates of a point in the phase space set of the Lorenz system at time t are equal to the values of the variables X(t), Y(t), and Z(t). As the system evolves in time, the values of these variables change. Thus the point moves in the phase space. The line traced out by the moving point is called a trajectory. These trajectories form the attractor. The values of the variables can be found by numerical integration of the Lorenz equations. That is, the values of the variables at the next point in time are computed from their values at the previous point in time.

		There are always some errors in the values of variables computed by numerical integration. Sensitivity to initial conditions means that the Lorenz system will amplify these small errors into large differences in the values of the variables. In fact, sensitivity to initial conditions means that after a while, the values of these variables are unpredictable. So how can we compute the shape of the attractor?

		The answer is subtle and beautiful. It is called the shadowing theorem. This theorem says that the errors introduced by the sensitivity to initial conditions mean that we did not accurately compute the trajectory that began with the initial conditions that we used. However, the values of the variables that we did compute, errors and all, are a good approximation of another "real" trajectory on the attractor. More formally, there is a "real" trajectory that "shadows'' (is close to) the one that we computed. This "real" trajectory is one that has a set of initial conditions different from the ones that we used.

		The reason for this wonderful result is that if the errors push the values of the variables off the attractor, they will be rapidly drawn back to their values on the attractor. If the errors push the values of the variables to another set of values on the attractor, then we just pick up one of the infinite set of other trajectories on the attractor.

 

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