Many physical systems of current interest are chaotic, which means that numerical errors in their simulation are exponentially magnified with the passage of time. This could mean that a numerical solution of a chaotic system is the result of nothing but magnified noise, which calls into question the value of such simulations. Although this fact has been well known for a long time, its impact on the validity of simulations is not well understood. The study of *shadowing* may provide an answer. A shadow is an *exact* trajectory of a chaotic map or ordinary differential equation that remains close to an approximate solution for a nontrivial duration of time. If it can be shown that a numerical solution has a shadow, then the validity of the solution is strong, in the sense that it can be viewed as an experimental observation of the shadow, which is an exact solution. We present a discussion of shadowing, including an algorithm to find shadows, using the gravitational $N$-body problem as an example.

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**f**to time $ti+1.$ The Jacobian $Df(yi)$ measures how $y\u2032$ changes if

**y**is changed by a small amount. The

*resolvent*$R(ti+1,ti)$ is the integral of $Df(y)$ along the path $y(t),$ and describes how a small perturbation δ

**y**of $yi$ at time $ti$ is mapped to a perturbation of $yi+1$ at time $ti+1.$ $R(ti+1,ti)$ is the solution of the

*variational equation*,

**y**at time $t0$ gets mapped to a perturbation at time $t2$ by the matrix–matrix and matrix–vector multiplication $R2\delta y=R1R0\delta y.$29 Finally, the linear map in the GHYS refinement procedure, if φ is the time-$h$ solution operator for Eq. (3), is $D\phi (yi)=R(ti+1,ti).$

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*The Physics Teacher*as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.