  To generalize the backpropagation method to both discrete-time and
continuous-time hyperbolic chaos, we introduce the adjoint shadowing operator
$\mathcal{S}$ acting on covector fields. We show that $\mathcal{S}$ can be
equivalently defined as:
  (a) $\mathcal{S}$ is the adjoint of the linear shadowing operator $S$;
  (b) $\mathcal{S}$ is given by a `split then propagate' expansion formula;
  (c) $\mathcal{S}(\omega)$ is the only bounded inhomogeneous adjoint solution
of $\omega$.
  By (a), $\mathcal{S}$ adjointly expresses the shadowing contribution, a
significant part of the linear response, where the linear response is the
derivative of the long-time statistics with respect to system parameters. By
(b), $\mathcal{S}$ also expresses the other part of the linear response, the
unstable contribution. By (c), $\mathcal{S}$ can be efficiently computed by the
nonintrusive shadowing algorithm in Ni and Talnikar (2019 J. Comput. Phys. 395
690-709), which is similar to the conventional backpropagation algorithm. For
continuous-time cases, we additionally show that the linear response admits a
well-defined decomposition into shadowing and unstable contributions.
