by $$\hat{T} = v \hat{U},$$ where $\hat{U}$ is a unitary operator acting on a space of dimension $N$ and the prefactor $v$, satisfying $0<v<1$, accounts for dissipation. In numerical investigations here and in following sections, we choose $\hat{U}$ to be the quantum analogue of a classical map on the unit torus (see \[app:B\] for further details). By choosing maps with different levels of chaotic behaviour, we aim to simulate the response of larger, complex systems governed by classical wave theories. In these models, the dimension $N$ takes on the role of the wave number $k$ as the large parameter.

In order to illustrate the average response set out in the preceding discussion, we let $\hat{U}$ take the form of a quantum kicked Harper (QKH) map in this section, see [@LKFA90; @GKP91; @LS91] or the review [@Art_Scholar] (other maps are used when discussing fluctuations about the mean later). The corresponding classical kicked Harper map exhibits a predominantly chaotic dynamics but with small regular islands. Although these islands are small, their presence leads to a significant degree of tanglement of the stable and unstable manifolds as demonstrated in Figs.  \[fig:kr:wig:a3\] and \[fig:kr:wig:a5\]. This gives rise to a strong dependence on initial source 