  The long-term average response of observables of chaotic systems to dynamical
perturbations can often be predicted using linear response theory, but not all
chaotic systems possess a linear response. Macroscopic observables of complex
dissipative chaotic systems, however, are widely assumed to have a linear
response even if the microscopic variables do not, but the mechanism for this
is not well-understood.
  We present a comprehensive picture for the linear response of macroscopic
observables in high-dimensional coupled deterministic dynamical systems, where
the coupling is via a mean field and the microscopic subsystems may or may not
obey linear response theory. We derive stochastic reductions of the dynamics of
these observables from statistics of the microscopic system, and provide
conditions for linear response theory to hold in finite dimensional systems and
in the thermodynamic limit. In particular, we show that for large systems of
finite size, linear response is induced via self-generated noise.
  We present examples in the thermodynamic limit where the macroscopic
observable satisfies LRT, although the microscopic subsystems individually
violate LRT, as well a converse example where the macroscopic observable does
not satisfy LRT despite all microscopic subsystems satisfying LRT when
uncoupled. This latter, maybe surprising, example is associated with emergent
non-trivial dynamics of the macroscopic observable. We provide numerical
evidence for our results on linear response as well as some analytical
intuition.
