  We analyze an intermediate collective regime where amplitude oscillators
distribute themselves along a closed, smooth, time-dependent curve
$\mathcal{C}$, thereby maintaining the typical ordering of (identical) phase
oscillators. This is achieved by developing a general formalism based on two
partial differential equations, which describe the evolution of the probability
density along $\mathcal{C}$ and of the shape of $\mathcal{C}$ itself. The
formalism is specifically developed for Stuart-Landau oscillators, but it is
general enough to be applied to other classes of amplitude oscillators. The
main achievements consist in: (i) identification and characterization of a new
transition to self-consistent partial synchrony (SCPS), which confirms the
crucial role played by higher Fourier hamonics in the coupling function; (ii)
an analytical treatment of SCPS, including a detailed stability analysis; (iii)
the discovery of a new form of collective chaos, which can be seen as a
generalization of SCPS and characterized by a multifractal probability density.
Finally, we are able to describe given dynamical regimes both at the
macroscopic as well as the microscopic level, thereby shedding further light on
the relationship between the two different levels of description.
