Stuart-Landau oscillators can be coupled so as to either preserve or destroy the rotational symmetry that the uncoupled system possesses. We examine some of the simplest cases of such couplings for a system of two nonidentical oscillators. When the coupling breaks the rotational invariance, there is a qualitative difference between oscillators wherein the phase velocity has the same sign (termed co-rotation) or opposite signs (termed counter-rotation). In the regime of oscillation death the relative sense of the phase rotations plays a major role. In particular, when rotational invariance is broken, counter-rotation or phase velocities of opposite signs appear to destabilize existing fixed points, thereby preserving and possibly extending the range of oscillatory behavior. The dynamical "frustration" induced by counter-rotations can thus suppress oscillation quenching when coupling breaks the symmetry.
The phenomenon of ageing in a population of autonomous oscillators, namely the increase in the number of inactive (or non-oscillatory) units due to coupling interactions is studied in a population of globally coupled Stuart–Landau oscillators. The initial populations are prepared either as a mixture of active and inactive oscillators or as an ensemble of active oscillators with a mixture of distinct frequencies. The ageing transition does not depend on whether the coupling breaks gauge symmetry or not, but is affected by the degree of diversity in the ensemble, namely the existence of different types of subsystems that can cause oscillation quenching when coupled. The scaling exponents depend on the nature of the coupling interaction.
We consider a heterogenous ensemble of dynamical systems in $\mathbb{R}^4$ that individually are either attracted 
to fixed points (and are termed inactive) or to limit cycles (in which case they are termed active). These distinct states are separated 
by bifurcations that are controlled by a single parameter. Upon coupling them globally, we find a {\em discontinuous} transition to 
global inactivity (or {\em stasis}) when the proportion of inactive components in the ensemble exceeds a threshold: 
there is a first--order phase transition from a globally oscillatory state to global oscillation death. There is hysteresis associated 
with these phase transitions. Numerical results for a representative system are supported by analysis using 
a system-reduction technique and different dynamical regimes can be rationalised through the corresponding bifurcation diagrams 
of the reduced set of equations. 
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