  Frequency plays a crucial role in exhibiting various collective dynamics in
the coexisting co- and counter-rotating (CR) systems. To illustrate the impact
of CR frequencies, we consider a network of non-identical and globally coupled
Stuart-Landau oscillators with additional perturbation. Primarily, we
investigate the dynamical transitions in the absence of perturbation,
demonstrating that the transition from desynchronized state to cluster
oscillatory state occurs through an interesting partial synchronization state.
Followed by this, the system dynamics transits to amplitude death and
oscillation death states. Importantly, we find that the observed dynamical
states do not preserve the parity(P) symmetry in the absence of perturbation.
When the perturbation is increased one can note that the system dynamics
exhibits a new kind of transition which corresponds to a change from incoherent
mixed synchronization to coherent mixed synchronization through chimera state.
In particular, incoherent mixed synchronization and coherent mixed
synchronization states completely preserve the P-symmetry, whereas the chimera
state preserves the P-symmetry only partially. To demonstrate the occurrence of
such partial symmetry breaking (chimera) state, we use basin stability analysis
and discover that PSB exists as a result of the coexistence of symmetry
preserving and symmetry breaking behavior in the initial state space. Further,
a measure of the strength of P-symmetry is established to quantify the
P-symmetry in the observed dynamical states. Finally, by increasing the network
size, the robustness of the chimera is also inspected and we find that the
chimera state is robust even in networks of larger sizes. We also show the
generality of the above results in the related phase reduced model as well as
in other coupled models such as the globally coupled van der Pol and R\"ossler
oscillators.
