Spontaneous symmetry breaking due to the trade-off between attractive and repulsive couplings
  Spontaneous symmetry breaking (SSB) is an important phenomenon observed in
various fields including physics and biology. In this connection, we here show
that the trade-off between attractive and repulsive couplings can induce
spontaneous symmetry breaking in a homogeneous system of coupled oscillators.
With a simple model of a system of two coupled Stuart-Landau oscillators, we
demonstrate how the tendency of attractive coupling in inducing in-phase
synchronized (IPS) oscillations and the tendency of repulsive coupling in
inducing out-of-phase synchronized (OPS) oscillations compete with each other
and give rise to symmetry breaking oscillatory (SBO) states and interesting
multistabilities. Further, we provide explicit expressions for synchronized and
anti-synchronized oscillatory states as well as the so called oscillation death
(OD) state and study their stability. If the Hopf bifurcation parameter
(${\lambda}$) is greater than the natural frequency ($\omega$) of the system,
the attractive coupling favours the emergence of an anti-symmetric OD state via
a Hopf bifurcation whereas the repulsive coupling favours the emergence of a
similar state through a saddle-node bifurcation. We show that an increase in
the repulsive coupling not only destabilizes the IPS state but also facilitates
the re-entrance of the IPS state.
