  The high-order synchronization was studied in systems driven by external
force and in autonomous systems with proper frequency mismatch. Differing from
the literature, in this article, we demonstrate the occurrence of high-order
(1:2) synchronization in an autonomous nonlinearly coupled (Stuart-Landau)
oscillators which admit a particular form of rotational symmetry.
Interestingly, the observed 1:2 synchronization happens not only for a
particular choice of natural frequencies but for all possible choices of
frequencies. We have observed such a behaviour in the case of 1:1
synchronization, where we have seen a variety of couplings in the literature
that forces the oscillators to have almost equal frequencies and makes the
system to oscillate with a common frequency independent of whether the
oscillators are identical or non-identical. Similarly, in this article we
observe that, whether the initial choice of frequencies is of the ratio 1:2 or
not, the given nonlinear coupling forces the system to oscillate with the
frequency ratio 1:2. Further, we present synchronization and other dynamical
behaviours of the system by considering different choices of natural
frequencies.
