  Chimera states arising in the classic Kuramoto system of two-dimensional
phase coupled oscillators are transient but they are "long" transients in the
sense that the average transient lifetime grows exponentially with the system
size. For reasonably large systems, e.g., those consisting of a few hundreds
oscillators, it is infeasible to numerically calculate or experimentally
measure the average lifetime, so the chimera states are practically permanent.
We find that small perturbations in the third dimension, which make system
"slightly" three-dimensional, will reduce dramatically the transient lifetime.
In particular, under such a perturbation, the practically infinite average
transient lifetime will become extremely short, because it scales with the
magnitude of the perturbation only logarithmically. Physically, this means that
a reduction in the perturbation strength over many orders of magnitude, insofar
as it is not zero, would result in only an incremental increase in the
lifetime. The uncovered type of fragility of chimera states raises concerns
about their observability in physical systems.
