  Chimeras are surprising yet important states in which domains of decoherent
(asynchronous) and coherent (synchronous) oscillations co-exist. In this
article, we report on the discovery of a new class of chimeras, called {\it
mixed-amplitude chimera states}, in which the structures, amplitudes, and
frequencies of the oscillations differ substantially in the decoherent and
coherent regions. These mixed-amplitude chimeras exhibit domains of decoherent
small-amplitude oscillations (phase waves) coexisting with domains of stable
and coherent large-amplitude or mixed-mode oscillations. They are observed in a
prototypical bistable partial differential equation with spatially homogeneous
kinetics and purely local, isotropic diffusion. New bifurcations are identified
in which the mixed-amplitude chimeras emerge from, or are annihilated in,
common large-amplitude solutions. Also, key singularities, folded nodes and
folded saddles, arising commonly in multi-scale, bistable systems play
important roles, and these have not previously been studied in systems with
chimeras. The discovery of these mixed-amplitude chimeras is an important
advance for understanding some processes in neuroscience, pattern formation,
and physics which involve both small-amplitude and large-amplitude
oscillations. It may also be of use for understanding some aspects of EEG
recordings from animals that exhibit unihemispheric slow-wave sleep.
