  Alternating patterns of small and large amplitude oscillations occur in a
wide variety of physical, chemical, biological and engineering systems. These
mixed-mode oscillations (MMOs) are often found in systems with multiple time
scales. Previous differential equation modeling and analysis of MMOs has mainly
focused on local mechanisms to explain the small oscillations. Numerical
continuation studies reported different MMO patterns based on parameter
variation. This paper aims at improving the link between local analysis and
numerical simulation. Our starting point is a numerical study of a singular
return map for the Koper model which is a prototypical example for MMOs that
also relates to local normal form theory. We demonstrate that many MMO patterns
can be understood geometrically by approximating the singular maps with affine
and quadratic maps. Motivated by our numerical analysis we use abstract affine
and quadratic return map models in combination with two local normal forms that
generate small oscillations. Using this decomposition approach we can reproduce
many classical MMO patterns and effectively decouple bifurcation parameters for
local and global parts of the flow. The overall strategy we employ provides an
alternative technique for understanding MMOs.
