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We explore the dynamics of FLRW cosmologies which consist of dark matter,
radiation and dark energy with a quadratic equation of state. Standard
cosmological singularities arise due to energy conditions which are violated by
dark energy, therefore we focus our analysis on non-singular bouncing and
cyclic cosmologies, in particular focusing on the possibility of closed models
always having a bounce for any initial conditions. We analyse the range of
dynamical behaviour admitted by the system, and find a class of closed models
that admit a non-singular bounce, with early- and late-time accelerated
expansion connected by a decelerating phase. In all cases, we find the bouncing
models are only relevant when dark matter and radiation appear at a certain
energy scale, and so require a period such as reheating. We then investigate
imposing an upper bound on the dark matter and radiation, such that their
energy densities cannot become infinite. We find that bounces are always the
general closed model, and a class of models exist with early- and late-time
acceleration, connected by a decelerating phase. We also consider parameter
values for the dark energy component, such that the discrepancy between the
observed value of $\Lambda$ and the theoretical estimates of the contributions
to the effective cosmological constant expected from quantum field theory would
be explained. However, we find that the class of models left does not allow for
an early- and late-time accelerated expansion, connected by a decelerating
period where large-scale structure could form. Nonetheless, our qualitative
analysis serves as a basis for the construction of more realistic models with
realistic quantitative behaviour.