  Low-order models obtained through Galerkin projection of several physically
important systems (e.g., Rayleigh-B\'enard convection, mid-latitude
quasi-geostrophic dynamics, and vorticity dynamics) appear in the form of
coupled gyrostats. Forced dissipative chaos is an important phenomenon in these
models, and this paper introduces and identifies 'minimal chaotic models'
(MCMs), in the sense of having the fewest external forcing and linear
dissipation terms, for the class of models arising from an underlying gyrostat
core. The identification of MCMs reveals common conditions for chaos across a
wide variety of physical systems. It is shown here that a critical distinction
is whether the gyrostat core (without forcing or dissipation) conserves energy,
depending on whether the sum of the quadratic coefficients is zero. The paper
demonstrates that, for the energy-conserving condition of the gyrostat core,
the requirement of a characteristic pair of fixed points that repel the chaotic
flow dictates placement of forcing and dissipation in the minimal chaotic
models. In contrast if the core does not conserve energy, the forcing can be
arranged in additional ways for chaos to appear in the subclasses where linear
feedbacks render fewer invariants in the gyrostat core. In all cases, the
linear mode must experience dissipation for chaos to arise. The Volterra
gyrostat presents a clear example where the arrangement of fixed points
circumscribes more complex dynamics.
