  This paper contributes to a theory of the behaviour of "finite-state" systems
that is generic in the system type. We propose that such systems are modelled
as coalgebras with a finitely generated carrier for an endofunctor on a locally
finitely presentable category. Their behaviour gives rise to a new fixpoint of
the coalgebraic type functor called locally finite fixpoint (LFF). We prove
that if the given endofunctor is finitary and preserves monomorphisms then the
LFF always exists and is a subcoalgebra of the final coalgebra (unlike the
rational fixpoint previously studied by Ad\'amek, Milius, and Velebil).
Moreover, we show that the LFF is characterized by two universal properties:
(1) as the final locally finitely generated coalgebra, and (2) as the initial
fg-iterative algebra. As instances of the LFF we first obtain the known
instances of the rational fixpoint, e.g. regular languages, rational streams
and formal power-series, regular trees etc. Moreover, we obtain a number of new
examples, e.g. (realtime deterministic resp. non-deterministic) context-free
languages, constructively S-algebraic formal power-series (in general, the
behaviour of finite coalgebras under the coalgebraic language semantics arising
from the generalized powerset construction by Silva, Bonchi, Bonsangue, and
Rutten), and the monad of Courcelle's algebraic trees.
