  An extension of coupled maps is given which allows for the growth of the
number of elements, and is inspired by the cell differentiation problem. The
growth of elements is made possible first by clustering the phases, and then by
differentiating roles. The former leads to the time sharing of resources, while
the latter leads to the separation of roles for the growth. The mechanism of
the differentiation of elements is studied. An extension to a model with
several internal phase variables is given, which shows differentiation of
internal states. The relevance of interacting dynamics with internal states
(``intra-inter" dynamics) to biological problems is discussed with an emphasis
on heterogeneity by clustering, macroscopic robustness by partial
synchronization and recursivity with the selection of initial conditions and
digitalization.
