  A system is invariant with respect to an input transformation if we can
transform any dynamic input by this function and obtain the same output
dynamics after adjusting the initial conditions appropriately. Often, the set
of all such input transformations forms a Lie group, the most prominent
examples being scale-invariant ($u\mapsto e^pu$, $p\in\mathbb{R}$) and
translational-invariant ($u\mapsto pu$) systems, the latter comprising linear
systems with transfer function zeros at the origin. Here, we derive a necessary
and sufficient normal form for invariant systems and, by analyzing this normal
form, provide a complete characterization of the mechanism by which invariance
can be achieved. In this normal form, all invariant systems (i) estimate the
applied input transformation by means of an integral feedback, and (ii) then
apply the inverse of this estimate to the input before processing it in any
other way. We demonstrate our results based on three examples: a
scale-invariant "feed-forward loop", a bistable switch, and a system resembling
the core of the mammalian circadian network.

Dynamical Systems;Adaptation and Self-Organizing Systems;Biological Physics