  We introduce a generalization of the Heisenberg algebra which is written in
terms of a functional of one generator of the algebra, $f(J_0)$, that can be
any analytical function. When $f$ is linear with slope $\theta$, we show that
the algebra in this case corresponds to $q$-oscillators for $q^2 = \tan
\theta$. The case where $f$ is a polynomial of order $n$ in $J_0$ corresponds
to a $n$-parameter deformed Heisenberg algebra. The representations of the
algebra, when $f$ is any analytical function, are shown to be obtained through
the study of the stability of the fixed points of $f$ and their composed
functions. The case when $f$ is a quadratic polynomial in $J_0$, the simplest
non-linear scheme which is able to create chaotic behavior, is analyzed in
detail and special regions in the parameter space give representations that
cannot be continuously deformed to representations of Heisenberg algebra.
