Multi-parametric deformed Heisenberg algebras: a route to complexity

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 an n-p arameter deformed Heisenberg algebra. The representations of the algebra, w hen f is any analytical function, are shown to be obtained through the stud y of the stability of the fixed points of f and their composed functions. T he case when f is a quadratic polynomial in J(0). the simplest nonlinear sc heme which is able to create chaotic behaviour, is analysed in detail and s pecial regions in the parameter space give representations that cannot be c ontinuously deformed to representations of Heisenberg algebra.