  A new, extended nonlinear framework of the ordinary real analysis
incorporating a novel concept of {\em duality structure} and its applications
into various nonlinear dynamical problems is presented. The duality structure
is an asymptotic property that should affect the late time asymptotic behaviour
of a nonlinear dynamical system in a nontrivial way leading naturally to
signatures generic to a complex system. We argue that the present formalism
would offer a natural framework to understand the abundance of complex systems
in natural, biological, financial and related problems. We show that the power
law attenuation of a dispersive, lossy wave equation, conventionally deduced
from fractional calculus techniques, could actually arise from the present
asymptotic duality structure. Differentiability on a Cantor type fractal set is
also formulated.
