  The question of a possible excitation and emergence of fractional type
dynamics, as a more realistic framework for understanding emergence of complex
systems, directly from a conventional integral order dynamics, in the form a
continuous transition or deformation, is of significant interest. Although
there have been a lot of activities in nonlinear, fractional or not, dynamical
systems, the above question appears yet to be addressed systematically in the
current literature. The present work may be considered to be a step forward in
this direction. Based on a novel concept of asymptotic duality structure, we
present here an extended analytical framework that would provide a scenario for
realizing the above stated continuous deformation of integral order dynamics to
a local fractional order dynamics on a fractal and fractional space. The
related concepts of self dual and strictly dual asymptotics are introduced and
there relevance in connection with smooth and nonsmooth deformation of the real
line are pointed out. The relationship of the duality structure and
renormalization group is examined. The ordinary derivation operator is shown to
be invariant under this duality enabled renormalization group transformation,
leading thereby to a {\em natural} realization of local fractional type
derivative in a fractal space. As an application we discuss linear wave
equation in one and two dimensions and show how the underlying integral order
wave equation could be deformed and renormalized suitably to yield meaningful
results for vibration of a fractal string or wave propagation in a region with
fractal boundary.
