  Transseries expansions build upon ordinary power series methods by including
additional basis elements such as exponentials and logarithms. Alternative
summation methods can then be used to "resum" series to obtain more efficient
approximations, and have been successfully widely applied in the study of
continuous linear and nonlinear, single and multidimensional problems. In
particular, a method known as transasymptotic resummation can be used to
describe continuous behaviour occurring on multiple scales without the need for
asymptotic matching. Here we apply transasymptotic resummation to discrete
systems and show that it may be used to naturally and efficiently describe
discrete delayed bifurcations, or "canards", in singularly-perturbed variants
of the logistic map which contain delayed period-doubling bifurcations. We use
transasymptotic resummation to approximate the solutions, and describe the
behaviour of the solution across the bifurcations. This approach has two
significant advantages: it may be applied in systematic fashion even across
multiple bifurcations, and the exponential multipliers encode information about
the bifurcations that are used to explain effects seen in the solution
behaviour.
