Capturing the cascade: a transseries approach to delayed bifurcations

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.