Capturing the cascade: a transseries approach to delayed bifurcations

Abstract

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.

The late to early time behaviour of an expanding plasma: hydrodynamisation from exponential asymptotics

“…The integer n in (37) parameterises the sequence of branch points. Note that (36) is the partial sum of a divergent asymptotic expansion in t, and thus equation ( 36) is only a good approximation for the branch points/zeros of f (w) when |t| is large enough.…”

“…The coefficient functions F r (τ ) are analytic at τ = 0 , and we will see that it is possible to systematically calculate them in closed form. This approach is called the transasymptotic summation [12,13], and has been shown to be a powerful tool in the study of non-linear problems [14,17,37]. This summation procedure allows us to probe regimes where |w| → ∞ but the exponentials are no longer small.…”

Resonant resurgent asymptotics from quantum field theory

“…These might be of help in evaluating the function associated to the initial perturbative factorially divergent power series explicitly via numerical methods. See for instance [108] for an explicit application of this method. The Lambert-W function previously appeared in the trans-asymptotic resummation of the similar Yukawa theory renormalon [82] and in many other problems that are associated to renormalization and Dyson-Schwinger equations [8,[99][100][101][102].…”

Locating complex singularities of Burgers’ equation using exponential asymptotics and transseries

“…The theory has been effectively used to extend models of relativistic hydrodynamics [26,48,49] and field theories [50] beyond their initial regime. Further, it has been used in previous analysis of the Burgers equation for small viscosity [5], and to predict the appearance of different scales in discrete bifurcation phenomena [12]. The starting point for our transseries method is constructing the asymptotic series of some solution Ufalse(ξfalse) in powers of a large parameter ξ.…”

“…The field of exponential asymptotics, including the associated techniques of analysing so-called beyond-all-orders terms in asymptotic expansions, has a variety of applications in physics and mathematics [1][2][3][4][5][6][7][8], in particular to understand Stokes phenomenon [9][10][11]. The more advanced use of transseries, transasymptotic summation and the theory of resurgence is less common in applied mathematics [11][12][13][14][15], although becoming more popular in theoretical physics (see e.g. [16][17][18][19][20][21], as well as [22] and references therein).…”