Multiple scales and matched asymptotic expansions for the discrete logistic equation

Abstract

In this paper, we combine the method of multiple scales and the method of matched asymptotic expansions to construct uniformly-valid asymptotic solutions to autonomous and non-autonomous difference equations in the neighbourhood of a period-doubling bifurcation. In each case, we begin by constructing multiple scales approximations in which the slow time scale is treated as a continuum variable, leading to difference-differential equations. The resultant approximations fail to be asymptotic at late time, due to behaviour on the slow time scale, it is necessary to eliminate the effects of the fast time scale in order to find the late time rescaling, but there are then no difficulties with applying the method of matched asymptotic expansions. The methods that we develop lead to a general strategy for obtaining asymptotic solutions to singularly-perturbed difference equations, and we discuss clear indicators of when multiple scales, matched asymptotic expansions, or a combined approach might be appropriate.

Capturing the cascade: a transseries approach to delayed bifurcations

“…The first is that the method we have described can be applied in systematic fashion to a wide range of problems, including both discrete and continuous systems. Whilst such advantages have already been seen elsewhere, in the context of the logistic equation it has been instructive to compare this to the multiple scales approach from [40], which required the careful comparison of asymptotic terms up to several orders. In order to capture the fast discrete scale, as well as both the inner and outer continuous scales near the bifurcation, asymptotic matching was performed through three scales.…”

“…This allows us to form a transseries that can be used to capture solutions which tend to a four-periodic stable manifold. In [40], this would have required solving a challenging multiple scales problem, as the asymptotic solution obtained therein is only valid for small ε. Using the transseries approach, we obtain a significant more general result.…”

“…The two solutions are visually almost indistinguishable. This figure shows the error in the dynamic system at the three points identified in [40] as belonging to the inner region, transition region, and outer region, shown as points , and in Figure 6. In each case, the error is shown as a blue curve.…”

“…Transasymptotic methods have been been used to determine the location of moveable poles in Painlevé equations directly from asymptotic solutions [5,15] (see also [16,17]). In this work, we show that these transseries and summation approaches can be used in a systematic fashion to identify complicated bifurcations in discrete systems that typically require careful application of multiple scales [40] or renormalisation methods [7].…”

Transient behavior of the solutions to the second order difference equations by the renormalization method based on Newton-Maclaurin expansion

“…Among those, the asymptotic behavior of the solutions is still an interesting subject from both of theory and practice. Many methods on the perturbation theory of difference equations have been proposed, and a lot of applications are given [4][5][6][7][8][9][10][11][12][13][14][15][16][17].…”