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.

arXiv comment20 pages, 8 figures

References in corpus (23)

• Hydrodynamics Beyond the Gradient Expansion: Resurgence and Resummation
• A Primer on Resurgent Transseries and Their Asymptotics
• Uniform WKB, Multi-instantons, and Resurgent Trans-Series
• Borel and Stokes Nonperturbative Phenomena in Topological String Theory and c=1 Matrix Models
• Nonperturbative Ambiguities and the Reality of Resurgent Transseries
• Resurgence and the Nekrasov-Shatashvili Limit: Connecting Weak and Strong Coupling in the Mathieu and Lam'e Systems
• Asymptotics of the instantons of Painleve I
• Resurgent Transseries and the Holomorphic Anomaly: Nonperturbative Closed Strings in Local CP2
• On the formation of singularities of solutions of nonlinear differential systems in antistokes directions
• The Resurgence of Instantons: Multi-Cut Stokes Phases and the Painleve II Equation
• Multi-Instantons and Multi-Cuts
• Resumming the string perturbation series
• Resurgence and Hydrodynamic Attractors in Gauss-Bonnet Holography
• Resurgent Extrapolation: Rebuilding a Function from Asymptotic Data. Painleve I
• The large proper-time expansion of Yang-Mills plasma as a resurgent transseries
• Resurgence in $\eta$-deformed Principal Chiral Models
• Finite N from Resurgent Large N
• Standing lattice solitons in the discrete NLS equation with saturation
• Argyres-Douglas theories, Painlevé II and quantum mechanics
• Resurgence and renormalons in the one-dimensional Hubbard model
• Exponential asymptotics and boundary value problems: keeping both sides happy at all orders
• Stokes Phenomena in Discrete Painlevé I
• Nonlinear $q$-Stokes phenomena for $q$-Painlevé I

Cited by in corpus (2)

• Resonant resurgent asymptotics from quantum field theory
• The late to early time behaviour of an expanding plasma: hydrodynamisation from exponential asymptotics