Noisy parametric sweep through a period-doubling bifurcation of the Hénon map

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.

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“…Instead, we expand (83) as a Taylor series about x = 0 using the form of Ω o,0 and Ω e,0 given in (23). This gives…”

Section: Computing Terms In the Resummed Transseriesmentioning

confidence: 99%

“…In addition to establishing specific results about the slowly-varying logistic map, this study established that delayed bifurcations can play an essential role in the behaviour of discrete systems. Similar methods were used to study delayed bifurcations in more general unimodal maps [21], as well as discrete maps with noise [7,22,23].…”

Section: Introductionmentioning

confidence: 99%

Capturing the cascade: a transseries approach to delayed bifurcations

et al. 2021

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“…To obtain the 4-periodic solution requires an adaptation of the previous process. We now take a four-periodic perturbation about the 2-periodic solution obtained in (22). This allows us to form a transseries that can be used to capture solutions which tend to a four-periodic stable manifold.…”

Section: Transseries Ansatzmentioning

confidence: 99%

“…where the asymptotic order of this expression can be obtained by rewriting (22) in powers of τ −1 0 and equating terms. We may now follow similar methods to the 2-periodic case, and formulate an ansatz for the solution in terms of ε and a new transseries parameter, denoted σ1.…”

Section: Transseries Ansatzmentioning

confidence: 99%

See 1 more Smart Citation

Capturing the cascade: a transseries approach to delayed bifurcations

Aniceto,

Hasenbichler,

Howls

et al. 2020

Preprint

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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.

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Oscillation death in coupled nonautonomous systems with parametrical modulation

Pisarchik

2003

Physics Letters A

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Noisy parametric sweep through a period-doubling bifurcation of the Hénon map

Cited by 5 publications

References 6 publications

Capturing the cascade: a transseries approach to delayed bifurcations

Capturing the cascade: a transseries approach to delayed bifurcations

Capturing the cascade: a transseries approach to delayed bifurcations

Oscillation death in coupled nonautonomous systems with parametrical modulation

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