Phase-amplitude reduction far beyond the weakly perturbed paradigm

Wilson, Dan

doi:10.1103/physreve.101.022220

Cited by 53 publications

(35 citation statements)

References 66 publications

(191 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Such higher-order interactions can be a source of intriguing complex dynamics in nonlinear oscillatory systems; see, e.g., Refs. [10,11,[35][36][37][38][39][40] for various types of higher-order descriptions and their consequences.…”

Section: Discussionmentioning

confidence: 99%

Phase and amplitude description of complex oscillatory patterns in reaction-diffusion systems

Nakao

2021

Preprint

View full text Add to dashboard Cite

Spontaneous rhythmic oscillations are widely observed in various real-world systems. In particular, biological rhythms, which typically arise via synchronization of many self-oscillatory cells, often play important functional roles in living systems. One of the standard theoretical methods for analyzing synchronization dynamics of oscillatory systems is the phase reduction for weakly perturbed limit-cycle oscillators, which allows us to simplify nonlinear dynamical models exhibiting stable limit-cycle oscillations to a simple one-dimensional phase equation. Recently, the classical phase reduction method has been generalized to infinite-dimensional oscillatory systems such as spatially extended systems and time-delayed systems, and also to include amplitude degrees of freedom representing deviations of the system state from the unperturbed limit cycle. In this chapter, we discuss the method of phase-amplitude reduction for spatially extended reaction-diffusion systems exhibiting stable oscillatory patterns. As an application, we analyze entrainment of a reaction-diffusion system exhibiting limit-cycle oscillations by an optimized periodic forcing and additional feedback stabilization.

show abstract

Section: Discussionmentioning

confidence: 99%

Phase and amplitude description of complex oscillatory patterns in reaction-diffusion systems

Nakao

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…In (5), the rapidly decaying amplitude coordinates are generally truncated so that only β ≤ N − 1 of the slowest decaying isostable coordinates are explicitly considered -isostables with rapid exponential decay can generally be assumed to be zero without adverse effects in the accuracy of the reduction. Numerical computation of Z, and each I j and g k can be performed using the 'adjoint method' [6] and related equations described in [57]. The so-called 'direct method' [21], [38] and related data-driven techniques [60], [56] have been developed for inferring the necessary terms of (5) from experimental data when the underlying model equations are unknown.…”

Section: Isostable Coordinate Reduced Frameworkmentioning

confidence: 99%

“…In this case, (5) is valid to leading order . Recent work has investigated isostable reduced equations that are valid to second order accuracy [58], [60] and beyond [57], however, these reduction frameworks still require the magnitude of the applied inputs to be sufficiently small. Other reduction frameworks have been developed that are valid for inputs with arbitrary magnitude provided that they are sufficiently slowly varying [28], [40] or rapidly varying [42], [52].…”

Section: Isostable Coordinate Reduced Frameworkmentioning

confidence: 99%

“…For example, the notions of entrainment maps [11], operational phase coordinates [59], functional phase response curves [9], local orthogonal rectification [29], stochastic phase [5], and average isophase [47] rely on different definitions of 'phase', with each being well-suited for specific applications. Other recent works have attempted to use phase-amplitude coordinate systems to identify reduced order equations that are valid beyond first order accuracy in , [58], [16], [57], [44], [61]. While such techniques generally provide better results than comparable first order accurate reduction methods, they still suffer from limitations that preclude the consideration of medium-to-large magnitude inputs.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Optimal Control of Oscillation Timing and Entrainment Using Large Magnitude Inputs: An Adaptive Phase-Amplitude-Coordinate-Based Approach

Wilson¹

2021

SIAM J. Appl. Dyn. Syst.

Self Cite

View full text Add to dashboard Cite

Given the high dimensionality and underlying complexity of many oscillatory dynamical systems, phase reduction is often an imperative first step in control applications where oscillation timing and entrainment are of interest. Unfortunately, most phase reduction frameworks place restrictive limitations on the magnitude of allowable inputs, limiting the practical utility of the resulting phase reduced models in many situations. In this work, motivated by the search for control strategies to hasten recovery from jet-lag caused by rapid travel through multiple time zones, the efficacy of the recently developed adaptive phase-amplitude reduction is considered for manipulating oscillation timing in the presence of a large magnitude entraining stimulus. The adaptive phase-amplitude reduced equations allow for a numerically tractable optimal control formulation and the associated optimal stimuli significantly outperform those resulting from from previously proposed optimal control formulations. Additionally, a data-driven technique to identify the necessary terms of the adaptive phase-amplitude reduction is proposed and validated using a model describing the aggregate oscillations of a large population of coupled limit cycle oscillators. Such data-driven model reduction algorithms are essential in situations where the underlying model equations are either unreliable or unavailable.

show abstract

“…[1,2]. In recent years, among other applications, the AM has been used in Robotics [3], Engineering [4][5][6][7][8], Biology [9] and Physics [10][11][12][13][14].…”

Section: Introductionmentioning

confidence: 99%

Averaging method and coherence applied to Rabi oscillations in a two-level system

Chalkopiadis,

Simserides

2021

Preprint

View full text Add to dashboard Cite

We study Rabi oscillations in a two-level system within the semiclassical approximation as an archetype test field of the Averaging Method (AM). The population transfer between the two levels is approached within the first and the second order AM. We systematically compare AM predictions with the rotating wave approximation (RWA) and with the complete numerical solution utilizing standard algorithms (NRWA). We study both the resonance (∆ = 0) and out-of-resonance (∆ = 0) cases, where ∆ = ω − Ω, and Ω = E2 − E1 is the two-level energetic separation, while ω is the (cyclic) frequency of the electromagnetic field. We introduce three types of dimensionless factors ǫ, i.e., ΩR/∆, ΩR/Σ, and ΩR/ω, where ΩR is the Rabi (cyclic) frequency and Σ = ω + Ω and explore the range of ǫ where the AM results are equivalent to NRWA. Finally, by allowing for a phase difference in the initial electron wave functions, we explore the prospects coherence can offer. We illustrate that even with equal initial probabilities at the two levels, but with phase difference, strong oscillations can be generated and manipulated.

show abstract

Phase-amplitude reduction far beyond the weakly perturbed paradigm

Cited by 53 publications

References 66 publications

Phase and amplitude description of complex oscillatory patterns in reaction-diffusion systems

Phase and amplitude description of complex oscillatory patterns in reaction-diffusion systems

Optimal Control of Oscillation Timing and Entrainment Using Large Magnitude Inputs: An Adaptive Phase-Amplitude-Coordinate-Based Approach

Averaging method and coherence applied to Rabi oscillations in a two-level system

Contact Info

Product

Resources

About