Measuring quasiperiodicity

Das, Suddhasattwa; Dock, Chris B.; Saiki, Yoshitaka; Salgado-Flores, Martin; Sander, Evelyn; Jin, Wanqin; Yorke, James A.

doi:10.1209/0295-5075/114/40005

Cited by 26 publications

(15 citation statements)

References 39 publications

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“…In the series of papers [2,[17][18][19] we have developed techniques to characterize quasiperiodic orbits. We summarize them in this section used for our computation of this paper.…”

Section: Methodsmentioning

confidence: 99%

Quasi-periodic Orbits in Siegel Disks/Balls and the Babylonian Problem

Saiki

Yorke

2018

Regul. Chaot. Dyn.

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We investigate numerically complex dynamical systems where a fixed point is surrounded by a disk or ball of quasiperiodic orbits, where there is a change of variables (or conjugacy) that converts the system into a linear map. We compute this "linearization" (or conjugacy) from knowledge of a single quasiperiodic trajectory. In our computations of rotation rates of the almost periodic orbits and Fourier coefficients of the conjugacy, we only use knowledge of a trajectory, and we do not assume knowledge of the explicit form of a dynamical system. This problem is called the Babylonian Problem: determining the characteristics of a quasiperiodic set from a trajectory. Our computation of rotation rates and Fourier coefficients depends on the very high speed of our computational method "the weighted Birkhoff average".

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“…In the series of papers [2,[17][18][19] we have developed techniques to characterize quasiperiodic orbits. We summarize them in this section used for our computation of this paper.…”

Section: Methodsmentioning

confidence: 99%

Quasi-periodic Orbits in Siegel Disks/Balls and the Babylonian Problem

Saiki

Yorke

2018

Regul. Chaot. Dyn.

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“…Quasiperiodicity in dynamical systems is typically studied with numerical methods including Birkhoff averages [11], periodic approximations [31,32], estimation of Lyapunov exponents [41], power spectra [43], and recurrence quantification analysis [40,45]. New techniques from applied topology have emerged recently as complements to these traditional approaches in the task of recurrence detectionspecifically for periodicity and quasiperiodicity quantification-in time series data [26,23,38].…”

Section: Introductionmentioning

confidence: 99%

Sliding Window Persistence of Quasiperiodic Functions

Gakhar,

Perea

2021

Preprint

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A function is called quasiperiodic if its fundamental frequencies are linearly independent over the rationals. With appropriate parameters, the sliding window point clouds of such functions can be shown to be dense in tori with dimension equal to the number of independent frequencies. In this paper, we develop theoretical and computational techniques to study the persistent homology of such sets. Specifically, we provide parameter optimization schemes for sliding windows of quasiperiodic functions, and present theoretical lower bounds on their Rips persistent homology. The latter leverages a recent persistent Künneth formula. The theory is illustrated via computational examples and an application to dissonance detection in music audio samples.

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“…Quasiperiodicity is one of the three types of commonly observed dynamics in both deterministic models and experiments: see e.g. [17,18]. Quasiperiodically forced systems are important objects in nonlinear dynamics and have been studied by many groups of researchers: phenomenology of chaotic attractors [19], analysis of dynamics described by the Schrödinger equations [20] and of quantum chaos [21], and dynamical systems analysis with applications to fluid mixing [22,23].…”

Section: Introductionmentioning

confidence: 99%

Invariant Sets in Quasiperiodically Forced Dynamical Systems

Susuki¹,

Mezić²

2020

SIAM J. Appl. Dyn. Syst.

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This paper addresses structures of state space in quasiperiodically forced dynamical systems. We develop a theory of ergodic partition of state space in a class of measurepreserving and dissipative flows, which is a natural extension of the existing theory for measure-preserving maps. The ergodic partition result is based on eigenspace at eigenvalue 0 of the associated Koopman operator, which is realized via time-averages of observables, and provides a constructive way to visualize a low-dimensional slice through a high-dimensional invariant set. We apply the result to the systems with a finite number of attractors and show that the time-average of a continuous observable is well-defined and reveals the invariant sets, namely, a finite number of basins of attraction. We provide a characterization of invariant sets in the quasiperiodically forced systems. A theorem on uniform boundedness of the invariant sets is proved. The series of analytical results enables numerical analysis of invariant sets in the quasiperiodically forced systems based on the ergodic partition and time-averages. Using this, we analyze a nonlinear model of complex power grids that represents the short-term swing instability, named the coherent swing instability. We show that our analytical results can be used to understand stability regions in such complex systems.

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Measuring quasiperiodicity

Cited by 26 publications

References 39 publications

Quasi-periodic Orbits in Siegel Disks/Balls and the Babylonian Problem

Quasi-periodic Orbits in Siegel Disks/Balls and the Babylonian Problem

Sliding Window Persistence of Quasiperiodic Functions

Invariant Sets in Quasiperiodically Forced Dynamical Systems

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