Dataset Information

Time-scales Herglotz type Noether theorem for delta derivatives of Birkhoffian systems.

ABSTRACT: The time-scales theory provides a powerful theoretical tool for studying differential and difference equations simultaneously. With regard to Herglotz type variational principle, this generalized variational principle can deal with non-conservative or dissipative problems. Combining the two tools, this paper aims to study time-scales Herglotz type Noether theorem for delta derivatives of Birkhoffian systems. We introduce the time-scales Herglotz type variational problem of Birkhoffian systems firstly and give the form of time-scales Pfaff-Herglotz action for delta derivatives. Then, time-scales Herglotz type Birkhoff's equations for delta derivatives are derived by calculating the variation of the action. Furthermore, time-scales Herglotz type Noether symmetry for delta derivatives of Birkhoffian systems are defined. According to this definition, time-scales Herglotz type Noether identity and Noether theorem for delta derivatives of Birkhoffian systems are proposed and proved, which can become the ones for delta derivatives of Hamiltonian systems or Lagrangian systems in some special cases. Therefore, it is shown that the results of Birkhoffian formalism are more universal than Hamiltonian or Lagrangian formalism. Finally, the time-scales damped oscillator and a non-Hamiltonian Birkhoffian system are given to exemplify the superiority of the results.

SUBMITTER: Tian X

PROVIDER: S-EPMC6894585 | biostudies-literature | 2019 Nov

REPOSITORIES: biostudies-literature

ACCESS DATA

Publications

Time-scales Herglotz type Noether theorem for delta derivatives of Birkhoffian systems.

Tian Xue X Zhang Yi Y

Royal Society open science 20191120 11

The time-scales theory provides a powerful theoretical tool for studying differential and difference equations simultaneously. With regard to Herglotz type variational principle, this generalized variational principle can deal with non-conservative or dissipative problems. Combining the two tools, this paper aims to study time-scales Herglotz type Noether theorem for delta derivatives of Birkhoffian systems. We introduce the time-scales Herglotz type variational problem of Birkhoffian systems fi ...[more]

PMID: 31827859

Similar Datasets

Project description:The ventriloquism aftereffect (VAE) refers to a shift in auditory spatial perception following exposure to a spatial disparity between auditory and visual stimuli. The VAE has been previously measured on two distinct time scales. Hundreds or thousands of exposures to a an audio-visual spatial disparity produces enduring VAE that persists after exposure ceases. Exposure to a single audio-visual spatial disparity produces immediate VAE that decays over seconds. To determine if these phenomena are two extremes of a continuum or represent distinct processes, we conducted an experiment with normal hearing listeners that measured VAE in response to a repeated, constant audio-visual disparity sequence, both immediately after exposure to each audio-visual disparity and after the end of the sequence. In each experimental session, subjects were exposed to sequences of auditory and visual targets that were constantly offset by +8° or -8° in azimuth from one another, then localized auditory targets presented in isolation following each sequence. Eye position was controlled throughout the experiment, to avoid the effects of gaze on auditory localization. In contrast to other studies that did not control eye position, we found both a large shift in auditory perception that decayed rapidly after each AV disparity exposure, along with a gradual shift in auditory perception that grew over time and persisted after exposure to the AV disparity ceased. We modeled the temporal and spatial properties of the measured auditory shifts using grey box nonlinear system identification, and found that two models could explain the data equally well. In the power model, the temporal decay of the ventriloquism aftereffect was modeled with a power law relationship. This causes an initial rapid drop in auditory shift, followed by a long tail which accumulates with repeated exposure to audio-visual disparity. In the double exponential model, two separate processes were required to explain the data, one which accumulated and decayed exponentially and the other which slowly integrated over time. Both models fit the data best when the spatial spread of the ventriloquism aftereffect was limited to a window around the location of the audio-visual disparity. We directly compare the predictions made by each model, and suggest additional measurements that could help distinguish which model best describes the mechanisms underlying the VAE.

Dataset Information

Time-scales Herglotz type Noether theorem for delta derivatives of Birkhoffian systems.

Publications

Time-scales Herglotz type Noether theorem for delta derivatives of Birkhoffian systems.

Similar Datasets

OmicsDI is part of the ELIXIR infrastructure

Tweets