A Technique for Determining Stability Regions for the Damped Mathieu Equation

Gunderson, H.; Rigas, H.; VanVleck, F. S.

doi:10.1137/0126032

Cited by 30 publications

(12 citation statements)

References 3 publications

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“…[45], where it is shown if the drive frequencies are integer multiples of one another, the same instability "tongue" patterns are obtained. Finally, let us mention in passing that with a finite phonon damping and for a fixed driving strength, the infinite cascade of ideal parametric resonances at Ω drv = ω 0 /(2n) will be truncated above a certain order [46]. For instance, only two resonances are noticeable in Fig.…”

Section: Parametric Amplification Of Phonon-mediated Electron-elmentioning

confidence: 99%

Theory of parametrically amplified electron-phonon superconductivity

et al. 2017

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Ultrafast optical manipulation of ordered phases in strongly correlated materials is a topic of significant theoretical, experimental, and technological interest. Inspired by a recent experiment on light-induced superconductivity in fullerenes [Mitrano et al., Nature 530, 2016], we develop a comprehensive theory of light-induced superconductivity in driven electron-phonon systems with lattice nonlinearities. In analogy with the operation of parametric amplifiers, we show how the interplay between the external drive and lattice nonlinearities lead to significantly enhanced effective electron-phonon couplings. We provide a detailed and unbiased study of the nonequilibrium dynamics of the driven system using the real-time Green's function technique. To this end, we develop a Floquet generalization of the Migdal-Eliashberg theory and derive a numerically tractable set of quantum Floquet-Boltzmann kinetic equations for the coupled electron-phonon system. We study the role of parametric phonon generation and electronic heating in destroying the transient superconducting state. Finally, we predict the transient formation of electronic Floquet bands in time-and angle-resolved photoemission spectroscopy experiments as a consequence of the proposed mechanism.

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Section: Parametric Amplification Of Phonon-mediated Electron-elmentioning

confidence: 99%

Theory of parametrically amplified electron-phonon superconductivity

et al. 2017

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“…It was shown that the shape and location of the n = 1 transition curve can be changed by changing the value of the order of the fractional derivative. In earlier work, a linear damping term was considered in Taylor & Narendra (1969), Gunderson et al (1974) and Richards (1976), and it was shown that adding a damping term increases the area of stability in parameter space. Insperger & Stépán (2002) studied equation (1.1) with a delay term and constructed a closedform three-dimensional stability chart for the delayed Mathieu equation.…”

Section: Introductionmentioning

confidence: 99%

Transition curves in a parametrically excited pendulum with a force of elliptic type

Sah

Mann

2012

Proc. R. Soc. A.

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This article investigates the equilibria and stability of a pendulum when the support has a prescribed motion defined by an elliptic function. Stability charts are generated in the parameter plane for different values of the elliptic function modulus. Numerical integration and Floquet theory are used to generate stability charts that are later obtained through harmonic balance analysis. It is shown that the size and location of the instability tongues is directly linked to the elliptic function modulus. Comparisons are also made between the stability charts of Mathieu's equation and those of the pendulum when the prescribed motion is defined by an elliptic function.

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“…Using a corollary of one of these theorems, conditions are also derived in Pradeep and Shrivastava (1986b) for the stability of the damped Mathieu equation. It is seen that the results obtained are superior to the existing, viz., those of Taylor and Narendra (1969) and Gunderson et al (1974) for some range of parameter values.…”

Section: Authors' Contributions To Multidimensional Linear Time Varyimentioning

confidence: 97%

“…Taylor and Narendra (1969) and Gunderson et al (1974) have developed analytic criteria for the anlaysis of this equation. Richards (1976Richards ( , 1977 approximated the equation based on the earlier work by Pipes(1953).…”

Section: Periodic Systemsmentioning

confidence: 98%

Stability of Time Varying Linear Systems

Shrivastava

Pradeep

1986

Astrophysics and Space Science Library

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ABSTRACT. The concept of stability is one of the most fundamental in the study of natural sciences. The complexity of the stability problem depends on the degree of mathematical idealization of the system. In general, time invariant systems are much easier to handle than time varying systems. Here, first a bird's eye view is presented on the important methods in constant parameter linear and nonlinear systems. Following this is a quick review of stability analysis of time varying systems touching upon periodic systems, second order systems, arbitrarily time varying systems, the absolute stability problem and the functional analytic approach. The paper also includes a summary of some important results of the authors' work on the stability of second order multidimensional linear arbitrarily time varying systems. The theorems derived generalize some existing theorems. Application of one of them to the damped Mathieu equation yields better results than those existing in literature for some values in the parameter space. The asymptotic behaviour and boundedness of almost constant coefficient systems have also been studied and some new theorems derived.

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A Technique for Determining Stability Regions for the Damped Mathieu Equation

Cited by 30 publications

References 3 publications

Theory of parametrically amplified electron-phonon superconductivity

Theory of parametrically amplified electron-phonon superconductivity

Transition curves in a parametrically excited pendulum with a force of elliptic type

Stability of Time Varying Linear Systems

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