Stability regions for Mathieu equation with imperfect periodicity

Bobryk, Roman V.; Chrzeszczyk, A.

doi:10.1016/j.physleta.2009.07.069

Cited by 20 publications

(15 citation statements)

References 15 publications

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“…Whereas the shortened set of equations identifies a solitary instability domain, modeling the original set of equations identifies a number of instability domains. Their boundaries, with increase of the noise intensity, merge into wider unstable regions, which has also been observed previously in [23]. Yet, applying the proper control strategy to the length of the pendulum provides a reliable way to shift the instability boundaries further upwards, thus ensuring for larger stable parameter regions.…”

Section: Discussionsupporting

confidence: 71%

Stability, control and reliability of a ship crane payload motion

Yurchenko

Alevras

2014

Probabilistic Engineering Mechanics

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Section: Discussionsupporting

confidence: 71%

Stability, control and reliability of a ship crane payload motion

Yurchenko

Alevras

2014

Probabilistic Engineering Mechanics

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“…In reference [35], the mean square stability for equation (8) is considered when the parameter b is small, in which the stability boundaries only near the principal resonance frequency are obtained by using asymptotic methods. In this paper, we employ the numerical procedure developed by Bobryk and Chrzeszczyk [26] to derive the mean square stability charts [36] for equation (8) in the case of arbitrary b and the higher-order resonance frequency ratios ω/Ω.…”

Section: Stability Chart Of Systemmentioning

confidence: 99%

“…From Bobryk and Chrzeszczyk [26], by using the Cameron-Martin formula [37] for the density of the Wiener measure, the following infinite hierarchy of linear differential equations can be obtained for the mean E[X(t)]:…”

Section: Stability Chart Of Systemmentioning

confidence: 99%

“…For computational purposes, we can obtain the closed system of linear differential equations of first order with constant coefficients by neglecting the terms u n+1 , v n+1 in the equations for u n , v n , where the index n is called the truncation index. From [26,[38][39][40], the procedure is quickly convergent. For sufficiently large truncation index, the asymptotic stability or instability of the system (21) determines the mean square stability or instability for equation (8).…”

Section: Stability Chart Of Systemmentioning

confidence: 99%

“…e randomness of the phase reduces when the lateral bridge amplitude increases, because the swaying bridge always makes pedestrians synchronize with the moving deck. By referring to some criteria on stability of nonlinear and stochastic equations [23,24,25,26], we will show the influence of randomness on stability of a suspension footbridge to understand the whole process that the lateral bridge amplitude increases from small to large. e most advantage of the proposed model is the unified form on description of the lateral force induced by pedestrians, which usually is discontinuously expressed for different lateral bridge amplitudes in other models.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A Suspension Footbridge Model under Crowd‐Induced Lateral Excitation

Ouyang

Ding

et al. 2019

Shock and Vibration

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In this paper, a plane pendulum model is proposed to investigate the lateral vibration of a suspension bridge under crowd excitation. The plane model consists of two strings and a rigid body, which represent cables and the bridge deck, respectively. The lateral force induced by crowd is expressed as a cosine function with random phase. Comparing with other existing pedestrian-footbridge interaction models, the proposed model has two features: one is that the structural characteristics of the suspension bridge are taken into account. The other is that the expression of the lateral force induced by crowd has a unified form for different lateral bridge amplitudes. By numerically analyzing the solution stability of the plane model, we exhibit the whole changing process how a suspension bridge increases its lateral amplitude from small to large. It is shown that the worst case occurs when the lateral natural frequency of the bridge is half the lateral step frequency of the pedestrians. Based on the analysis results, the plane pendulum model can be easily used to explain why the central span of the London Millennium Bridge has large lateral oscillations at about 0.48 Hz.

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Multifaceted Kinetics of Immuno-Evasion from Tumor Dormancy

d’Onofrio

2012

Advances in Experimental Medicine and Biology

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Tumor progression is subject to modulation by the immune system. The immune system can eliminate tumors or keep them at a dormant equilibrium size, while some tumors escape immunomodulation and advance to malignancy. Herein, we discuss some aspects of immune evasion of dormant tumors from a theoretical biophysics point of view that can be modeled mathematically. We go on to analyze the mathematical system on multiple timescales. First, we consider a long timescale where tumor evasion is likely due to adaptive (and somewhat deterministic) immuno-editing. Then, we consider the temporal mesoscale and hypothesize that extrinsic noise could be a major factor in induction of immuno-evasion. Implications of immuno-evasive mechanisms for the outcome of immunotherapies are also discussed. In addition, we discuss the ideas that population level tumor dormancy may not be a quiescence phenomenon and that dormant tumors can, at least if modulated by the immune system, live a very active and noisy life!

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Stability regions for Mathieu equation with imperfect periodicity

Cited by 20 publications

References 15 publications

Stability, control and reliability of a ship crane payload motion

Stability, control and reliability of a ship crane payload motion

A Suspension Footbridge Model under Crowd‐Induced Lateral Excitation

Multifaceted Kinetics of Immuno-Evasion from Tumor Dormancy

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