Fractional delayed damped Mathieu equation

Afshin Mesbahi; Mohammad Haeri; Morad Nazari; Eric A. Butcher

doi:10.1080/00207179.2014.971430

Fractional delayed damped Mathieu equation

Afshin Mesbahi, Mohammad Haeri, Morad Nazari, Eric A. Butcher

Research output: Contribution to journal › Article › peer-review

14 Scopus citations

Abstract

This paper investigates the dynamical behaviour of the fractional delayed damped Mathieu equation. This system includes three different phenomena (fractional order, time delay, parametric resonance). The method of harmonic balance is employed to achieve approximate expressions for the transition curves in the parameter plane. The n = 0 and n = 1 transition curves (both lower and higher order approximations) are obtained. The dependencies of these curves on the system parameters and fractional orders are determined. Previous results for the transition curves reported for the damped Mathieu equation, delayed second-order oscillator, and fractional Mathieu equation are confirmed as special cases of the results for the current system.

Original language	English (US)
Pages (from-to)	622-630
Number of pages	9
Journal	International Journal of Control
Volume	88
Issue number	3
DOIs	https://doi.org/10.1080/00207179.2014.971430
State	Published - Mar 4 2015

Keywords

Mathieu equation
fractional order
harmonic balance
parametric resonance
time delay
transition curves

ASJC Scopus subject areas

Control and Systems Engineering
Computer Science Applications

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10.1080/00207179.2014.971430

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title = "Fractional delayed damped Mathieu equation",

abstract = "This paper investigates the dynamical behaviour of the fractional delayed damped Mathieu equation. This system includes three different phenomena (fractional order, time delay, parametric resonance). The method of harmonic balance is employed to achieve approximate expressions for the transition curves in the parameter plane. The n = 0 and n = 1 transition curves (both lower and higher order approximations) are obtained. The dependencies of these curves on the system parameters and fractional orders are determined. Previous results for the transition curves reported for the damped Mathieu equation, delayed second-order oscillator, and fractional Mathieu equation are confirmed as special cases of the results for the current system.",

keywords = "Mathieu equation, fractional order, harmonic balance, parametric resonance, time delay, transition curves",

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AU - Mesbahi, Afshin

AU - Haeri, Mohammad

AU - Nazari, Morad

AU - Butcher, Eric A.

PY - 2015/3/4

Y1 - 2015/3/4

N2 - This paper investigates the dynamical behaviour of the fractional delayed damped Mathieu equation. This system includes three different phenomena (fractional order, time delay, parametric resonance). The method of harmonic balance is employed to achieve approximate expressions for the transition curves in the parameter plane. The n = 0 and n = 1 transition curves (both lower and higher order approximations) are obtained. The dependencies of these curves on the system parameters and fractional orders are determined. Previous results for the transition curves reported for the damped Mathieu equation, delayed second-order oscillator, and fractional Mathieu equation are confirmed as special cases of the results for the current system.

AB - This paper investigates the dynamical behaviour of the fractional delayed damped Mathieu equation. This system includes three different phenomena (fractional order, time delay, parametric resonance). The method of harmonic balance is employed to achieve approximate expressions for the transition curves in the parameter plane. The n = 0 and n = 1 transition curves (both lower and higher order approximations) are obtained. The dependencies of these curves on the system parameters and fractional orders are determined. Previous results for the transition curves reported for the damped Mathieu equation, delayed second-order oscillator, and fractional Mathieu equation are confirmed as special cases of the results for the current system.

KW - Mathieu equation

KW - fractional order

KW - harmonic balance

KW - parametric resonance

KW - time delay

KW - transition curves

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Fractional delayed damped Mathieu equation

Abstract

Keywords

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