Fractional Mathieu equation

Rand, Richard H.; Sah, Si Mohamed; Suchorsky, M. K.

doi:10.1016/j.cnsns.2009.12.009

Cited by 59 publications

(14 citation statements)

References 21 publications

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“…Rand et al (2010) investigated the effect of fractional damping on the transition curves in Mathieu's equation (1.1). 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.…”

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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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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“…It is found that the fractional order system can generate regular oscillation just like integer-order systems. Many phenomena found in the integer-order dynamic system, such as different kinds of resonance, hysteresis, bifurcation and chaos can also be exhibited in the fractional differential system [Baleanu & Trujillo, 2010;Rand et al, 2010;Sheu et al, 2007;Barbosa et al, 2007;Chen et al, 2009;Yu et al, 2009]. In general, most fractional differential equations do not have exact analytical solutions, so there are several analytical approximation methods [Odibat, 2010;He & Wu, 2007;Abdulaziz et al, 2008;Zurigat et al, 2010;Rabtah et al, 2010;Ray et al, 2005] which were developed to solve the fractional differential equations.…”

Section: Introductionmentioning

confidence: 98%

“…In recent years, the study of oscillatory behaviors in fractional order systems has been a subject of increasing attention [Tavazoei & Haeri, 2008;Guo et al, 2011;Debnath, 2003;Rossikhin & Shitikova, 2010;Baleanu & Trujillo, 2010;Rand et al, 2010;Sheu et al, 2007;Barbosa et al, 2007;Chen et al, 2009;Yu et al, 2009]. It is found that the fractional order system can generate regular oscillation just like integer-order systems.…”

Section: Introductionmentioning

confidence: 99%

Effects of Fractional Differentiation Orders on a Coupled Duffing Circuit

Leung

Guo

2013

Int. J. Bifurcation Chaos

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Classical electrical circuits consist of resistors and capacitors and are governed by integer-order model. Circuits may have so-called fractance which represents an electrical element with fractional order impedance. Therefore, fractional order derivative is important to study the dynamical behaviors of circuits. This paper extends the classical coupled Duffing circuits to cover a new fractional order Duffing system consisting of two identical periodic forced circuits coupled by a linear resistor. The fundamental resonance responses under various paired and unpaired fractional orders are investigated in detail using the harmonic balance in combination with polynomial homotopy continuation. The approximate solutions having a high degree of accuracy in the steady state response are sought. There exist different shapes of frequency versus response and excitation amplitude versus response curves under various fractional orders. Multiple-valued solutions and nonclassical bifurcations are observed analytically and verified numerically. The influence of coupling intensity on the fundamental resonance response is also examined. New contributions include the innovative introduction of fractional order to the coupled Duffing circuits, the explicit integration of fractional order and the linear consideration of higher harmonics to improve the nonlinear solutions of the lower harmonics without increasing the computational complexity.

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“…One of the reasons of investigating fractional oscillators is that complex structures can be reduced to vibrations of a simple set of oscillators that can capture the salient features effectively [5][6][7][8][9][10]. Many phenomena are found in the integer order dynamic system, such as different kind of resonance, hysteresis, symmetry breaking, bifurcations and chaos can also be exhibited in the fractional derivative system [5][6][7][8][9][10][11][12][13][14]. On the other hand, owing to the difficulties in extracting solutions of the nonlinear fractional differential systems, investigating in accurate and efficient methods for solving fractional derivative systems has been an active research topic.…”

Section: Introductionmentioning

confidence: 99%

“…In the past decades, fractional differential equations are increasingly applied in modeling many physical and engineering problems [1][2][3][4][5][6][7][8][9][10]. It has been found that the behavior of many dynamic systems in various fields including thermal engineering, acoustics, electromagnetism, control, viscoelasticity, turbulence, robotics, signal processing and some other physical processes can be properly modeled by fractional order derivative systems.…”

Section: Introductionmentioning

confidence: 99%