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Parametric Frequency Analysis of Mathieu–Duffing Equation
International Journal of Bifurcation and Chaos, 2021 The classic linear Mathieu equation is one of the archetypical differential equations which has been studied frequently by employing different analytical and numerical methods. The Mathieu equation with cubic nonlinear term, also known as Mathieu–Duffing equation, is one of the many extensions of the classic Mathieu equation. Nonlinear characteristics
Azimi, Mohsen
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Applications of the Mathieu equation
American Journal of Physics, 1996The properties of the Mathieu equation are reviewed in order to discuss some of the applications that have appeared in recent years. Those mentioned are: vibrations in an elliptic drum, the inverted pendulum, the radio frequency quadrupole, frequency modulation, stability of a floating body, alternating gradient focusing, the Paul trap for charged ...
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Transition Curves for the Quasi-Periodic Mathieu Equation
SIAM Journal on Applied Mathematics, 1998The quasi-periodic Mathieu equation \[ \ddot\psi+ \bigl[\delta+ \varepsilon(\cos t+\cos\omega t)\bigr]\psi=0 \] is investigated for small \(\varepsilon\) and irrational \(\omega\). The aim is to obtain a stability diagram in the \(\delta f\)-\(\omega\) plane (for fixed \(\varepsilon)\) for which all solutions are bounded.
Richard Rand
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Communications in Nonlinear Science and Numerical Simulation, 2010
After reviewing the concept of fractional derivative, we derive expressions for the transition curves separating regions of stability from regions of instability in the ODE: x″+(δ+εcost)x+cDαx=0 where Dαx is the order α derivative of x(t), where 0 < α < 1.
Richard H. Rand +2 more
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After reviewing the concept of fractional derivative, we derive expressions for the transition curves separating regions of stability from regions of instability in the ODE: x″+(δ+εcost)x+cDαx=0 where Dαx is the order α derivative of x(t), where 0 < α < 1.
Richard H. Rand +2 more
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An Application of the Infinite Matrix Theory to Mathieu Equation [PDF]
Computers and Mathematics With Applications, 2006 In this paper we study the infinite linear system MμX=0 equivalent to the Mathieu equation. Applying some results in summability we determine the Floquet exponents corresponding to the solutions of the differential equation.
Bruno de Malafosse
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Periodic analytic approximate solutions for the Mathieu equation [PDF]
Applied Mathematics and Computation, 2015 International audienceWe propose two methods to find analytic periodic approximations intended for differential equations of Hill type. Here, we apply these methods on the simplest case of the Mathieu equation.
Manuel Gadella
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Stability and bifurcation of Mathieu–Duffing equation
International Journal of Non-Linear Mechanics, 2022 Various phenomena in science, physics, and engineering result in the Mathieu equation with cubic nonlinear term, known as the Mathieu–Duffing equation.
Mohsen Azimi
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Vibrational control of Mathieu's equation
2013 IEEE/ASME International Conference on Advanced Intelligent Mechatronics, 2013The vertically driven inverted pendulum-sometimes called the “Kapitza pendulum”-is a well-known example of an unstable system that can be stabilized by oscillatory forcing. Averaging methods and asymptotic stability results can be applied to develop a general framework for designing suitable inputs.
I. P. M. Wickramasinghe, Jordan M. Berg
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On the double points of a Mathieu equation [PDF]
Journal of Computational and Applied Mathematics, 1999 For a Mathieu equation with parameter q, the eigenvalues can be regarded as functions of the variable q. Our aim is to find q when adjacent eigenvalues of the same type become equal yielding double points of the given Mathieu equation.
Jungong Xue
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Damped equations of Mathieu type
Applied Mathematics and Computation, 2014zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Anindya Ghose Choudhury, Partha Guha
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