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Mathieu functions and numerical solutions of the Mathieu equation
2009 IEEE International Workshop on Open-source Software for Scientific Computation (OSSC), 2009We review the full spectrum of solutions to the Mathieu differential equation y'' + [a - 2q cos(2z)]y = 0, and we describe a numerical algorithm which allows a flexible approach to the computation of all the Mathieu functions. We use an elegant and compact matrix notation which can be readily implemented on any computing platform. We give some explicit
Roberto Coisson +2 more
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Fractional delayed damped Mathieu equation
International Journal of Control, 2014This 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.
Afshin Mesbahi +3 more
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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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Damped equations of Mathieu type
Applied Mathematics and Computation, 2014zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Ghose Choudhury, A., Guha, Partha
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Stability of damped mathieu equation
IEE-IERE Proceedings - India, 1972Following a simple method, explicit stability of lyapunov function has been obtained to study the Stability of damped Mathieu equation, The method may also be used to other time-varying problems.
R. Ganguli, A.K. Mandal, A.K. Chowdhary
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Parametric Frequency Analysis of Mathieu–Duffing Equation
International Journal of Bifurcation and Chaos, 2021The 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
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1983
The Mathieu equation in its standard form $$\ddot x + (a - 2q\cos 2t)x = 0$$ (6.1) is the most widely known and, in the past, most extensively treated Hill equation. In many ways this is curious since the equation eludes solution in a usable closed form; yet many investigators have sought to describe experiments in terms of a Mathieu equation,
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The Mathieu equation in its standard form $$\ddot x + (a - 2q\cos 2t)x = 0$$ (6.1) is the most widely known and, in the past, most extensively treated Hill equation. In many ways this is curious since the equation eludes solution in a usable closed form; yet many investigators have sought to describe experiments in terms of a Mathieu equation,
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On Mathieu equation with damping
Journal of Mathematical Physics, 1980A direct variational method is applied to the linear and nonlinear Mathieu equation with damping. It is found that the nature of the periodic solutions and the characteristic curves are modified due to the presence of the damping. A threshold value of β is required to overcome the damping for the existence of the periodic solutions.
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Numerical solution of Mathieu's equation
Journal of Computational Physics, 1971Abstract A method is presented for the numerical solution of Mathieu's equation. The power of the method lies in the fact that it can be used equally for ordinary and extremely asymptotic problems, making possible the computation of Mathieu functions for large values of the parameter with an accuracy heretofore unattainable.
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Variational method and Mathieu equation
Journal of Mathematical Physics, 1978A variational method is developed to study the linear and nonlinear Mathieu equations. For the linear Mathieu equation, various modes of the Mathieu functions, the characteristic curves, and the stability regions are found, which agree with the established results. The variational method is then applied to the nonlinear Mathieu equation.
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