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A Quasiperiodic Mathieu–Hill Equation
SIAM Journal on Applied Mathematics, 1980A study is made of a generalized Mathieu–Hill equation \[u'' + [ \delta + \varepsilon (\cos 2t + \alpha \cos 2(\lambda + \varepsilon )t) ]u = 0,\] where $\delta$, $\varepsilon$, $\alpha$, $\lambda$ are constants, with $| \varepsilon | \ll 1$ and $\lambda $ a rational fraction.
Davis, Stephen H., Rosenblat, S.
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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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THE NONLINEAR MATHIEU EQUATION
International Journal of Bifurcation and Chaos, 1994The purpose of this paper is to classify the different sequences of bifurcation that can occur for small amplitude solutions to the nonlinear Mathieu equation near to the Mathieu regions of instability. We do this by using the Lindstedt-Poincare perturbation method to construct a vector field which interpolates the successive iterations of the ...
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Analysis of Mathieu Equation Stable Solutions in the First Zone of Stability [PDF]
Procedia Engineering, 2016 The paper presents the results of a homogeneous Mathieu equation studies. Mathieu equation solutions are oscillations, modulated in amplitude and frequency.
Nesterov, S.V. +2 more
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The stochastic Mathieu's equation
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2008In this manuscript, we consider generalizations of the classical Mathieu's equation to stochastic systems. Unlike previous works, we focus on internal frequencies that vary continuously between periodic and stochastic variables. By numerically integrating the system of equations using a symplectic method, we determine the Lyapunov exponents for a wide ...
Poulin, Francis J., Flierl, Glenn R.
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Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2006
An inverted pendulum with asymmetric elastic restraints (e.g. a one-sided spring), when subjected to harmonic vertical base excitation, on linearizing trigonometric terms, is governed by an asymmetric Mathieu equation. This system is parametrically forced and strongly nonlinear (linearization for small motions is not possible ...
Amol Marathe, Anindya Chatterjee
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An inverted pendulum with asymmetric elastic restraints (e.g. a one-sided spring), when subjected to harmonic vertical base excitation, on linearizing trigonometric terms, is governed by an asymmetric Mathieu equation. This system is parametrically forced and strongly nonlinear (linearization for small motions is not possible ...
Amol Marathe, Anindya Chatterjee
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A Quasiperiodic Mathieu Equation
Volume 3A: 15th Biennial Conference on Mechanical Vibration and Noise — Vibration of Nonlinear, Random, and Time-Varying Systems, 1995Abstract In this work we investigate the following quasiperiodic Mathieu equation: x ¨ + ( δ + ϵ cos t + ϵ cos ω t ) x = 0 We use numerical integration to determine regions of stability in the δ–ω plane for fixed ϵ. Graphs of these stability regions are presented, based on extensive computation.
Richard Rand, Rachel Hastings
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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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Nonoscillation of Mathieu equations with two frequencies
Applied Mathematics and Computation, 2019zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Jitsuro Sugie, Kazuki Ishibashi
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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,
openaire +1 more source