Bounds for the Width of the Instability Intervals in the Mathieu Equation

1996 ◽  
pp. 348-357 ◽  
Author(s):  
P. N. Shivakumar ◽  
Qiang Ye
Keyword(s):  
Mathieu Equation

Mathieu equation with application to analysis of dynamic characteristics of resonant inertial sensors

2013 ◽  
Vol 18 (2) ◽  
pp. 401-410 ◽  
Author(s):  
Yan Li ◽  
Shangchun Fan ◽  
Zhanshe Guo ◽  
Jing Li ◽  
Le Cao ◽  
...  
Keyword(s):  
Dynamic Characteristics ◽  
Inertial Sensors ◽  
Mathieu Equation

Observations Regarding Numerical Results Obtained by the Floquet-Method

2013 ◽  
Author(s):  
Till J. Kniffka ◽  
Horst Ecker
Keyword(s):  
Numerical Methods ◽  
Monodromy Matrix ◽  
Mathieu Equation ◽  
Numerical Results ◽  
The Other ◽  
Benchmark Problem ◽  
Stability Studies ◽  
Floquet Method ◽  
System Equations ◽  
Time Periodic

Stability studies of parametrically excited systems are frequently carried out by numerical methods. Especially for LTP-systems, several such methods are known and in practical use. This study investigates and compares two methods that are both based on Floquet’s theorem. As an introductary benchmark problem a 1-dof system is employed, which is basically a mechanical representation of the damped Mathieu-equation. The second problem to be studied in this contribution is a time-periodic 2-dof vibrational system. The system equations are transformed into a modal representation to facilitate the application and interpretation of the results obtained by different methods. Both numerical methods are similar in the sense that a monodromy matrix for the LTP-system is calculated numerically. However, one method uses the period of the parametric excitation as the interval for establishing that matrix. The other method is based on the period of the solution, which is not known exactly. Numerical results are computed by both methods and compared in order to work out how they can be applied efficiently.

A Quasiperiodic Mathieu Equation

1995 ◽  
Author(s):  
Richard Rand ◽  
Rachel Hastings
Keyword(s):  
Numerical Integration ◽  
Mathieu Equation ◽  
Stability Regions ◽  
Regions Of Stability ◽  
Extensive Computation ◽  
The Stability ◽  
Direct Numerical Integration

Abstract 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. In addition, we use perturbations to obtain approximations for the stability regions near δ=14 for small ω, and we compare the results with those of direct numerical integration.

Use of diagrams in computing the stability boundaries for the Mathieu equation

1977 ◽  
Vol 17 (3) ◽  
pp. 360-360
Author(s):  
R. Banene ◽  
V. Lesauskis
Keyword(s):  
Mathieu Equation ◽  
Stability Boundaries ◽  
The Stability

Parametric instability of internal gravity waves

1975 ◽  
Vol 67 (4) ◽  
pp. 667-687 ◽  
Author(s):  
A. D. McEwan ◽  
R. M. Robinson
Keyword(s):  
Internal Wave ◽  
Large Scale ◽  
Wave Spectrum ◽  
Parametric Instability ◽  
Mathieu Equation ◽  
Internal Gravity Waves ◽  
Small Scale ◽  
Preliminary Calculation ◽  
Theoretical Predictions ◽  
Oceanic Internal Wave

A continuously stratified fluid, when subjected to a weak periodic horizontal acceleration, is shown to be susceptible to a form of parametric instability whose time dependence is described, in its simplest form, by the Mathieu equation. Such an acceleration could be imposed by a large-scale internal wave field. The growth rates of small-scale unstable modes may readily be determined as functions of the forcing-acceleration amplitude and frequency. If any such mode has a natural frequency near to half the forcing frequency, the forcing amplitude required for instability may be limited in smallness only by internal viscous dissipation. Greater amplitudes are required when boundaries constrain the form of the modes, but for a given bounding geometry the most unstable mode and its critical forcing amplitude can be defined.An experiment designed to isolate the instability precisely confirms theoretical predictions, and evidence is given from previous experiments which suggest that its appearance can be the penultimate stage before the traumatic distortion of continuous stratifications under internal wave action.A preliminary calculation, using the Garrett & Munk (197%) oceanic internal wave spectrum, indicates that parametric instability could occur in the ocean at scales down to that of the finest observed microstructure, and may therefore have a significant role to play in its formation.

Distribution of Sub and Super Harmonic Solution of Mathieu Equation Within Stable Zones

2005 ◽  
Author(s):  
G. Nakhaie Jazar ◽  
M. Mahinfalah ◽  
M. Rastgaar Aagaah ◽  
N. Mahmoudian
Keyword(s):  
Spectral Density ◽  
Periodic Solutions ◽  
Mathieu Equation ◽  
Stable Region ◽  
Transition Curve ◽  
Harmonic Solution ◽  
The Third ◽  
Power Spectral ◽  
Stability Chart ◽  
Spectral Density Method

The third stable region of the Mathieu stability chart, surrounded by one π-transition and one 2π-transition curve is investigated. It is known that the solution of Mathieu equation is either periodic or quasi-periodic when its parameters are within stable regions. Periodic responses occur when they are on a “splitting curve”. Splitting curves are within stable regions and are corresponding to coexisting of periodic curves where an instability tongue closes. Distributions of sub and super-harmonics, as well as quasi-periodic solutions are analyzed using power spectral density method.

scholarly journals Duality and singular continuous spectrum in the almost Mathieu equation

1997 ◽  
Vol 178 (2) ◽  
pp. 169-183 ◽  
Author(s):  
A. Y. Gordon ◽  
S. Jitomirskaya ◽  
Y. Last ◽  
B. Simon
Keyword(s):  
Continuous Spectrum ◽  
Mathieu Equation ◽  
Singular Continuous Spectrum

Lyapunov and marginal instability in Hamiltonian systems

1999 ◽  
Vol 77 (8) ◽  
pp. 603-633 ◽  
Author(s):  
J Grindlay
Keyword(s):  
Lyapunov Exponents ◽  
Nonlinear Oscillator ◽  
Linear Chain ◽  
Mathieu Equation ◽  
Singular Values ◽  
Space Velocity ◽  
Singular Value ◽  
The Stability ◽  
Value Decomposition ◽  
Simple Systems

The variational equations and the evolution matrix are introduced and used to discuss the stability of a bound Hamiltonian trajectory. Singular-value decomposition is applied to the evolution matrix. Singular values and Lyapunov exponents are defined and their properties described. The singular-value expansion of the phase-space velocity is derived. Singular values and Lyapunov exponents are used to characterize the stability behaviour of five simple systems, namely, the nonlinear oscillator with cubic anharmonicity, the quasi-periodic Mathieu equation, the Hénon-Heilesmodel, the 4+2 linear chain with cubic anharmonicity, and an integrable system of arbitrary order.PACS Nos.: 03.20, 05.20

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