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Mathieu function
About: Mathieu function is a research topic. Over the lifetime, 1724 publications have been published within this topic receiving 28994 citations.
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TL;DR: In this article, the authors present a numerical method for the stability analysis of linear delayed systems based on a special kind of discretization technique with respect to the past effect only.
Abstract: SUMMARY The paper presents an ecient numerical method for the stability analysis of linear delayed systems. The method is based on a special kind of discretization technique with respect to the past eect only. The resulting approximate system is delayed and also time periodic, but still, it can be transformed analytically into a high-dimensional linear discrete system. The method is applied to determine the stability charts of the Mathieu equation with continuous time delay. Copyright ? 2002 John Wiley & Sons, Ltd.
583 citations
TL;DR: In this article, the alternating gradient focusing was applied to a confined region of space by means of alternating and static electric fields, where the motion was governed by Mathieu's equation.
Abstract: Electrically charged iron and aluminum particles having diameters of a few microns have been contained in a confined region of space by means of alternating and static electric fields. The theory is essentially that of alternating gradient focusing; here the motion is governed by Mathieu's equation. Under certain circumstances when many particles are confined the three dimensional focusing force and the Coulomb repulsion results in a ``crystaline'' array which can be ``melted'' and reformed.
577 citations
TL;DR: In this paper, an updated version of the semi-discretization method is presented for periodic systems with a single discrete time delay, where the delayed term is approximated as a weighted sum of two neighboring discrete delayed state values and the transition matrix over a single period is determined.
Abstract: An updated version of the semi-discretization method is presented for periodic systems with a single discrete time delay. The delayed term is approximated as a weighted sum of two neighbouring discrete delayed state values and the transition matrix over a single period is determined. Stability charts are constructed for the damped and delayed Mathieu equation for different time-period/time-delay ratios. The convergence of the method is investigated by examples. Stability charts are constructed for 1 and 2 degree of freedom milling models. The codes of the algorithm are also attached in the appendix. Copyright © 2004 John Wiley & Sons, Ltd.
566 citations
TL;DR: In this article, two methods leading to general prescriptions for the construction of the time independent perturbation expansions for bound states to all orders are discussed, and the accuracy of the expansions are tested by a calculation of two eigenvalues of the Mathieu equation.
562 citations