An innovative efficient approach to solving damped Mathieu–Duffing equation with the non-perturbative technique

El-Dib, Yusry O.

doi:10.1016/j.cnsns.2023.107590

Cited by 10 publications

(6 citation statements)

References 26 publications

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“…Hence, many terms can be derived in the perturbation expansion of both the eigenvalues and Mathieu functions. What is more, the perturbative approach theory [17] is also fitted to investigate Hill's equation. We express the Mathieu equation as…”

Section: Appendix a Hill's Methods To Find The Solution Of The Mathie...mentioning

confidence: 99%

“…As the Mathieu equation does not possess closed-form analytic solution, its applications are affected with analytical [10] and numerical approximation schemes [11], along with nonlinear analysis of the associated stability charts [12,13]. Therefore, analytical solutions of Mathieu (Hill) or Duffing equations are generally investigated by means of various perturbation techniques [14][15][16], while other approaches also use non-perturbative techniques to characterize the parametric damping nonlinear Mathieu-Duffing oscillator [17].…”

Section: Introductionmentioning

confidence: 99%

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Mathieu-Hill Equation Stability Analysis for Trapped Ions. Anharmonic Corrections for Nonlinear Electrodynamic Traps

Mihalcea

2024

Preprint

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The stability properties of the Hill equation are discussed, and especially those of the Mathieu equation that characterize ion motion in electrodynamic traps. The solutions of the Mathieu equation for a trapped ion are characterized by using the Floquet theory and Hill's Method solution, which yields an infinite system of linear and homogeneous equations whose coefficients are recursively determined. Stability is discussed for parameters $a$ and $q$ that are real. Characteristic curves are introduced naturally by the Sturm-Liouville problem for the well known even and odd Mathieu equations $ce_m(z,q)$ and $se_m(z,q)$. We illustrate the stability diagram for a combined (Paul and Penning) trap and represent the frontiers of the stability domains for axial and radial motion. In case of a Paul trap the stable solution corresponds to a superposition of harmonic motions. The problem of evaluating the maximum amplitudes of stable oscillations for the ideal conditions (taken into consideration) is also approached. Anharmonic corrections are discussed within the frame of the perturbation theory, while the frontiers of the modified stability domains are determined as a function of the chosen perturbation parameter. The results apply to 2D and 3D ion traps used for different applications in quantum engineering, among which optical clocks, quantum logic and quantum metrology, but not restricted only to these.

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Section: Appendix a Hill's Methods To Find The Solution Of The Mathie...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Mathieu-Hill Equation Stability Analysis for Trapped Ions. Anharmonic Corrections for Nonlinear Electrodynamic Traps

Mihalcea

2024

Preprint

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“…According to Equation (17) the holomorphic function ξ exhibits a single pole at a = (2s − iµ) 2 , so that the function…”

Section: Appendix a Hill's Methods To Find The Solution Of The Mathie...mentioning

confidence: 99%

“…Therefore, analytical solutions of Mathieu-Hill (MH) or Duffing equations are generally investigated by means of various perturbation techniques [14][15][16], while other approaches also use non-perturbative techniques to characterize the parametric damping nonlinear Mathieu-Duffing oscillator [17].…”

Section: Introductionmentioning

confidence: 99%

Mathieu–Hill Equation Stability Analysis for Trapped Ions: Anharmonic Corrections for Nonlinear Electrodynamic Traps

Mihalcea

2024

Photonics

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The stability properties of the Hill equation are discussed, especially those of the Mathieu equation that characterize ion motion in electrodynamic traps. The solutions of the Mathieu-Hill equation for a trapped ion are characterized by employing the Floquet theory and Hill’s method solution, which yields an infinite system of linear and homogeneous equations whose coefficients are recursively determined. Stability is discussed for parameters a and q that are real. Characteristic curves are introduced naturally by the Sturm–Liouville problem for the well-known even and odd Mathieu equations cem(z,q) and sem(z,q). In the case of a Paul trap, the stable solution corresponds to a superposition of harmonic motions. The maximum amplitude of stable oscillations for ideal conditions (taken into consideration) is derived. We illustrate the stability diagram for a combined (Paul and Penning) trap and represent the frontiers of the stability domains for both axial and radial motion, where the former is described by the canonical Mathieu equation. Anharmonic corrections for nonlinear Paul traps are discussed within the frame of perturbation theory, while the frontiers of the modified stability domains are determined as a function of the chosen perturbation parameter and we demonstrate they are shifted towards negative values of the a parameter. The applications of the results include but are not restricted to 2D and 3D ion traps used for different applications such as mass spectrometry (including nanoparticles), high resolution atomic spectroscopy and quantum engineering applications, among which we mention optical atomic clocks and quantum frequency metrology.

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“…Notably, El-Dib's frequency formula has been effectively used to solve the nonlinear Mathieu equation without resorting to a perturbative approach, showcasing its utility in complex oscillator analysis. 16…”

Section: Introductionmentioning

confidence: 99%

Short remarks on the paper: A novel methodology for a time-delayed controller to prevent nonlinear system oscillations

El-Dib

2023

Journal of Low Frequency Noise, Vibration and Active Control

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In this brief communication, we highlight several critical inaccuracies in the paper entitled “A novel methodology for a time-delayed controller to prevent nonlinear system oscillations,” identified by the DOI: 10.1177/14613484231195276. The core of the non-perturbative approach discussed in this paper is the introduction of a trial solution with an indeterminate nonlinear frequency. We elucidate the technical missteps and provide necessary corrections. The scrutinized article presents ample opportunities for enhancement, and we propose several amendments. Additionally, we reveal that the numerical outcomes showcased in the mentioned paper expose deficiencies in the proposed approximate solution. With the amendments implemented herein, a more precise numerical solution is achieved.

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An innovative efficient approach to solving damped Mathieu–Duffing equation with the non-perturbative technique

Cited by 10 publications

References 26 publications

Mathieu-Hill Equation Stability Analysis for Trapped Ions. Anharmonic Corrections for Nonlinear Electrodynamic Traps

Mathieu-Hill Equation Stability Analysis for Trapped Ions. Anharmonic Corrections for Nonlinear Electrodynamic Traps

Mathieu–Hill Equation Stability Analysis for Trapped Ions: Anharmonic Corrections for Nonlinear Electrodynamic Traps

Short remarks on the paper: A novel methodology for a time-delayed controller to prevent nonlinear system oscillations

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