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Stability regions for coupled Hill's equations

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  • Mahmoud, Gamal M.

Abstract

In this paper we extend well-known results for one Hill's equation and present the stability analysis of two coupled Hill's equations for which the general theory is not readily available. Approximate expressions are derived in the context of peturbation theory for the boundaries between bounded and unbounded periodic solutions with frequencies ω = n/m (n and m are positive integers) of both linear and nonlinear coupled Mathieu equations as examples. Excellent agreement is found between theoretical predictions and numerical computations over large ranges of parameter values and initial conditions. These periodic solutions are important because they correspond to some of the lowest-order resonances of the system and when they are stable, they turn out to have large regions of regular motion around them in phase space. Coupled Mathieu equations appear in numerous important physical applications, in problems of accelerator dynamics, electrohydrodynamics and mechanics.

Suggested Citation

  • Mahmoud, Gamal M., 1997. "Stability regions for coupled Hill's equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 242(1), pages 239-249.
    • Handle: RePEc:eee:phsmap:v:242:y:1997:i:1:p:239-249
    DOI: 10.1016/S0378-4371(97)00194-5
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    Cited by:

    1. Cveticanin, L., 2001. "Analytic approach for the solution of the complex-valued strong non-linear differential equation of Duffing type," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 297(3), pages 348-360.
    2. Xu, Yong & Xu, Wei & Mahmoud, Gamal M. & Lei, Youming, 2005. "Beam–beam interaction models under narrow-band random excitation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 346(3), pages 372-386.
    3. Mahmoud, Gamal M. & Bountis, Tassos & Ahmed, Sayed A., 2000. "Stability analysis for systems of nonlinear Hill's equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 286(1), pages 133-146.
    4. Mahmoud, Gamal M. & Aly, Shaban A.H., 2000. "On periodic solutions of parametrically excited complex non-linear dynamical systems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 278(3), pages 390-404.
    5. Xu, Yong & Xu, Wei & Mahmoud, Gamal M, 2004. "On a complex beam–beam interaction model with random forcing," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 336(3), pages 347-360.
    6. Xu, Yong & Zhang, Huiqing & Xu, Wei, 2007. "On stochastic complex beam–beam interaction models with Gaussian colored noise," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 384(2), pages 259-272.

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