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On a complex beam–beam interaction model with random forcing

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  • Xu, Yong
  • Xu, Wei
  • Mahmoud, Gamal M

Abstract

In recent years, many studies have been devoted to complex differential equations (CDE), which appear in very important applications in physics and engineering. This paper aims to investigate one such CDE, containing a random forcing term: (∗)z̈+ωo2z+ε2λg(ż)+ε2f(z,z̄)p(ωot)=αn(t),where z(t)=x(t)+iy(t) a complex function, i=−1, n(t) is a broad-band process with zero mean and ε2 and λ small real parameters. In particular, we use Eq. (∗) to model the interaction between two colliding beams in particle accelerators, setting g(ż)=ż, f(z,z̄)=z|z|2 and p(ωot)=cosωot and extend the work we had started in an earlier publication (Mahmoud, Physica A 216 (1995) 445). We apply the stochastic averaging method to derive a Fokker–Planck–Kolmogorov equation for this equation and obtain analytically the exact stationary probability density function and the first and second moments in the amplitude of the solutions. Numerical simulations are carried out to compare with the theoretical ones and excellent agreement is found.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:phsmap:v:336:y:2004:i:3:p:347-360
    DOI: 10.1016/j.physa.2003.12.030
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    References listed on IDEAS

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    1. 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.
    2. 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.
    3. 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.
    4. Mahmoud, Gamal M., 1995. "Beam-beam interaction models with a small stochastic perturbation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 216(4), pages 445-451.
    5. Mahmoud, Gamal M., 1998. "Approximate solutions of a class of complex nonlinear dynamical systems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 253(1), pages 211-222.
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    Cited by:

    1. Li, Wei & Li, Jiaorui & Chen, Weisheng, 2012. "The reliability of a stochastically complex dynamical system," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(13), pages 3556-3565.
    2. Li, Wei & Xu, Wei & Zhao, Junfeng & Wu, Haibo, 2007. "The study on stationary solution of a stochastically complex dynamical system," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 385(2), pages 465-472.
    3. Xu, Yong & Xu, Wei & Mahmoud, Gamal M., 2008. "On a complex Duffing system with random excitation," Chaos, Solitons & Fractals, Elsevier, vol. 35(1), pages 126-132.
    4. 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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