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Nonlinear Dynamics

, Volume 4, Issue 4, pp 373–387 | Cite as

Stochastic averaging using elliptic functions to study nonlinear stochastic systems

  • Win-Min Tien
  • N. Sri Namachchivaya
  • V. T. Coppola
Article

Abstract

In this paper, a new scheme of stochastic averaging using elliptic functions is presented that approximates nonlinear dynamical systems with strong cubic nonlinearities in the presence of noise by a set of Itô differential equations. This is an extension of some recent results presented in deterministic dynamical systems. The second order nonlinear differential equation that is examined in this work can be expressed as % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWexLMBb50ujb% qeguuDJXwAKbacfiGaf8hEaGNbamaacqGHRaWkcaWGJbadcaaIXaGc% cqWF4baEcqGHRaWkcaWGJbadcaaIZaGccqWF4baEdaahaaWcbeqaai% aaiodaaaGccqGHRaWkcqaH1oqzcaWGMbGaaiikaiab-Hha4jaacYca% cqWFGaaicuWF4baEgaGaaiaacMcacqGHRaWkcqaH1oqzdaahaaWcbe% qaaiaaigdacaGGVaGaaGOmaaaaruWrL9MCNLwyaGGbcOGaa43zaiaa% cIcacqWF4baEcaGGSaGae8hiaaIaf8hEaGNbaiaacaGGSaGae8hiaa% IaeqOVdGNaaeikaiaadshacaqGPaGaaiykaiabg2da9iaaicdaaaa!645D!\[\ddot x + c1x + c3x^3 + \varepsilon f(x, \dot x) + \varepsilon ^{1/2} g(x, \dot x, \xi {\text{(}}t{\text{)}}) = 0\] where c1 and c3 are given constants, ξ(t) is stationary stochastic process with zero mean and ε≪1 is a small parameter. This method involves the laborious manipulation of Jacobian elliptic functions such as cn, dn and sn rather than the usual trigonometric functions. The use of a symbolic language such as Mathematica reduces the computational effort and allows us to express the results in a convenient form. The resulting equations are Markov approximations of amplitude and phase involving integrals of elliptic functions. Finally, this method was applied to study some standard second order systems.

Key words

Stochastic averaging nonlinear systems elliptic functions probability density 

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Copyright information

© Kluwer Academic Publishers 1993

Authors and Affiliations

  • Win-Min Tien
    • 1
  • N. Sri Namachchivaya
    • 1
  • V. T. Coppola
    • 2
  1. 1.Aeronautical and Astronautical Engineering DepartmentUniversity of Illinois at Urbana-ChampaignUrbanaU.S.A.
  2. 2.Department of Aerospace EngineeringUniversity of MichiganAnn ArborU.S.A.

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