Stochastic averaging using elliptic functions to study nonlinear stochastic systems

Win Min Tien, N. Sri Namachchivaya, V. T. Coppola

Research output: Contribution to journalArticlepeer-review


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 {Mathematical expression} 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.

Original languageEnglish (US)
Pages (from-to)373-387
Number of pages15
JournalNonlinear Dynamics
Issue number4
StatePublished - Aug 1993
Externally publishedYes


  • Stochastic averaging
  • elliptic functions
  • nonlinear systems
  • probability density

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Aerospace Engineering
  • Ocean Engineering
  • Mechanical Engineering
  • Applied Mathematics
  • Electrical and Electronic Engineering


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