Stochastic stability of nonlinear oscillators

M. M. Klosek-Dygas*, B. J. Matkowsky, Z. Schuss

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


The stochastic stability of a nonlinear oscillator parametrically excited by a stationary Markov process is considered. The stochastic stability problem in terms of a mean first passage time is formulated. Specifically, if the mean first passage time of the energy E of the oscillator to a given energy level is finite, then the oscillator is unstable. A method of averaging is used to derive a Fokker-Planck equation in the energy variable. The stability criterion depends on the nature of the boundary points E=0 and E=∞ and is expressed in terms of a Feller-type criterion. The stability condition is derived for various types of nonlinearities, including Coulomb friction. In contrast, it is observed that the standard criterion, in terms of Lyapunov exponents, is inconclusive for this type of problem.

Original languageEnglish
Pages (from-to)1115-1127
Number of pages13
JournalSIAM Journal on Applied Mathematics
Issue number5
StatePublished - 1988


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