Accession Number:

ADA058537

Title:

Oscillations, Fluctuations, and the Hopf Bifurcation.

Descriptive Note:

Professional paper,

Corporate Author:

CENTER FOR NAVAL ANALYSES ARLINGTON VA

Personal Author(s):

Report Date:

1978-06-01

Pagination or Media Count:

47.0

Abstract:

Consider the effects of small random perturbations on deterministic systems of differential equations. The deterministic systems of interest have oscillatory dynamics and may undergo a bifurcation the Hopf bifurcation. A first exit problem is formulated for experiments beginning near stable and unstable limit cycles. The unstable limit cycle is surrounded by an annulus. Of interest is the probability of first exit from the annulus through a specified boundary, conditioned on initial position. The diffusion approximation is used, so that the conditional probability satisfies a backward diffusion equation. Appropriate solutions on the backward equation are constructed by an asymptotic method. The behavior of the stochastic system in the vicinity of stable and unstable limit cycles is compared. When the deterministic system exhibits the Hopf bifurcation, the above analysis must be modified.

Subject Categories:

  • Statistics and Probability

Distribution Statement:

APPROVED FOR PUBLIC RELEASE