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Preprint
The Poincare map of randomly perturbed periodic motion
arXiv.org
02 Jun 2012
Abstract
A system of autonomous differential equations with a stable limit cycle and
perturbed by small white noise is analyzed in this work. In the vicinity of the
limit cycle of the unperturbed deterministic system, we define, construct, and
analyze the Poincare map of the randomly perturbed periodic motion. We show
that the time of the first exit from a small neighborhood of the fixed point of
the map, which corresponds to the unperturbed periodic orbit, is well
approximated by the geometric distribution. The parameter of the geometric
distribution tends zero together with the noise intensity. Therefore, our
result can be interpreted as an estimate of stability of periodic motion to
random perturbations.
In addition, we show that the geometric distribution of the first exit times
translates into statistical properties of solutions of important differential
equation models in applications. To this end, we demonstrate three examples
from mathematical neuroscience featuring complex oscillatory patterns
characterized by the geometric distribution. We show that in each of these
models the statistical properties of emerging oscillations are fully explained
by the general properties of randomly perturbed periodic motions identified in
this paper.
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Details
- Title
- The Poincare map of randomly perturbed periodic motion
- Creators
- Pawel HitczenkoGeorgi S Medvedev
- Publication Details
- arXiv.org
- Resource Type
- Preprint
- Language
- English
- Academic Unit
- Mathematics
- Other Identifier
- 991021863168804721