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Journal article
The Poincare Map of Randomly Perturbed Periodic Motion
Journal of nonlinear science, v 23(5), pp 835-861
01 Oct 2013
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 Poincar, 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 to zero together with the noise intensity. Therefore, our result can be interpreted as an estimate of the 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 distinct 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 Hitczenko - Drexel UniversityGeorgi S. Medvedev - Drexel University
- Publication Details
- Journal of nonlinear science, v 23(5), pp 835-861
- Publisher
- Springer Nature
- Number of pages
- 27
- Grant note
- 208766 / Simons Foundation DMS 1109367 / NSF; National Science Foundation (NSF) 1109367 / Direct For Mathematical & Physical Scien; National Science Foundation (NSF); NSF - Directorate for Mathematical & Physical Sciences (MPS)
- Resource Type
- Journal article
- Language
- English
- Academic Unit
- Mathematics
- Web of Science ID
- WOS:000324332300006
- Scopus ID
- 2-s2.0-84884673287
- Other Identifier
- 991019167882304721
InCites Highlights
Data related to this publication, from InCites Benchmarking & Analytics tool:
- Web of Science research areas
- Mathematics, Applied
- Mechanics
- Physics, Mathematical