### Abstract

We investigate the pattern of optimal paths along which a dynamical system driven by weak noise moves, with overwhelming probability, when it fluctuates far away from a stable state. Our emphasis is on systems that perform self-sustained periodic vibrations, and have an unstable focus inside a stable limit cycle. We show that in the vicinity of the unstable focus, the flow field of optimal paths generically displays a pattern of singularities. In particular, it contains a switching line that separates areas to which the system arrives along optimal paths of topologically different types. The switching line spirals into the focus and has a self-similar structure. Depending on the behavior of the system near the focus, it may be smooth, or have finite-length branches. Our results are based on an analysis of the topology of the Lagrangian manifold for an auxiliary, purely dynamical, problem that determines the optimal paths. We illustrate our theory by studying, both theoretically and nu-merically, a van der Pol oscillator driven by weak white noise.

Original language | English (US) |
---|---|

Pages (from-to) | 2369-2391 |

Number of pages | 23 |

Journal | Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics |

Volume | 55 |

Issue number | 3 SUPPL. A |

State | Published - 1997 |

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### ASJC Scopus subject areas

- Mathematical Physics
- Physics and Astronomy(all)
- Condensed Matter Physics
- Statistical and Nonlinear Physics

### Cite this

*Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics*,

*55*(3 SUPPL. A), 2369-2391.

**Topological features of large fluctuations to the interior of a limit cycle.** / Smelyanskiy, V. N.; Dykman, M. I.; Maier, Robert S.

Research output: Contribution to journal › Article

*Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics*, vol. 55, no. 3 SUPPL. A, pp. 2369-2391.

}

TY - JOUR

T1 - Topological features of large fluctuations to the interior of a limit cycle

AU - Smelyanskiy, V. N.

AU - Dykman, M. I.

AU - Maier, Robert S

PY - 1997

Y1 - 1997

N2 - We investigate the pattern of optimal paths along which a dynamical system driven by weak noise moves, with overwhelming probability, when it fluctuates far away from a stable state. Our emphasis is on systems that perform self-sustained periodic vibrations, and have an unstable focus inside a stable limit cycle. We show that in the vicinity of the unstable focus, the flow field of optimal paths generically displays a pattern of singularities. In particular, it contains a switching line that separates areas to which the system arrives along optimal paths of topologically different types. The switching line spirals into the focus and has a self-similar structure. Depending on the behavior of the system near the focus, it may be smooth, or have finite-length branches. Our results are based on an analysis of the topology of the Lagrangian manifold for an auxiliary, purely dynamical, problem that determines the optimal paths. We illustrate our theory by studying, both theoretically and nu-merically, a van der Pol oscillator driven by weak white noise.

AB - We investigate the pattern of optimal paths along which a dynamical system driven by weak noise moves, with overwhelming probability, when it fluctuates far away from a stable state. Our emphasis is on systems that perform self-sustained periodic vibrations, and have an unstable focus inside a stable limit cycle. We show that in the vicinity of the unstable focus, the flow field of optimal paths generically displays a pattern of singularities. In particular, it contains a switching line that separates areas to which the system arrives along optimal paths of topologically different types. The switching line spirals into the focus and has a self-similar structure. Depending on the behavior of the system near the focus, it may be smooth, or have finite-length branches. Our results are based on an analysis of the topology of the Lagrangian manifold for an auxiliary, purely dynamical, problem that determines the optimal paths. We illustrate our theory by studying, both theoretically and nu-merically, a van der Pol oscillator driven by weak white noise.

UR - http://www.scopus.com/inward/record.url?scp=0001375031&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0001375031&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0001375031

VL - 55

SP - 2369

EP - 2391

JO - Physical review. E

JF - Physical review. E

SN - 2470-0045

IS - 3 SUPPL. A

ER -