### Abstract

In stochastic systems, one is often interested in finding the optimal path that maximizes the probability of escape from a metastable state or of switching between metastable states. Even for simple systems, it may be impossible to find an analytic form of the optimal path, and in highdimensional systems, this is almost always the case. In this article, we formulate a constructive methodology that is used to compute the optimal path numerically. The method utilizes finite-time Lyapunov exponents, statistical selection criteria, and a Newton-based iterative minimizing scheme. The method is applied to four examples. The first example is a two-dimensional system that describes a single population with internal noise. This model has an analytical solution for the optimal path. The numerical solution found using our computational method agrees well with the analytical result. The second example is a more complicated four-dimensional system where our numerical method must be used to find the optimal path. The third example, although a seemingly imple two-dimensional system, demonstrates the success of our method in finding the optimal path where other numerical methods are known to fail. In the fourth example, the optimal path lies in six-dimensional space and demonstrates the power of our method in computing paths in higherdimensional spaces.

Original language | English |
---|---|

Article number | 083101 |

Journal | Chaos |

Volume | 26 |

Issue number | 8 |

DOIs | |

State | Published - 1 Aug 2016 |

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*Chaos*,

*26*(8), [083101]. https://doi.org/10.1063/1.4958926

}

*Chaos*, vol. 26, no. 8, 083101. https://doi.org/10.1063/1.4958926

**Computing the optimal path in stochastic dynamical systems CHAOS 26, 083101 (2016).** / Bauver, Martha; Forgoston, Eric; Billings, Lora.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Computing the optimal path in stochastic dynamical systems CHAOS 26, 083101 (2016)

AU - Bauver, Martha

AU - Forgoston, Eric

AU - Billings, Lora

PY - 2016/8/1

Y1 - 2016/8/1

N2 - In stochastic systems, one is often interested in finding the optimal path that maximizes the probability of escape from a metastable state or of switching between metastable states. Even for simple systems, it may be impossible to find an analytic form of the optimal path, and in highdimensional systems, this is almost always the case. In this article, we formulate a constructive methodology that is used to compute the optimal path numerically. The method utilizes finite-time Lyapunov exponents, statistical selection criteria, and a Newton-based iterative minimizing scheme. The method is applied to four examples. The first example is a two-dimensional system that describes a single population with internal noise. This model has an analytical solution for the optimal path. The numerical solution found using our computational method agrees well with the analytical result. The second example is a more complicated four-dimensional system where our numerical method must be used to find the optimal path. The third example, although a seemingly imple two-dimensional system, demonstrates the success of our method in finding the optimal path where other numerical methods are known to fail. In the fourth example, the optimal path lies in six-dimensional space and demonstrates the power of our method in computing paths in higherdimensional spaces.

AB - In stochastic systems, one is often interested in finding the optimal path that maximizes the probability of escape from a metastable state or of switching between metastable states. Even for simple systems, it may be impossible to find an analytic form of the optimal path, and in highdimensional systems, this is almost always the case. In this article, we formulate a constructive methodology that is used to compute the optimal path numerically. The method utilizes finite-time Lyapunov exponents, statistical selection criteria, and a Newton-based iterative minimizing scheme. The method is applied to four examples. The first example is a two-dimensional system that describes a single population with internal noise. This model has an analytical solution for the optimal path. The numerical solution found using our computational method agrees well with the analytical result. The second example is a more complicated four-dimensional system where our numerical method must be used to find the optimal path. The third example, although a seemingly imple two-dimensional system, demonstrates the success of our method in finding the optimal path where other numerical methods are known to fail. In the fourth example, the optimal path lies in six-dimensional space and demonstrates the power of our method in computing paths in higherdimensional spaces.

UR - http://www.scopus.com/inward/record.url?scp=84982743604&partnerID=8YFLogxK

U2 - 10.1063/1.4958926

DO - 10.1063/1.4958926

M3 - Article

AN - SCOPUS:84982743604

VL - 26

JO - Chaos

JF - Chaos

SN - 1054-1500

IS - 8

M1 - 083101

ER -