### Abstract

An important component of the mathematical definition of chaos is sensitivity to initial conditions. Sensitivity to initial conditions is usually measured in a deterministic model by the dominant Lyapunov exponent (LE), with chaos indicated by a positive LE. The sensitivity measure has been extended to stochastic models; however, it is possible for the stochastic Lyapunov exponent (SLE) to be positive when the LE of the underlying deterministic model is negative, and vice versa. This occurs because the LE is a long-term average over the deterministic attractor while the SLE is the long-term average over the stationary probability distribution. The property of sensitivity to initial conditions, uniquely associated with chaotic dynamics in deterministic systems, is widespread in stochastic systems because of time spent near repelling invariant sets (such as unstable equilibria and unstable cycles). Such sensitivity is due to a mechanism fundamentally different from deterministic chaos. Positive SLE's should therefore not be viewed as a hallmark of chaos. We develop examples of ecological population models in which contradictory LE and SLE values lead to confusion about whether or not the population fluctuations are primarily the result of chaotic dynamics. We suggest that "chaos" should retain its deterministic definition in light of the origins and spirit of the topic in ecology. While a stochastic system cannot then strictly be chaotic, chaotic dynamics can be revealed in stochastic systems through the strong influence of underlying deterministic chaotic invariant sets.

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

Pages (from-to) | 329-339 |

Number of pages | 11 |

Journal | Oikos |

Volume | 102 |

Issue number | 2 |

DOIs | |

State | Published - Aug 1 2003 |

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### ASJC Scopus subject areas

- Ecology

### Cite this

*Oikos*,

*102*(2), 329-339. https://doi.org/10.1034/j.1600-0706.2003.12387.x

**Can noise induce chaos?** / Dennis, Brian; Desharnais, Robert A.; Cushing, Jim M; Henson, Shandelle M.; Costantino, Robert F.

Research output: Contribution to journal › Article

*Oikos*, vol. 102, no. 2, pp. 329-339. https://doi.org/10.1034/j.1600-0706.2003.12387.x

}

TY - JOUR

T1 - Can noise induce chaos?

AU - Dennis, Brian

AU - Desharnais, Robert A.

AU - Cushing, Jim M

AU - Henson, Shandelle M.

AU - Costantino, Robert F

PY - 2003/8/1

Y1 - 2003/8/1

N2 - An important component of the mathematical definition of chaos is sensitivity to initial conditions. Sensitivity to initial conditions is usually measured in a deterministic model by the dominant Lyapunov exponent (LE), with chaos indicated by a positive LE. The sensitivity measure has been extended to stochastic models; however, it is possible for the stochastic Lyapunov exponent (SLE) to be positive when the LE of the underlying deterministic model is negative, and vice versa. This occurs because the LE is a long-term average over the deterministic attractor while the SLE is the long-term average over the stationary probability distribution. The property of sensitivity to initial conditions, uniquely associated with chaotic dynamics in deterministic systems, is widespread in stochastic systems because of time spent near repelling invariant sets (such as unstable equilibria and unstable cycles). Such sensitivity is due to a mechanism fundamentally different from deterministic chaos. Positive SLE's should therefore not be viewed as a hallmark of chaos. We develop examples of ecological population models in which contradictory LE and SLE values lead to confusion about whether or not the population fluctuations are primarily the result of chaotic dynamics. We suggest that "chaos" should retain its deterministic definition in light of the origins and spirit of the topic in ecology. While a stochastic system cannot then strictly be chaotic, chaotic dynamics can be revealed in stochastic systems through the strong influence of underlying deterministic chaotic invariant sets.

AB - An important component of the mathematical definition of chaos is sensitivity to initial conditions. Sensitivity to initial conditions is usually measured in a deterministic model by the dominant Lyapunov exponent (LE), with chaos indicated by a positive LE. The sensitivity measure has been extended to stochastic models; however, it is possible for the stochastic Lyapunov exponent (SLE) to be positive when the LE of the underlying deterministic model is negative, and vice versa. This occurs because the LE is a long-term average over the deterministic attractor while the SLE is the long-term average over the stationary probability distribution. The property of sensitivity to initial conditions, uniquely associated with chaotic dynamics in deterministic systems, is widespread in stochastic systems because of time spent near repelling invariant sets (such as unstable equilibria and unstable cycles). Such sensitivity is due to a mechanism fundamentally different from deterministic chaos. Positive SLE's should therefore not be viewed as a hallmark of chaos. We develop examples of ecological population models in which contradictory LE and SLE values lead to confusion about whether or not the population fluctuations are primarily the result of chaotic dynamics. We suggest that "chaos" should retain its deterministic definition in light of the origins and spirit of the topic in ecology. While a stochastic system cannot then strictly be chaotic, chaotic dynamics can be revealed in stochastic systems through the strong influence of underlying deterministic chaotic invariant sets.

UR - http://www.scopus.com/inward/record.url?scp=0042259193&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0042259193&partnerID=8YFLogxK

U2 - 10.1034/j.1600-0706.2003.12387.x

DO - 10.1034/j.1600-0706.2003.12387.x

M3 - Article

AN - SCOPUS:0042259193

VL - 102

SP - 329

EP - 339

JO - Oikos

JF - Oikos

SN - 0030-1299

IS - 2

ER -