### Abstract

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

Article number | 043102 |

Journal | Chaos |

Volume | 28 |

Issue number | 4 |

Early online date | Apr 2018 |

DOIs | |

Publication status | Published - 2018 |

### Fingerprint

### Keywords

- nlin.CD
- Monte Carlo methods
- stochastic processes
- metric spaces
- solar energy
- neuronal network dynamics
- vector fields
- control theory

### Cite this

*Chaos*,

*28*(4), [043102 ]. https://doi.org/10.1063/1.5013127

**Bounding the first exit from the basin : Independence times and finite-time basin stability.** / Schultz, Paul; Hellmann, Frank; Webster, Kevin N.; Kurths, Jürgen.

Research output: Contribution to journal › Article

*Chaos*, vol. 28, no. 4, 043102 . https://doi.org/10.1063/1.5013127

}

TY - JOUR

T1 - Bounding the first exit from the basin

T2 - Independence times and finite-time basin stability

AU - Schultz, Paul

AU - Hellmann, Frank

AU - Webster, Kevin N.

AU - Kurths, Jürgen

N1 - 9 pages, 3 figures

PY - 2018

Y1 - 2018

N2 - We study the stability of deterministic systems given sequences of large, jump-like perturbations. Our main result is to dervie a lower bound for the probability of the system to remain in the basin, given that perturbations are rare enough. This bound is efficient to evaluate numerically. To quantify rare enough, we define the notion of the independence time of such a system. This is the time after which a perturbed state has probably returned close to the attractor, meaning that subsequent perturbations can be considered separately. The effect of jump-like perturbations that occur at least the independence time apart is thus well described by a fixed probability to exit the basin at each jump, allowing us to obtain the bound. To determine the independence time, we introduce the concept of finite-time basin stability, which corresponds to the probability that a perturbed trajectory returns to an attractor within a given time. The independence time can then be determined as the time scale at which the finite-time basin stability reaches its asymptotic value. Besides that, finite-time basin stability is a novel probabilistic stability measure on its own, with potential broad applications in complex systems.

AB - We study the stability of deterministic systems given sequences of large, jump-like perturbations. Our main result is to dervie a lower bound for the probability of the system to remain in the basin, given that perturbations are rare enough. This bound is efficient to evaluate numerically. To quantify rare enough, we define the notion of the independence time of such a system. This is the time after which a perturbed state has probably returned close to the attractor, meaning that subsequent perturbations can be considered separately. The effect of jump-like perturbations that occur at least the independence time apart is thus well described by a fixed probability to exit the basin at each jump, allowing us to obtain the bound. To determine the independence time, we introduce the concept of finite-time basin stability, which corresponds to the probability that a perturbed trajectory returns to an attractor within a given time. The independence time can then be determined as the time scale at which the finite-time basin stability reaches its asymptotic value. Besides that, finite-time basin stability is a novel probabilistic stability measure on its own, with potential broad applications in complex systems.

KW - nlin.CD

KW - Monte Carlo methods

KW - stochastic processes

KW - metric spaces

KW - solar energy

KW - neuronal network dynamics

KW - vector fields

KW - control theory

U2 - 10.1063/1.5013127

DO - 10.1063/1.5013127

M3 - Article

VL - 28

JO - Chaos

JF - Chaos

SN - 1054-1500

IS - 4

M1 - 043102

ER -