Survivability of Deterministic Dynamical Systems

Frank Hellmann; Paul Schultz; Carsten Grabow; Jobst Heitzig; Jürgen Kurths

doi:10.1038/srep29654

Survivability of Deterministic Dynamical Systems

Frank Hellmann^*, Paul Schultz, Carsten Grabow, Jobst Heitzig, Jürgen Kurths

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

56 Citations (Scopus)

8 Downloads (Pure)

Abstract

The notion of a part of phase space containing desired (or allowed) states of a dynamical system is important in a wide range of complex systems research. It has been called the safe operating space, the viability kernel or the sunny region. In this paper we define the notion of survivability: Given a random initial condition, what is the likelihood that the transient behaviour of a deterministic system does not leave a region of desirable states. We demonstrate the utility of this novel stability measure by considering models from climate science, neuronal networks and power grids. We also show that a semi-analytic lower bound for the survivability of linear systems allows a numerically very efficient survivability analysis in realistic models of power grids. Our numerical and semi-analytic work underlines that the type of stability measured by survivability is not captured by common asymptotic stability measures.

Original language	English
Article number	29654
Pages (from-to)	1-12
Number of pages	12
Journal	Scientific Reports
Volume	6
DOIs	https://doi.org/10.1038/srep29654
Publication status	Published - 13 Jul 2016

Bibliographical note

Acknowledgements
We acknowledge gratefully the support of BMBF, CoNDyNet, FK. 03SF0472A, of the EIT Climate-KIC project SWIPO and Nora Molkenthin for illustrating our illustration of the concept of survivability using penguins. We thank Martin Rohden for providing us with the UK high-voltage transmission grid topology and Yang Tang for very useful discussions. The publication of this article was funded by the Open Access Fund of the Leibniz Association.

Access to Document

10.1038/srep29654Licence: CC BY

Survivability of Deterministic Dynamical Systems
This work is licensed under a Creative Commons Attribution 4.0 International License. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in the credit line; if the material is not included under the Creative Commons license, users will need to obtain permission from the license holder to reproduce the material. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/
Final published version, 1.79 MBLicence: CC BY

https://arxiv.org/abs/1506.01257v2Licence: Unspecified

Cite this

@article{ca8ba7ad08d541f4827d6ed357832041,

title = "Survivability of Deterministic Dynamical Systems",

abstract = "The notion of a part of phase space containing desired (or allowed) states of a dynamical system is important in a wide range of complex systems research. It has been called the safe operating space, the viability kernel or the sunny region. In this paper we define the notion of survivability: Given a random initial condition, what is the likelihood that the transient behaviour of a deterministic system does not leave a region of desirable states. We demonstrate the utility of this novel stability measure by considering models from climate science, neuronal networks and power grids. We also show that a semi-analytic lower bound for the survivability of linear systems allows a numerically very efficient survivability analysis in realistic models of power grids. Our numerical and semi-analytic work underlines that the type of stability measured by survivability is not captured by common asymptotic stability measures.",

author = "Frank Hellmann and Paul Schultz and Carsten Grabow and Jobst Heitzig and J{\"u}rgen Kurths",

note = "Acknowledgements We acknowledge gratefully the support of BMBF, CoNDyNet, FK. 03SF0472A, of the EIT Climate-KIC project SWIPO and Nora Molkenthin for illustrating our illustration of the concept of survivability using penguins. We thank Martin Rohden for providing us with the UK high-voltage transmission grid topology and Yang Tang for very useful discussions. The publication of this article was funded by the Open Access Fund of the Leibniz Association.",

year = "2016",

month = jul,

day = "13",

doi = "10.1038/srep29654",

language = "English",

volume = "6",

pages = "1--12",

journal = "Scientific Reports",

issn = "2045-2322",

publisher = "Nature Publishing Group",

}

TY - JOUR

T1 - Survivability of Deterministic Dynamical Systems

AU - Hellmann, Frank

AU - Schultz, Paul

AU - Grabow, Carsten

AU - Heitzig, Jobst

AU - Kurths, Jürgen

N1 - Acknowledgements We acknowledge gratefully the support of BMBF, CoNDyNet, FK. 03SF0472A, of the EIT Climate-KIC project SWIPO and Nora Molkenthin for illustrating our illustration of the concept of survivability using penguins. We thank Martin Rohden for providing us with the UK high-voltage transmission grid topology and Yang Tang for very useful discussions. The publication of this article was funded by the Open Access Fund of the Leibniz Association.

PY - 2016/7/13

Y1 - 2016/7/13

N2 - The notion of a part of phase space containing desired (or allowed) states of a dynamical system is important in a wide range of complex systems research. It has been called the safe operating space, the viability kernel or the sunny region. In this paper we define the notion of survivability: Given a random initial condition, what is the likelihood that the transient behaviour of a deterministic system does not leave a region of desirable states. We demonstrate the utility of this novel stability measure by considering models from climate science, neuronal networks and power grids. We also show that a semi-analytic lower bound for the survivability of linear systems allows a numerically very efficient survivability analysis in realistic models of power grids. Our numerical and semi-analytic work underlines that the type of stability measured by survivability is not captured by common asymptotic stability measures.

AB - The notion of a part of phase space containing desired (or allowed) states of a dynamical system is important in a wide range of complex systems research. It has been called the safe operating space, the viability kernel or the sunny region. In this paper we define the notion of survivability: Given a random initial condition, what is the likelihood that the transient behaviour of a deterministic system does not leave a region of desirable states. We demonstrate the utility of this novel stability measure by considering models from climate science, neuronal networks and power grids. We also show that a semi-analytic lower bound for the survivability of linear systems allows a numerically very efficient survivability analysis in realistic models of power grids. Our numerical and semi-analytic work underlines that the type of stability measured by survivability is not captured by common asymptotic stability measures.

U2 - 10.1038/srep29654

DO - 10.1038/srep29654

M3 - Article

SN - 2045-2322

VL - 6

SP - 1

EP - 12

JO - Scientific Reports

JF - Scientific Reports

M1 - 29654

ER -

Survivability of Deterministic Dynamical Systems

Abstract

Bibliographical note

Access to Document

Fingerprint

Cite this