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.
authors.affiliation: Potsdam Institute for Climate Impact Research, P.O. Box 60 12 03, 14412 Potsdam, Germany.
authors.forename: Frank
authors.initials: F
authors.lastname: Hellmann
authors.affiliation: Potsdam Institute for Climate Impact Research, P.O. Box 60 12 03, 14412 Potsdam, Germany.|Department of Physics, Humboldt University of Berlin, Newtonstr. 15, 12489 Berlin, Germany.
authors.forename: Paul
authors.initials: P
authors.lastname: Schultz
authors.affiliation: Potsdam Institute for Climate Impact Research, P.O. Box 60 12 03, 14412 Potsdam, Germany.
authors.forename: Carsten
authors.initials: C
authors.lastname: Grabow
authors.affiliation: Potsdam Institute for Climate Impact Research, P.O. Box 60 12 03, 14412 Potsdam, Germany.
authors.forename: Jobst
authors.initials: J
authors.lastname: Heitzig
authors.affiliation: Potsdam Institute for Climate Impact Research, P.O. Box 60 12 03, 14412 Potsdam, Germany.|Department of Physics, Humboldt University of Berlin, Newtonstr. 15, 12489 Berlin, Germany.|Institute for Complex Systems and Mathematical Biology, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom.|Department of Control Theory, Nizhny Novgorod State University, Gagarin Avenue 23, 606950 Nizhny Novgorod, Russia.
authors.forename: Jürgen
authors.initials: J
authors.lastname: Kurths