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.
Highlights
The notion of a part of phase space containing desired states of a dynamical system is important in a wide range of complex systems research
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
Summary
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 almost all dynamical systems applicable to the real world, the stability of the system’s stationary states (periodic orbits, chaotic attractors, etc.) is of key interest, because perturbations are never truly absent and initial data is never exactly determined. Real-world systems typically are multistable[4,5,6] They have more than one stable attractor[7], and potentially exhibit a wide range of different asymptotic behaviours. Most work so far focused on the geometry of the basin of attraction[8] of desirable attractors, e.g. by finding Lyapunov functions[9,10,11]
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