Abstract
The abundance of multistable dynamical systems calls for an appropriate quantification of the respective stability of the (stable) states of such systems. Motivated by the concept of ecological resilience, we propose a novel and pragmatic measure called ‘integral stability’ which integrates different aspects commonly addressed separately by existing local and global stability concepts. We demonstrate the potential of integral stability by using exemplary multistable dynamical systems such as the damped driven pendulum, a model of Amazonian rainforest as a known climate tipping element and the Daisyworld model. A crucial feature of integral stability lies in its potential of arresting a gradual loss of the stability of a system when approaching a tipping point, thus providing a potential early-warning signal sufficiently prior to a qualitative change of the system’s dynamics.
Highlights
Hand, ecological resilience presumes the existence of multiple stable states and the tolerance of the system to disturbances that facilitate transitions among the stable states
Given our interest in quantifying the stability of multistable dynamical systems, we primarily focus and build upon ecological resilience
We demonstrate the potential of the proposed measure of IS in quantifying the stability of the attractors of exemplary multistable dynamical systems
Summary
Hand, ecological resilience presumes the existence of multiple stable states and the tolerance of the system to disturbances that facilitate transitions among the stable states. Building upon our qualitative definition of resistance, and combining it with engineering resilience, we here define resistance at a particular point in the state space as the instantaneous rate at which the system converges to the unperturbed trajectory following a perturbation. We associate the resistance at a particular point in the state space with the local Lyapunov exponents evaluated at the respective point in state space[15]. The negative of the local Lyapunov exponents measure the rate of convergence of nearby trajectories to the trajectory starting at the state space point in question. We quantify the resistance at any point in state space as the negative of the largest local Lyapunov exponent, evaluated at the respective point. The detailed procedure for calculating the local Lyapunov exponents is described in the Methods section
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