How to find simple nonlocal stability and resilience measures
Nonlinear Dyn
How to find simple nonlocal stability and resilience measures
Niklas L. P. Lundstr?m 0
0 N. L. P. Lundstro?m (
Stability of dynamical systems is a central topic with applications in widespread areas such as economy, biology, physics and mechanical engineering. The dynamics of nonlinear systems may completely change due to perturbations forcing the solution to jump from a safe state into another, possibly dangerous, attractor. Such phenomena cannot be traced by the widespread local stability and resilience measures, based on linearizations, accounting only for arbitrary small perturbations. Using numerical estimates of the size and shape of the basin of attraction, as well as the systems returntime to the attractor after given a perturbation, we construct simple nonlocal stability and resilience measures that record a systems ability to tackle both large and small perturbations. We demonstrate our approach on the Solow-Swan model of economic growth, an electro-mechanical system, a stage-structured population model as well as on a highdimensional system, and conclude that the suggested measures detect dynamic behavior, crucial for a systems stability and resilience, which can be completely missed by local measures. The presented measures are also easy to implement on a standard laptop computer. We believe that our approach will constitute an important step toward filling a current gap in the literature by putting forward and explaining simple ideas and methods, and by delivering explicit constructions of several promising nonlocal stability and resilience measures.
Measure of stability; Measure of resilience; Sensitivity; Initial conditions; Random testing; Bistability; Returntime; Recovery rate; Global stability; Basin stability
1 Introduction
Understanding stability of dynamical systems is
important for widespread areas of research such as
control theory, economy, biology, physics,
hydrodynamics and mechanical engineering. As computers and
computational tools have become more and more
efficient, simulations and numerical investigations of
realistic, high-dimensional, advanced mathematical
models have become easier and also increasingly popular.
As such advanced models account for numerous factors
and their interplay, the dynamics are often nontrivial,
nonlinear and coexisting attractors (bistability) often
exist or may be hard to rule out. Such systems
algebraic structure are usually also complicated and
classical mathematical analysis is therefore often difficult.
This calls for further research in mathematical analysis,
as well as in numerical methods, for finding efficient
ways of quantifying the stability of dynamical systems.
The analysis of the stability of attractors (e.g.,
equilibrium points, limit cycles, quasiperiodic or strange
attractors) in dynamical systems naturally split into
local analysis and nonlocal analysis. The local stability
approach is usually based on linearizations and yields
information in a small neighborhood of the attractor,
saying little or nothing about the systems behavior a
bit away from the attractor. Measuring stability in
nonlinear dynamical systems using local methods (such
as eigenvalues of the Jacobian matrix at an
equilibrium) delivers information of how the system reacts
only on arbitrary small perturbations. A perturbation of
a given size may push a locally stable state into another
attractor having a completely different, possibly
dangerous, behavior. For example, researchers believe that
the crash of the aircraft YF-22 Boeing in April 1992 was
caused by a sudden switch to an unsafe attractor [
1,2
].
Figure 1 shows three systems having completely
different abilities to withstand perturbations, but exactly
the same local stability and local resilience.
The local stability analysis should therefore be
complemented by, or replaced by, a nonlocal approach that
considers properties of the basin of attraction
(henceforth basin) to the attractor under investigation. The
size of basin (its volume) constitutes a natural
candidate for a nonlocal stability measure as a large basin
indicates that the system comes back to the attractor
with a high probability, given a random perturbation.
It is also easy to numerically estimate the size of the
basin. However, if the distance from the attractor to the
boundary of the basin is short in some direction, then a
small perturbation in this direction can push the system
to another, possibly dangerous, attractor, even though
the basin is large, see Fig. 1c. Therefore, the shape of
basin is another natural candidate to be included in a
(a)
(b)
(c)
Fig. 1 Three identical balls rest at equilibrium on three
different stands. All stands have the same shape in a neighborhood
of the equilibrium and hence a local approach would rank the
systems in a?c as equally stable and equally resilient. The basin
of attraction is largest in (a), followed by (c) while it is smallest
in (b). Therefore, measuring only the size of th (...truncated)