Abstract
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 high-dimensional 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.
Highlights
Understanding stability of dynamical systems is important for widespread areas of research such as control theory, economy, biology, physics, hydrodynamics and mechanical engineering
All suggested stability and resilience measures are easy to implement on a standard laptop computer, and together, they yield an explaining picture of the systems dynamics in a “large” neighborhood of the attractor
We compare and discuss properties of the suggested nonlocal stability and resilience measures on four different models: the Solow-Swan model of economic growth, a simple electro-mechanical model, a stage-structured population model and a highdimensional model related to evolution
Summary
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 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 complicated and classical mathematical analysis is often difficult.
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