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Stability landscapes are useful for understanding the properties of dynamical systems. These landscapes can be calculated from the system's dynamical equations using the physical concept of scalar potential. Unfortunately, for most…
Stochastic feedback systems give rise to a variety of notions of stability. The conditions for the stability of the median, mean, and variance stability conditions differ. These conditions can be stated explicitly for scalar discrete-time…
We combine geometric data analysis and stochastic modeling to describe the collective dynamics of complex systems. As an example we apply this approach to financial data and focus on the non-stationarity of the market correlation structure.…
Dynamical systems, that are used to model power grids, the brain, and other physical systems, can exhibit coexisting stable states known as attractors. A powerful tool to understand such systems, as well as to better predict when they may…
We develop a generalized stability framework for stochastic discrete-time systems, where the generality pertains to the ways in which the distribution of the state energy can be characterized. We use tools from finance and operations…
We search for steady states in a class of fluctuating and driven physical systems that exhibit sustained currents. We find that the physical concept of a steady state, well known for systems at equilibrium, must be generalised to describe…
We introduce the notion of a quasistatic dynamical system, which generalizes that of an ordinary dynamical system. Quasistatic dynamical systems are inspired by the namesake processes in thermodynamics, which are idealized processes where…
Multistability is a phenomenon prevalent in many natural systems. In climate, for example, it allows the possibility of irreversible consequences on planetary scale as a result of climate change. Indeed, a climate ``tipping element'' is a…
The stability of solutions to evolution equations with respect to small stochastic perturbations is considered. The stability of a stochastic dynamical system is characterized by the local stability index. The limit of this index with…
Employing model predictive control to systems with unbounded, stochastic disturbances poses the challenge of guaranteeing safety, i.e., repeated feasibility and stability of the closed-loop system. Especially, there are no strict repeated…
There has been a long-standing and at times fractious debate whether complex and large systems can be stable. In ecology, the so-called `diversity-stability debate' arose because mathematical analyses of ecosystem stability were either…
We study long-range interacting systems driven by external stochastic forces that act collectively on all the particles constituting the system. Such a scenario is frequently encountered in the context of plasmas, self-gravitating systems,…
Locomotion in the real world involves unexpected perturbations, and therefore requires strategies to maintain stability to successfully execute desired behaviours. Ensuring the safety of locomoting systems therefore necessitates a…
The stability analysis of socioeconomic systems has been centered on answering whether small perturbations when a system is in a given quantitative state will push the system permanently to a different quantitative state. However, typically…
Multiple stable states - the coexistence of two or more distinct ecological configurations under identical environmental conditions - have attracted sustained interest in ecology, yet the field still lacks a unified framework connecting…
We propose a new procedure to monitor and forecast the onset of transitions in high dimensional complex systems. We describe our procedure by an application to the Tangled Nature model of evolutionary ecology. The quasi-stable…
An attractor of a dynamical system may represent the system's 'desirable' state. Perturbations to the system may push the system out of the basin of attraction of the desirable attractor and into undesirable states. Hence, it is important…
Entropy notions for $\varepsilon$-incremental practical stability and incremental stability of deterministic nonlinear systems under disturbances are introduced. The entropy notions are constructed via a set of points in state space which…
Multistability, the coexistence of multiple stable states, is a cornerstone of nonlinear dynamical systems, governing their equilibrium, tunability, and emergent complexity. Recently, the concept of hidden multistability, where certain…
We consider the problem of state selection for a stochastic system, initially in an unstable stationary state, when multiple metastable states compete for occupation. Using path-integral techniques we derive remarkably simple and accurate…