Related papers: Climbing Escher's stairs: a way to approximate sta…
The article under discussion, titled "Climbing Escher's stairs: A way to approximate stability landscapes in multidimensional systems" (doi: 10.1371/journal.pcbi.1007788), has captured our attention due to important methodological…
When a system has more than one stable state, how can the stability of these states be compared? This deceptively simple question has important consequences for ecosystems, because systems with alternative stable states can undergo dramatic…
The construction of effective and informative landscapes for stochastic dynamical systems has proven a long-standing and complex problem. In many situations, the dynamics may be described by a Langevin equation while constructing a…
Quantifying stochastic processes is essential to understand many natural phenomena, particularly in biology, including cell-fate decision in developmental processes as well as genesis and progression of cancers. While various attempts have…
Developmental dynamics of multicellular organism is a process that takes place in a multi-stable system in which each attractor state represents a cell type and attractor transitions correspond to cell differentiation paths. This new…
The quasipotential is a natural generalization of the concept of energy functions to non-equilibrium systems. In the analysis of rare events in stochastic dynamics, it plays a central role in characterizing the statistics of transition…
The reduction of dynamical systems has a rich history, with many important applications related to stability, control and verification. Reduction of nonlinear systems is typically performed in an exact manner - as is the case with…
A new measure to characterize stability of complex dynamical systems against large perturbation is suggested, the stability threshold (ST). It quantifies the magnitude of the weakest perturbation capable to disrupt the system and switch it…
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…
We propose a combination of cluster analysis and stochastic process analysis to characterize high-dimensional complex dynamical systems by few dominating variables. As an example, stock market data are analyzed for which the dynamical…
Many systems in biology, physics and engineering can be described by systems of ordinary differential equation containing many parameters. When studying the dynamic behavior of these large, nonlinear systems, it is useful to identify and…
Stability is among the most important concepts in dynamical systems. Local stability is well-studied, whereas determining how "globally stable" a nonlinear system is very challenging. Over the last few decades, many different ideas have…
A wide body of work has applied the concept of critical slowing down to estimate the stability of different Earth system components. Most of them -- such as global vegetation -- are inherently non-stationary, for example due to strong…
In this paper, we consider a system of partial differential equations modeling the evolution of a landscape. A ground surface is eroded by the flow of water over it, either by sedimentation or dilution. The system is composed by three…
A novel notion for constructing a well-balanced scheme - a gradient-robust scheme - is introduced and a showcase application for a steady compressible, isothermal Stokes equations is presented. Gradient-robustness means that arbitrary…
We explore how to build a vector field from the various functions involved in a given mathematical program, and show that locally-stable equilibria of the underlying dynamical system are precisely the local solutions of the optimization…
Estimation of the degree of stability and the bounds of solutions to non-autonomous nonlinear systems present major concerns in numerous applied problems. Yet, current techniques are frequently yield overconservative conditions which are…
Changes in environmental or system parameters often drive major biological transitions, including ecosystem collapse, disease outbreaks, and tumor development. Analyzing the stability of steady states in dynamical systems provides critical…
The complex dynamics of an increasing number of systems is attributed to the emergence of a rugged energy landscape with an exponential number of metastable states. To develop this picture into a predictive dynamical theory I discuss how to…
The quasistatic approximation is a useful but questionable simplification for analyzing step instabilities during the growth/evaporation of vicinal surfaces. Using this approximation, we characterized in Part I of this work the effect on…