arXiv subjects

Björn S. Rüffer

Publications and source records attributed to Björn S. Rüffer.

9 recordsLinked to original sources

arXiv2026

A mathematical model for depression and resilience

A basic dynamical model for (clinical) depression is presented to describe the time evolution of two coupled states: a resilience level and a depression symptom. The resilience level can also be interpreted in terms of the memory of past symptoms. The model consists of a system of two coupled first order differential equations without free parameters that qualitatively captures different courses of illness, without the overhead of a derivation from neuroscientific first principles. A comprehensive mathematical analysis of the model is provided, including equilibria, stability properties, and monotonicity. The model can reproduce chronic, delayed, recovery, and resilience scenarios that are prominent in the literature, as well as scenarios such as improvement from pre-existing conditions, burnout from low-grade adversity, or isolated and recurrent depressive episodes. Supplementary to the manuscript are computational tools to replicate the figures and, without prior mathematical or programming skills, to experiment with the model interactively in the web browser (at https://rsmodel.org/).

math.DS↗

arXiv2018

Convergence Properties for Discrete-time Nonlinear Systems

Three similar convergence notions are considered. Two of them are the long established notions of convergent dynamics and incremental stability. The other is the more recent notion of contraction analysis. All three convergence notions require that all solutions of a system converge to each other. In this paper, we investigate the differences between these convergence properties for discrete-time, time-varying nonlinear systems by comparing the properties in pairs and using examples. We also demonstrate a time-varying smooth Lyapunov function characterization for each of these convergence notions, and, with appropriate assumptions, we provide several sufficient conditions to establish relationships between these properties in terms of Lyapunov functions.

math.OC↗

arXiv2018

A Perron-Frobenius type result for integer maps and applications

It is shown that for certain maps, including concave maps, on the $d$-dimensional lattice of positive integer points, 'approximate' eigenvectors can be found. Applications in epidemiology as well as distributed resource allocation are discussed as examples.

math.DS↗

arXiv2018

A Lyapunov function construction for a non-convex Douglas-Rachford iteration

While global convergence of the Douglas-Rachford iteration is often observed in applications, proving it is still limited to convex and a handful of other special cases. Lyapunov functions for difference inclusions provide not only global or local convergence certificates, but also imply robust stability, which means that the convergence is still guaranteed in the presence of persistent disturbances. In this work, a global Lyapunov function is constructed by combining known local Lyapunov functions for simpler, local sub-problems via an explicit formula that depends on the problem parameters. Specifically, we consider the scenario where one set consists of the union of two lines and the other set is a line, so that the two sets intersect in two distinct points. Locally, near each intersection point, the problem reduces to the intersection of just two lines, but globally the geometry is non-convex and the Douglas-Rachford operator multi-valued. Our approach is intended to be prototypical for addressing the convergence analysis of the Douglas-Rachford iteration in more complex geometries that can be approximated by polygonal sets through the combination of local, simple Lyapunov functions.

math.OC↗

arXiv2017

Non-conservative discrete-time ISS small-gain conditions for closed sets

This paper presents a unification and a generalization of the small-gain theory subsuming a wide range of existing small-gain theorems. In particular, we introduce small-gain conditions that are necessary and sufficient to ensure input-to-state stability (ISS) with respect to closed sets. Toward this end, we first develop a Lyapunov characterization of $ω$ISS via finite-step $ω$ISS Lyapunov functions. Then, we provide the small-gain conditions to guarantee $ω$ISS of a network of systems. Finally, applications of our results to partial input-to-state stability, ISS of time-varying systems, synchronization problems, incremental stability, and distributed observers are given.

math.OC↗

arXiv2016

Nonlinear left and right eigenvectors for max-preserving maps

It is shown that max-preserving maps (or join-morphisms) on the positive orthant in Euclidean $n$-space endowed with the component-wise partial order give rise to a semiring. This semiring admits a closure operation for maps that generate stable dynamical systems. For these monotone maps, the closure is used to define suitable notions of left and right eigenvectors that are characterized by inequalities. Some explicit examples are given and applications in the construction of Lyapunov functions are described.

math.OC↗

arXiv2010

Stability verification for monotone systems using homotopy algorithms

A monotone self-mapping of the nonnegative orthant induces a monotone discrete-time dynamical system which evolves on the same orthant. If with respect to this system the origin is attractive then there must exists points whose image under the monotone map is strictly smaller than the original point, in the component-wise partial ordering. Here it is shown how such points can be found numerically, leading to a recipe to compute order intervals that are contained in the region of attraction and where the monotone map acts essentially as a contraction. An important application is the numerical verification of so-called generalized small-gain conditions that appear in the stability theory of large-scale systems.

math.NA↗

arXiv2009

Small gain theorems for large scale systems and construction of ISS Lyapunov functions

We consider interconnections of n nonlinear subsystems in the input-to-state stability (ISS) framework. For each subsystem an ISS Lyapunov function is given that treats the other subsystems as independent inputs. A gain matrix is used to encode the mutual dependencies of the systems in the network. Under a small gain assumption on the monotone operator induced by the gain matrix, a locally Lipschitz continuous ISS Lyapunov function is obtained constructively for the entire network by appropriately scaling the individual Lyapunov functions for the subsystems. The results are obtained in a general formulation of ISS, the cases of summation, maximization and separation with respect to external gains are obtained as corollaries.

math.OC↗

arXiv2005

An ISS Small-Gain Theorem for General Networks

We provide a generalized version of the nonlinear small-gain theorem for the case of more than two coupled input-to-state stable (ISS) systems. For this result the interconnection gains are described in a nonlinear gain matrix and the small-gain condition requires bounds on the image of this gain matrix. The condition may be interpreted as a nonlinear generalization of the requirement that the spectral radius of the gain matrix is less than one. We give some interpretations of the condition in special cases covering two subsystems, linear gains, linear systems and an associated artificial dynamical system.

math.OC↗