arXiv subjects

Jan Sieber

Publications and source records attributed to Jan Sieber.

At least 19 recordsLinked to original sources

arXiv2026

Local interaction of two systems with saddle-node bifurcations: mutualistic and mixed cases

The saddle-node bifurcation is the simplest example of a generic bifurcation in smooth ordinary differential equations, and is associated with the creation or destruction of a pair of equilibria. In this paper we examine the unfolding of the dynamics that occur when two generically coupled systems have simultaneous saddle-node bifurcations. We note that four parameters are required to generically unfold the interactions, and the dynamics are surprisingly complicated relative to the simplicity of a single saddle-node bifurcation. In the unfolding, in addition to saddle-node, Hopf and codimension-two local bifurcations, we also find a variety of global bifurcations, including homoclinic, SNIC, SNICeroclinic and non-central SNIC bifurcations. The latter two are codimension-two bifurcations that occur at the termination of a curve of SNIC bifurcations. A further contribution of this work is the development of numerical continuation techniques for the tracking of these codimension-two bifurcations through parameter space.

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arXiv2025

Dynamics of singularly perturbed sliding flow in Filippov systems

In this article, we present an analysis of the effects of singular perturbations on the sliding motion in Filippov systems. We show that singular perturbations may lead to qualitatively distinct topologies of phase space on the switching manifold, which we classify into six distinct topologies. Five of these topologies imply that singularly perturbed trajectory includes a segment (or segments) of sliding, and one topology, which we study here, implies the evolution characterised by switchings between trajectory segments along the switching surface, but without any sliding. In particular, we show that in the case of $n$-dimensional Filippov systems with one switching surface and $m$-dimensional fast dynamics, which plays the role of a stable singular perturbation, the flow follows sliding motion of the reduced system, but the perturbation becomes time dependent and may create a micro chaotic behaviour. However, a significant change in the flow direction is not possible. In the particular case of $1$-dimensional fast dynamics, the singular perturbation implies regular perturbation of $O(\varepsilon)$ of the perturbed flow at the points of switching between vector fields.

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arXiv2025

Spectral element methods for boundary-value problems of functional differential equations

We prove convergence of the spectral element method for piecewise polynomial collocation applied to periodic boundary value problems for functional differential equations. In particular, we prove that the numerical collocation solution approximates the true solution with accuracy of order $\mathrm{e}^{-\eta m}$ for some $\eta>0$ and increasing degree $m$ of the polynomials for a case that is common in applications: differential equations where the right-hand side depends on a finite number of delayed arguments with parametric delays and real analytic coefficients. For state-dependent delays the spectral element method also converges under mild regularity assumptions, but the geometric convergence of the collocation solution depends on the properties of the true solution, which may in general not be real analytic even for analytic coefficients. However, in those cases the convergence rate is still higher than all finite orders.

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arXiv2025

Predicting Perceptual Boundaries in Auditory Streaming using Delay Differential Equations

Auditory streaming enables the brain to organize sequences of sounds into perceptually distinct sources, such as following a conversation in a noisy environment. A typical experiment for investigating perceptual boundaries and bistability is to present a subject with a stream containing two alternating tone stimuli. We investigate a model for the processing of such a stream consisting of two identical neural populations of excitatory and inhibitory neurons. The populations are coupled via delayed cross-inhibition and periodically forced with sharp step-type signals (the two-tone stream). We track how the perception boundary depends on threshold selection and establish how boundaries between three different auditory perceptions (single tone versus two tones versus bistability between both perceptions) relate to bifurcations such as symmetry breaking. We demonstrate that these transitions are governed by symmetry-breaking bifurcations and that the perceptual classification based on neural thresholds is highly sensitive to threshold choice. Our analysis reveals that a fixed threshold is insufficient to capture the true perceptual boundaries and proposes a variable-threshold criterion, informed by the amplitude dynamics of neural responses. Finally, we illustrate how key stimulus parameters such as tone duration, delay, and internal time scale shape the boundaries of auditory perceptual organization in the plane of the two most commonly varied experimental parameters, the representation rate, and the difference in tone frequency. These findings offer mechanistic insight into auditory perception dynamics and provide a refined framework for linking neural activity to perceptual organization.

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arXiv2024

Alpha-Delta Transitions in Cortical Rhythms as grazing bifurcations

The Jansen-Rit model of a cortical column in the cerebral cortex is widely used to simulate spontaneous brain activity (EEG) and event-related potentials. It couples a pyramidal cell population with two interneuron populations, of which one is fast and excitatory and the other slow and inhibitory. Our paper studies the transition between alpha and delta oscillations produced by the model. Delta oscillations are slower than alpha oscillations and have a more complex relaxation-type time profile. In the context of neuronal population activation dynamics, a small threshold means that neurons begin to activate with small input or stimulus, indicating high sensitivity to incoming signals. A steep slope signifies that activation increases sharply as input crosses the threshold. Accordingly in the model the excitatory activation thresholds are small and the slopes are steep. Hence, a singular limit replacing the excitatory activation function with all-or-nothing switches, eg. a Heaviside function, is appropriate. In this limit we identify the transition between alpha and delta oscillations as a discontinuity-induced grazing bifurcation. At the grazing the minimum of the pyramidal-cell output equals the threshold for switching off the excitatory interneuron population, leading to a collapse in excitatory feedback.

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arXiv2024

Computational Bifurcation Analysis

Bifurcation analysis collects techniques for characterizing the dependence of certain classes of solutions of a dynamical system on variations in problem parameters. Common solution classes of interest include equilibria and periodic orbits, the number and stability of which may vary as parameters vary. Continuation techniques generate continuous families of such solutions in the combined state and parameter space, e.g., curves (branches) of periodic orbits or surfaces of equilibria. Their advantage over simulation-based approaches is the ability to map out such families independently of the dynamic stability of the equilibria or periodic orbits. Bifurcation diagrams represent families of equilibria and periodic orbits as curves or surfaces in appropriate coordinate systems. Special points, such as bifurcations, are often highlighted in such diagrams. This article provides an illustration of this paradigm of synergy between theoretical derivations and computational analysis for several characteristic examples of bifurcation analysis in commonly encountered classes of problems. General theoretical principles are deduced from these illustrations and collected for the reader's subsequent reference.

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arXiv2024

Boundary-value problems of functional differential equations with state-dependent delays

We prove convergence of piecewise polynomial collocation methods applied to periodic boundary value problems for functional differential equations with state-dependent delays. The state dependence of the delays leads to nonlinearities that are not locally Lipschitz continuous preventing the direct application of general abstract discretization theoretic frameworks. We employ a weaker form of differentiability, which we call mild differentiability, to prove that a locally unique solution of the functional differential equation is approximated by the solution of the discretized problem with the expected order.

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arXiv2023

Finding periodic orbits in state-dependent delay differential equations as roots of algebraic equations

In this paper we prove that periodic boundary-value problems (BVPs) for delay differential equations are locally equivalent to finite-dimensional algebraic systems of equations. We rely only on regularity assumptions that follow those of the review by Hartung et al. (2006). Thus, the equivalence result can be applied to differential equations with state-dependent delays (SD-DDEs), transferring many results of bifurcation theory for periodic orbits to this class of systems. We demonstrate this by using the equivalence to give an elementary proof of the Hopf bifurcation theorem for differential equations with state-dependent delays. This is an alternative and extension to the original Hopf bifurcation theorem for SD-DDEs by Eichmann (2006).

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arXiv2023

Emergent structure and dynamics of tropical forest-grassland landscapes

Previous work indicates that tropical forest can exist as an alternative stable state to savanna. Therefore, perturbation by climate change or human impact may lead to crossing of a tipping point beyond which there is rapid forest dieback that is not easily reversed. A hypothesised mechanism for such bistability is a feedback between fire and vegetation, where fire spreads as a contagion process on grass patches. Theoretical models have largely implemented this mechanism implicitly, by assuming a threshold dependence of fire spread on flammable vegetation. Here, we show how the nonlinear dynamics and bistability emerge spontaneously, without assuming equations or thresholds for fire spread. We find that the forest geometry causes the nonlinearity that induces bistability. We demonstrate this in three steps. First, we model forest and fire as interacting contagion processes on grass patches, %using a cellular automaton showing that spatial structure emerges due to two counteracting effects on the forest perimeter: forest expansion by dispersal, and forest erosion by fires originating in adjacent grassland. Then, we derive a landscape-scale balance equation in which these two effects link forest geometry and dynamics: forest expands proportionally to its perimeter, while it shrinks proportionally to its perimeter weighted by adjacent grassland area. Finally, we show that these perimeter quantities introduce nonlinearity in our balance equation and lead to bistability. Relying on the link between structure and dynamics, we propose a forest resilience indicator that could be used for targeted conservation or restoration.

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arXiv2023

Observing hidden neuronal states in experiments

In this article we demonstrate a general protocol for constructing systematically experimental steady-state bifurcation diagrams for electrophysiologically active cells. We perform our experiments on entorhinal cortex neurons, both excitatory (pyramidal neurons) and inhibitiory (interneurons). A slowly ramped voltage-clamp electrophysiology protocol serves as closed-loop feedback controlled experiment for the subsequent current-clamp open-loop protocol on the same cell. In this way, the voltage-clamped experiment determines dynamically stable and unstable (hidden) steady states of the current-clamp experiment. The transitions between observable steady states and observable spiking states in the current-clamp experiment provide partial evidence for stability and bifurcations of the steady states. This technique for completing steady-state bifurcation diagrams in a model-independent way expands support for model validation to otherwise inaccessible regions of the phase space. Overlaying the voltage-clamp and current-clamp protocols leads to an experimental validation of the classical slow-fast dissection method introduced by J. Rinzel in the 1980s and routinely applied ever since in order to analyse slow-fast neuronal models. Our approach opens doors to observing further complex hidden states with more advanced control strategies, allowing to control real cells beyond pharmacological manipulations.

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arXiv2023

Numerical analysis of a multistable capsule system under the delayed feedback control with a constant delay

The vibro-impact capsule system is a self-propelled mechanism that has abundant coexisting attractors and moves rectilinearly under periodic excitation when overcoming environmental resistance. In this paper, we study the control of coexisting attractors in this system by using a delayed feedback controller (DFC) with a constant delay. The aim of our control is to steer this complex system toward an attractor with preferable performance characteristics among multiple coexisting attractors, e.g., a periodically fast forward progression. For this purpose, we give an example of a feedback-controlled transition from a period-3 motion with low progression speed to a period-1 motion with high progression speed at the system parameters where both responses coexist. The effectiveness of this controller is investigated numerically by considering its convergence time and the required control energy input to achieve transition. We combine pseudo-spectral approximation of the delay, event detection for the discontinuities and path-following (continuation) techniques for non-smooth delay dynamical systems to carry out bifurcation analysis. We systematically study the dynamical performance of the controlled system when varying its control gain and delay time. Our numerical simulations show the effectiveness of DFC under a wide range of system parameters. We find that the desired period-1 motion is achievable in a range of control delays between a period-doubling and a grazing bifurcation. Therefore, two-parameter continuation of these two bifurcations with respect to the control delay and control gain is conducted to identify the delay-gain parameter region where the period-1 motion is stable. The findings of this work can be used for tuning control parameters in experiments, and similar analysis can be carried out for other non-smooth dynamical systems with a constant delay term.

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arXiv2022

Mean-field models of dynamics on networks via moment closure: an automated procedure

In the study of dynamics on networks, moment closure is a commonly used method to obtain low-dimensional evolution equations amenable to analysis. The variables in the evolution equations are mean counts of subgraph states and are referred to as moments. Due to interaction between neighbours, each moment equation is a function of higher-order moments, such that an infinite hierarchy of equations arises. Hence, the derivation requires truncation at a given order, and, an approximation of the highest-order moments in terms of lower-order ones,known as a closure formula. Recent systematic approximations have either restricted focus to closed moment equations for SIR epidemic spreading or to unclosed moment equations for arbitrary dynamics. In this paper, we develop a general procedure that automates both derivation and closure of arbitrary order moment equations for dynamics with nearest-neighbour interactions on undirected networks. Automation of the closure step was made possible by our generalised closure scheme,which systematically decomposes the largest subgraphs into their smaller components.We show that this decomposition is exact if these components form a tree, there is independence at distances beyond their graph diameter, and there is spatial homogeneity. Testing our method for SIS epidemic spreading on lattices and random networks confirms that biases are larger for networks with many short cycles in regimes with long-range dependence. A Mathematica package that automates the moment closure is available for download.

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arXiv2022

Time series analysis and modelling of the freezing of gait phenomenon

Freezing of Gait (FOG) is one of the most debilitating symptoms of Parkinson's Disease and is associated with falls and loss of independence. The patho-physiological mechanisms underpinning FOG are currently poorly understood. In this paper we combine time series analysis and mathematical modelling to study the FOG phenomenon's dynamics. We focus on the transition from stepping in place into freezing and treat this phenomenon in the context of an escape from an oscillatory attractor into an equilibrium attractor state. We extract a discrete-time discrete-space Markov chain from experimental data and divide its state space into communicating classes to identify the transition into freezing. This allows us to develop a methodology for computationally estimating the time to freezing as well as the phase along the oscillatory (stepping) cycle of a patient experiencing Freezing Episodes (FE). The developed methodology is general and could be applied to any time series featuring transitions between different dynamic regimes including time series data from forward walking in people with FOG.

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arXiv2022

Continuation with Non-invasive Control Schemes: Revealing Unstable States in a Pedestrian Evacuation Scenario

This paper presents a framework to perform bifurcation analysis in laboratory experiments or simulations. We employ control-based continuation to study the dynamics of a macroscopic variable of a microscopically defined model, exploring the potential viability of the underlying feedback control techniques in an experiment. In contrast to previous experimental studies that used iterative root-finding methods on the feedback control targets, we propose a feedback control law that is inherently non-invasive. That is, the control discovers the location of equilibria and stabilizes them simultaneously. We call the proposed control zero-in-equilibrium feedback control and we prove that it is able to stabilize branches of equilibria, except at singularities of codimension n+1, where n is the number of state space dimensions the feedback can depend on. We apply the method to a simulated evacuation scenario were pedestrians have to reach an exit after maneuvering left or right around an obstacle. The scenario shows a hysteresis phenomenon with bistability and tipping between two possible steady pedestrian flows in microscopic simulations. We demonstrate for the evacuation scenario that the proposed control law is able to uniformly discover and stabilize steady flows along the entire branch, including points where other non-invasive approaches to feedback control become singular.

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arXiv2022

Sensitivity Analysis for Periodic Orbits and Quasiperiodic Invariant Tori Using the Adjoint Method

This paper presents a rigorous framework for the continuation of solutions to nonlinear constraints and the simultaneous analysis of the sensitivities of test functions to constraint violations at each solution point using an adjoint-based approach. By the linearity of a problem Lagrangian in the associated Lagrange multipliers, the formalism is shown to be directly amenable to analysis using the COCO software package, specifically its paradigm for staged problem construction. The general theory is illustrated in the context of algebraic equations and boundary-value problems, with emphasis on periodic orbits in smooth and hybrid dynamical systems, and quasiperiodic invariant tori of flows. In the latter case, normal hyperbolicity is used to prove the existence of continuous solutions to the adjoint conditions associated with the sensitivities of the orbital periods to parameter perturbations and constraint violations, even though the linearization of the governing boundary-value problem lacks a bounded inverse, as required by the general theory. An assumption of transversal stability then implies that these solutions predict the asymptotic phases of trajectories based at initial conditions perturbed away from the torus. Example COCO code is used to illustrate the minimal additional investment in setup costs required to append sensitivity analysis to regular parameter continuation.

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arXiv2022

Methods of Continuation and their Implementation in the COCO Software Platform with Application to Delay Differential Equations

This paper treats comprehensively the construction of problems from nonlinear dynamics and constrained optimization amenable to parameter continuation techniques and with particular emphasis on multi-segment boundary-value problems with delay. The discussion is grounded in the context of the COCO software package and its explicit support for community-driven development. To this end, the paper first formalizes the COCO construction paradigm for augmented continuation problems compatible with simultaneous analysis of implicitly defined manifolds of solutions to nonlinear equations and the corresponding adjoint variables associated with optimization of scalar objective functions along such manifolds. The paper uses applications to data assimilation from finite time histories and phase response analysis of periodic orbits to identify a universal paradigm of construction that permits abstraction and generalization. It then details the theoretical framework for a COCO-compatible toolbox able to support the analysis of a large family of delay-coupled multi-segment boundary-value problems, including periodic orbits, quasiperiodic orbits, connecting orbits, initial-value problems, and optimal control problems, as illustrated in a suite of numerical examples. The paper aims to present a pedagogical treatment that is accessible to the novice and inspiring to the expert by appealing to the many senses of the applied nonlinear dynamicist. Sprinkled among a systematic discussion of problem construction, graph representations of delay-coupled problems, and vectorized formulas for problem discretization, the paper includes an original derivation using Lagrangian sensitivity analysis of phase-response functionals for periodic-orbit problems in abstract Banach spaces, as well as a demonstration of the regularizing benefits of multi-dimensional manifold continuation for near-singular problems.

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arXiv2022

Nonlinear effects of instantaneous and delayed state dependence in a delayed feedback loop

We study a scalar, first-order delay differential equation (DDE) with instantaneous and state-dependent delayed feedback, which itself may be delayed. The state dependence introduces nonlinearity into an otherwise linear system. We investigate the ensuing nonlinear dynamics with the case of instantaneous state dependence as our starting point. We present the bifurcation diagram in the parameter plane of the two feedback strengths showing how periodic orbits bifurcate from a curve of Hopf bifurcations and disappear along a curve where both period and amplitude grow beyond bound as the orbits become saw-tooth shaped. We then `switch on' the delay within the state-dependent feedback term, reflected by a parameter $b>0$. Our main conclusion is that the new parameter $b$ has an immediate effect: as soon as $b>0$ the bifurcation diagram for $b=0$ changes qualitatively and, specifically, the nature of the limiting saw-tooth shaped periodic orbits changes. Moreover, we show $-$ numerically and through center manifold analysis $-$ that a degeneracy at $b=1/3$ of an equilibrium with a double real eigenvalue zero leads to a further qualitative change and acts as an organizing center for the bifurcation diagram. Our results demonstrate that state dependence in delayed feedback terms may give rise to new dynamics and, moreover, that the observed dynamics may change significantly when the state-dependent feedback depends on past states of the system. This is expected to have implications for models arising in different application contexts, such as models of human balancing and conceptual climate models of delayed action oscillator type.

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arXiv2021

Controlling coexisting attractors of a class of non-autonomous dynamical systems

This paper studies a control method for switching stable coexisting attractors of a class of non-autonomous dynamical systems. The central idea is to introduce a continuous path for the system's trajectory to transition from its original undesired stable attractor to a desired one by varying one of the system parameters according to the information of the desired attractor. The behaviour of the control is studied numerically for both non-smooth and smooth dynamical systems, using a sof-impact and a Duffing oscillator as examples. Special attention is given to identify the regions where the proposed control strategy is applicable by using the path-following methods implemented via the continuation platform COCO. It is shown that the proposed control concept can be implemented through either using an external control input or varying a system parameter. Finally, extensive numerical results are presented to validate the proposed control methods.

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