Formal normal form of Ak slow fast systems

An Ak slow fast system is a particular type of singularly perturbed ODE. The corresponding slow manifold is defined by the critical points of a universal unfolding of an Ak singularity. In this note we propose a formal normal form of Ak slow fast systems

On fast-slow consensus networks with a dynamic weight

We study dynamic networks under an undirected consensus communication protocol and with one state-dependent weighted edge. We assume that the aforementioned dynamic edge can take values over the whole real numbers, and that its behaviour depends on the nodes it connects and on an extrinsic slow variable. We show that, under mild conditions on the weight, there exists a reduction such that the dynamics of the network are organized by a transcritical singularity. As such, we detail a slow passage through a transcritical singularity for a simple network, and we observe that an exchange between consensus and clustering of the nodes is possible. In contrast to the classical planar fast-slow transcritical singularity, the network structure of the system under consideration induces the presence of a maximal canard. Our main tool of analysis is the blow-up method. Thus, we also focus on tracking the effects of the blow-up transformation on the network's structure. We show that on each blow-up chart one recovers a particular dynamic network related to the original one. We further indicate a numerical issue produced by the slow passage through the transcritical singularity

A topological perspective on singular canards for critical sets with transverse intersections

This paper gives a new perspective on singular canards, which is topological in flavour. One key feature is that our construction does not rely on coordinates; consequently, the conditions for the existence of singular canards that we provide are purely geometric. The singularities we study originate at the self-intersection of curves of equilibria of the unperturbed system. Our contribution even allows us to consider degenerate cases of multiple pairwise transverse intersecting branches of the critical set. We employ stratification theory and algebraic geometric properties to provide sufficient conditions leading to the presence of singular canards. By means of two examples, we corroborate our findings using the well-known blow-up technique

Nonlinear Laplacian Dynamics:Symmetries, Perturbations, and Consensus

In this paper, we study a class of dynamic networks called Absolute Laplacian Flows under small perturbations. Absolute Laplacian Flows are a type of nonlinear generalisation of classical linear Laplacian dynamics. Our main goal is to describe the behaviour of the system near the consensus space. The nonlinearity of the studied system gives rise to potentially intricate structures of equilibria that can intersect the consensus space, creating singularities. For the unperturbed case, we characterise the sets of equilibria by exploiting the symmetries under group transformations of the nonlinear vector field. Under perturbations, Absolute Laplacian Flows behave as a slow-fast system. Thus, we analyse the slow-fast dynamics near the singularities on the consensus space. In particular, we prove a theorem that provides existence conditions for a maximal canard, that coincides with the consensus subspace, by using the symmetry properties of the network. Furthermore, we provide a linear approximation of the intersecting branches of equilibria at the singular points; as a consequence, we show that, generically, the singularities on the consensus space turn out to be transcritical

On network dynamical systems with a nilpotent singularity

Network dynamics is nowadays of extreme relevance to model and analyze complex systems. From a dynamical systems perspective, understanding the local behavior near equilibria is of utmost importance. In particular, equilibria with at least one zero eigenvalue play a crucial role in bifurcation analysis. In this paper, we want to shed some light on nilpotent equilibria of network dynamical systems. As a main result, we show that the blow-up technique, which has proven to be extremely useful in understanding degenerate singularities in low-dimensional ordinary differential equations, is also suitable in the framework of network dynamical systems. Most importantly, we show that the blow-up technique preserves the network structure. The further usefulness of the blow-up technique, especially with regard to the desingularization of a nilpotent point, is showcased through several examples including linear diffusive systems, systems with nilpotent internal dynamics, and an adaptive network of Kuramoto oscillators

Classification of constrained differential equations embedded in the theory of slow fast systems:A_k singularities and geometric desingularization

Veel natuurlijke fenomenen spelen zich af op verschillende tijdschalen. Denk bijvoorbeeld aan de hartslag, zenuwactiviteit, scheikundige reacties of het weer. Dergelijke fenomenen kunnen daarom worden gemodelleerd door middel van zogenaamde “slow-fast” systemen. Dit zijn gewone differentiaalvergelijkingen die op een singuliere manier afhangen van een kleine parameter. Door deze parameter gelijk aan nul te stellen ontstaat een differentiaalvergelijking met een algebraïsche beperking. De Groningse wiskundige Floris Takens (1940-2010) heeft in 1975 belangrijke bijdragen geleverd aan de theorie van differentiaalvergelijkingen met algebraïsche beperkingen en hun relatie tot slow-fast systemen. Zijn resultaten zijn in het bijzonder bruikbaar als men de meer gecompliceerde dynamica van slow-fast systemen wil bestuderen. Dit proefschrift is een studie naar de dynamica en locale eigenschappen van slow-fast systemen en de daaraan gerelateerde differentiaalvergelijkingen met algebraïsche beperkingen. Het belangrijkste resultaat van het onderzoek is een uitbreiding van de resultaten van Takens met betrekking tot de classificatie van differentiaalvergelijkingen met algebraïsche beperkingen. Op basis van deze uitbreiding ontwikkelen we op eendrachtige wijze een methode om de dynamica van een grote klasse van slow-fast systemen te bestuderen.Many natural phenomena take place at different time scales. Think of the heartbeat, nerve activity, chemical reactions or the weather. Such phenomena can therefore be modeled by so-called "slow-fast" systems. These are ordinary differential equations that depend singularly on a small parameter. When this parameter is set to zero, a differential equation with an algebraic constraint arises. The Groningen mathematician Floris Takens (1940-2010) made major contributions in 1975 to the theory of differential equations with algebraic constraints and their relationship to slow-fast systems. His results are particularly useful if one wants to study the more complicated dynamics of slow-fast systems. This thesis is a study of the dynamics and local properties of slow-fast systems and the related differential equations with algebraic constraints. The main result of this research is an extension of the results of Takens in relation to the classification of differential equations with algebraic constraints. Based on this extension, we have developed a unified method to study the dynamics of a large class of slow-fast systems