Day: 13th of November, 14h30
Speaker: Alexander Lohse (Hamburg University)
Title: Switching dynamics
Abstract: In every heteroclinic network there exists at least one node (equilibrium) with more than one outgoing connection towards another node. Trajectories in a neighbourhood of the network may therefore "choose" which of these connections to follow. For a given network, it is possible to analyze systematically for which sequences of heteroclinic connections there are initial conditions near the network such that the solution follows exactly the prescribed sequence of connections. This helps to understand the long-term dynamics of a system given through an ODE. Studying such questions has led to the notion of switching dynamics and a wide range of dynamic behaviour has been unveiled subsequently, related among other things to the eigenvalues of the linearization of the vector field at the nodes.
In this talk we introduce different levels of switching and give a basic idea of conditions under which they may or may not occur. We start by looking at a simple example and finish with some questions which currently draw considerable attention.
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Day: 15th of November, 16h30
Speaker: Soeren von der Gracht (Paderborn University)
Title: Strange symmetries and strange bifurcations in network dynamical systems
Abstract: Many dynamical systems in fields such as neuroscience (the workings of the brain), systems biology (metabolic systems) and robotics (robot swarms) exhibit the structure of a network: they consist of nodes (neurons, proteins, robots) with connections between them. It usually does not suffice to understand the nature of the individual nodes to deduce the behavior of the network, as the specific interaction structure of a network can produce remarkable dynamics. Prominent examples include synchronization (e.g., the simultaneous firing of neurons) and highly complex branching behavior in bifurcations, phenomena that are not found in dynamical systems without the structure of a network.
Network dynamical systems are not well understood mathematically, which makes it hard to quantify and control their behavior. The reason is that most of the established machinery of dynamical systems theory fails to distinguish between networks and general dynamical systems. Thus, we need mathematical tools that are tailor-made for network problems. Several techniques have been proposed recently, and they strikingly have one thing in common: they exploit the algebraic nature of networks.
In this talk, I will give an overview over some recent results regarding the question which dynamical behavior and generic bifurcations are dictated by the network structure of a system. In particular, I will illustrate how structural and algebraic properties culminate in symmetries of the governing equations and how these can be exploited for (partial) answers. This includes classical symmetries but also more exotic concepts such as monoid and quiver representations.
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Day: 16th of November, 14h30
Speaker: Haibo Ruan (Hamburg University of Technology)
Title: Synchrony Patterns in Gene Regulatory Networks
Abstract: Motivated by studying synchronization mechanisms in gene regulatory networks (GRNs) and their relation to evolutionary events such as genetic duplication and genetic redundancy, we consider two mathematical dynamical models of GRNs. We obtain results on robust synchronization on these dynamical models inspired by the existing theoretical results in the coupled cell network formalisms. We also explore the concepts of quotient networks and network lifting in the context of GRNs which are related to the process of gene duplication and the phenomenon of subfunctionalization as an outcome of functional divergence.
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Day: 17th of November, 14h30
Speaker: Telmo Peixe (REM, CEMAPRE, ISEG -- University of Lisbon)
Title: Dynamics along the heteroclinic network of polymatrix replicators
Abstract: The polymatrix replicator is a system of ordinary differential equations that can be used to model the time evolution of behavioural strategies of in- dividuals in a stratified population. The flow of these systems evolve on a prism (polytope) given by the product of simplices. In this talk we present a new method to analyse the asymptotic dynamics of a flow on a polytope along its edge-vertex heteroclinic network. For this purpose we will explore the dynamics of some examples of polymatrix replicators. A significant part of this work is joint work with Alexandre Rodrigues (ISEG, CMUP).
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