Can a fish with limited velocity capabilities reach any point in the (possibly unbounded) ocean? In a recent paper by D. Burago, S. Ivanov and A. Novikov, "A survival guide for feeble fish", an affirmative answer has been given under the condition that the fluid velocity field is incompressible, bounded and has vanishing mean drift. This brilliant result extends some known point-to-point global controllability theorems though being substantially non constructive. We will give a fish a different recipe of how to survive in a turbulent ocean, and show how this is related to structural stability of dynamical systems by providing a constructive way to change slightly a divergence free vector field with vanishing mean drift to produce a non dissipative dynamics. This immediately leads to closing lemmas for dynamical systems, in particular to C. Pugh's closing lemma, saying also that the fish can eventually return home.
Joint work with Sergey Kryzhevich (Nova Gorica and St. Petersburg).
Relatore:Stefan Schreieder (Leibniz Universität Hannover)
Luogo: Aula 5
Data e ora: 29 settembre 2022, ore 14:30
The Gauss--Green formulas are of significant relevance in many areas of analysis and mathematical physics. This motivated several investigations to extend such formulas to more general classes of integration domains and weakly differentiable vector fields, and thus led to the definition of the divergence-measure fields. These are $L^{p}$-summable vector fields on $\mathbb{R}^{n}$ whose distributional divergence is a Radon measure, and so they form a new family of function spaces, which in a sense generalize the $BV$ fields. The divergence-measure fields were introduced at first by Anzellotti in 1983 for the case $p = \infty$, and then they have been rediscovered in the early 2000s by many authors interested in various applications.
In this talk, we shall present an overview of such researches, with a particular focus on the results concerning essentially bounded divergence-measure fields. For such a family of vector fields, Silhavy (2005), Chen, Torres and Ziemer (2009) and Comi and Payne (2017) showed that Gauss--Green formulas hold for sets of finite perimeter, by proving the existence of essentially bounded interior and exterior normal traces on the reduced boundary of the given set.
Subsequent extensions of these results and generalizations to non-Euclidean geometries will be also briefly discussed.
]]>Network dynamics preserves the sum of all incoming pairwise coupling strengths and is designed to adaptively interact with system dynamics.For adaptive couplings, we use two adaptive coupling laws for the pairwise coupling strength. Kuramoto oscillators are assumed to be on the nodes of the networks.
We present frameworks that guarantee the emergence of synchronization for various coupling feedback laws. Our results generalize earlier work on the synchronization of Kuramoto oscillators in fixed and symmetric networks.
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