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Register a new userDimension Groups and Dynamical Systems
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Added by
Dominique Perrin
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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We give a description of the link between topological dynamical systems and their dimension groups. The focus is on minimal systems and, in particular, on substitution shifts. We describe in detail the various classes of systems including Sturmian shifts and interval exchange shifts. This is a preliminary version of a book which will be published by Cambridge University Press. Any comments are of course welcome.
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Neural Networks (NNs) have been identified as a potentially powerful tool in the study of complex dynamical systems. A good example is the NN differential equation (DE) solver, which provides closed form, differentiable, functional approximations for the evolution of a wide variety of dynamical systems. A major disadvantage of such NN solvers can be the amount of computational resources needed to achieve accuracy comparable to existing numerical solvers. We present new strategies for existing dynamical system NN DE solvers, making efficient use of the textit{learnt} information, to speed up their training process, while still pursuing a completely unsupervised approach. We establish a fundamental connection between NN theory and dynamical systems theory via Koopman Operator Theory (KOT), by showing that the usual training processes for Neural Nets are fertile ground for identifying multiple Koopman operators of interest. We end by illuminating certain applications that KOT might have for NNs in general.
Toric dynamical systems are known as complex balancing mass action systems in the mathematical chemistry literature, where many of their remarkable properties have been established. They include as special cases all deficiency zero systems and all detailed balancing systems. One feature is that the steady state locus of a toric dynamical system is a toric variety, which has a unique point within each invariant polyhedron. We develop the basic theory of toric dynamical systems in the context of computational algebraic geometry and show that the associated moduli space is also a toric variety. It is conjectured that the complex balancing state is a global attractor. We prove this for detailed balancing systems whose invariant polyhedron is two-dimensional and bounded.
In this paper we define unstable topological entropy for any subsets (not necessarily compact or invariant) in partially hyperbolic systems as a Carath{e}odory dimension characteristic, motivated by the work of Bowen and Pesin etc. We then establish some basic results in dimension theory for Bowen unstable topological entropy, including an entropy distribution principle and a variational principle in general setting. As applications of this new concept, we study unstable topological entropy of saturated sets and extend some results in cite{Bo, PS2007}. Our results give new insights to the multifractal analysis for partially hyperbolic systems.
We study two coupled discrete-time equations with different (asynchronous) periodic time scales. The coupling is of the type sample and hold, i.e., the state of each equation is sampled at its update times and held until it is read as an input at the next update time for the other equation. We construct an interpolating two-dimensional complex-valued system on the union of the two time scales and an extrapolating four-dimensional system on the intersection of the two time scales. We discuss stability by several results, examples and counterexamples in various frameworks to show that the asynchronicity can have a significant impact on the dynamical properties.
Evolutionary games on graphs play an important role in the study of evolution of cooperation in applied biology. Using rigorous mathematical concepts from a dynamical systems and graph theoretical point of view, we formalize the notions of attractor, update rules and update orders. We prove results on attractors for different utility functions and update orders. For complete graphs we characterize attractors for synchronous and sequential update rules. In other cases (for $k$-regular graphs or for different update orders) we provide sufficient conditions for attractivity of full cooperation and full defection. We construct examples to show that these conditions are not necessary. Finally, by formulating a list of open questions we emphasize the advantages of our rigorous approach.
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