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Register a new userAn algebra of open continuous time dynamical systems and networks
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Many systems of interest in science and engineering are made up of interacting subsystems. These subsystems, in turn, could be made up of collections of smaller interacting subsystems and so on. In a series of papers David Spivak with collaborators formalized these kinds of structures (systems of systems) as algebras over presentable colored operads. It is also very useful to consider maps between dynamical systems. This amounts to viewing dynamical systems as objects in an appropriate category. This is the point taken by DeVille and Lerman in the study of dynamics on networks. The goal of this paper is to describe an algebraic structure that encompasses both approaches to systems of systems. To this end we replace the monoidal category of wiring diagrams by a monoidal double category whose objects are surjective submersions. This allows us, on one hand, build new large open systems out of collections of smaller open subsystems and on the other keep track of maps between open systems. As a special case we recover the results of DeVille and Lerman on fibrations of networks of manifolds.
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We generalize the results of Networks of open systems by the first author to the setting of hybrid systems. In particular we introduce the notions of hybrid open systems, their networks and maps between networks. A network of systems is a blueprint for building a larger system out of smaller subsystems by specifying a pattern of interactions between subsystems --- an interconnection map. Maps between networks allow us to produce maps between complex hybrid dynamical systems by specifying maps between their subsystems
Many systems of interest in science and engineering are made up of interacting subsystems. These subsystems, in turn, could be made up of collections of smaller interacting subsystems and so on. In a series of papers David Spivak with collaborators formalized these kinds of structures (systems of systems) as algebras over presentable colored operads. It is also very useful to consider maps between dynamical systems, which in effect amounts to viewing dynamical systems as objects in an appropriate category. This is the point of view taken by DeVille and Lerman in the study of dynamics on networks. The goal of this paper is to describe an algebraic structure that encompasses both approaches to systems of systems. This allows us, on one hand, build new large open systems out of collections of smaller open subsystems and on the other keep track of maps between open systems. Consequently we obtain synchrony results for open systems which generalize the synchrony results of Golubitsky, Stewart and their collaborators for groupoid invariant vector fields on coupled cell networks.
We show that there is a natural restriction on the smoothness of spaces where the transfer operator for a continuous dynamical system has a spectral gap. Such a space cannot be embedded in a Holder space with Holder exponent greater than 1/2 unless it consists entirely of coboundaries.
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.
We propose a new framework for the study of continuous time dynamical systems on networks. We view such dynamical systems as collections of interacting control systems. We show that a class of maps between graphs called graph fibrations give rise to maps between dynamical systems on networks. This allows us to produce conjugacy between dynamical systems out of combinatorial data. In particular we show that surjective graph fibrations lead to synchrony subspaces in networks. The injective graph fibrations, on the other hand, give rise to surjective maps from large dynamical systems to smaller ones. One can view these surjections as a kind of fast/slow variable decompositions or as abstractions in the computer science sense of the word.
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