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Register a new userConleys fundamental theorem for a class of hybrid systems
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Matthew Kvalheim
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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In the first part of this paper, we generalize the results of the author cite{Liu,Liu2} from the random flow case to the random semiflow case, i.e. we obtain Conley decomposition theorem for infinite dimensional random dynamical systems. In the second part, by introducing the backward orbit for random semiflow, we are able to decompose invariant random compact set (e.g. global random attractor) into random Morse sets and connecting orbits between them, which generalizes the Morse decomposition of invariant sets originated from Conley cite{Con} to the random semiflow setting and gives the positive answer to an open problem put forward by Caraballo and Langa cite{CL}.
We propose a definition of the category of hybrid systems in which executions are special types of morphisms. Consequently morphisms of hybrid systems send executions to executions. We plan to use this result to define and study networks of hybrid systems.
We consider a class of interacting particle systems with values in $[0,8)^{zd}$, of which the binary contact path process is an example. For $d ge 3$ and under a certain square integrability condition on the total number of the particles, we prove a central limit theorem for the density of the particles, together with upper bounds for the density of the most populated site and the replica overlap.
For a topological group G, we show that a compact metric G-space is tame if and only if it can be linearly represented on a separable Banach space which does not contain an isomorphic copy of $l_1$ (we call such Banach spaces, Rosenthal spaces). With this goal in mind we study tame dynamical systems and their representations on Banach spaces.
Although the spatially continuous version of the reaction-diffusion equation has been well studied, in some instances a spatially-discretized representation provides a more realistic approximation of biological processes. Indeed, mathematically the discretized and continuous systems can lead to different predictions of biological dynamics. It is well known in the continuous case that the incorporation of diffusion can cause diffusion-driven blow-up with respect to the $L^{infty}$ norm. However, this does not imply diffusion-driven blow-up will occur in the discretized version of the system. For example, in a continuous reaction-diffusion system with Dirichlet boundary conditions and nonnegative solutions, diffusion-driven blow up occurs even when the total species concentration is non-increasing. For systems that instead have homogeneous Neumann boundary conditions, it is currently unknown whether this deviation between the continuous and discretized system can occur. Therefore, it is worth examining the discretized system independently of the continuous system. Since no criteria exist for the boundedness of the discretized system, the focus of this paper is to determine sufficient conditions to guarantee the system with diffusion remains bounded for all time. We consider reaction-diffusion systems on a 1D domain with homogeneous Neumann boundary conditions and non-negative initial data and solutions. We define a Lyapunov-like function and show that its existence guarantees that the discretized reaction-diffusion system is bounded. These results are considered in the context of three example systems for which Lyapunov-like functions can and cannot be found.
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