Computability and dynamical systems

In this paper we explore results that establish a link between dynamical systems and computability theory (not numerical analysis). In the last few decades, computers have increasingly been used as simulation tools for gaining insight into dynamical behavior. However, due to the presence of errors inherent in such numerical simulations, with few exceptions, computers have not been used for the nobler task of proving mathematical results. Nevertheless, there have been some recent developments in the latter direction. Here we introduce some of the ideas and techniques used so far, and suggest some lines of research for further work on this fascinating topic

Interplay of Analysis and Probability in Physics

The main purpose of this workshop was to foster interaction between researchers in the fields of analysis and probability with the aim of joining forces to understand difficult problems from physics rigorously. 52 researchers of all age groups and from many parts of Europe and overseas attended. The talks and discussions evolved around five topics on the interface between analysis and probability. The main goal of the workshop, the systematic encouragement of intense discussions between the two communities, was achieved to a high extent

Parameter inference for stochastic biological models

PhD ThesisParameter inference is the field concerned with estimating reliable model parameters from data. In recent years there has been a trend in the biology community toward single cell technologies such as fluorescent flow cytometry, transcriptomics and mass cytometry: providing a rich array of stochastic time series and temporal distribution data for analysis. Deterministically, there are a wide range of parameter inference and global optimisation techniques available. However, these do not always scale well to non-deterministic (i.e., stochastic) settings — whereby the temporal evolution of the system can be described by a chemical master equation for which the solution is nearly always intractable, and the dynamic behaviour of a system is hard to predict. For systems biology, the inference of stochastic parameters remains a bottleneck for accurate model simulation. This thesis is concerned with the parameter inference problem for stochastic chemical reaction networks. Stochastic chemical reaction networks are most frequently modelled as a continuous time discretestate Markov chain using Gillespie’s stochastic simulation algorithm. Firstly, I present a new parameter inference algorithm, SPICE, that combines Gillespie’s algorithm with the cross-entropy method. The cross-entropy method is a novel approach for global optimisation inspired from the field of rare-event probability estimation. I then present recent advances in utilising the generalised method of moments for inference, and seek to provide these approaches with a direct stochastic simulation based correction. Subsequently, I present a novel use of a recent multi-level tau-leaping approach for simulating population moments efficiently, and use this to provide a simulation based correction to the generalised method of moments. I also propose a new method for moment closures based on the use of Padé approximants. The presented algorithms are evaluated on a number of challenging case studies, including bistable systems — e.g., the Schlögl System and the Genetic Toggle Switch — and real experimental data. Experimental results are presented using each of the given algorithms. We also consider ‘realistic’ data — i.e., datasets missing model species, multiple datasets originating from experiment repetitions, and datasets containing arbitrary units (e.g., fluorescence values). The developed approaches are found to be viable alternatives to existing state-ofthe-art methods, and in certain cases are able to outperform other methods in terms of either speed, or accuracyNewcastle/Liverpool/Durham BBSRC Doctoral Training Partnership for financial suppor

Typicalness of chaotic fractal behaviour of integral vortexes in Hamiltonian systems with discontinuous right hand side

We consider a linear-quadratic deterministic optimal control problem where the control takes values in a two-dimensional simplex. The phase portrait of the optimal synthesis contains second-order singular extremals and exhibits modes of infinite accumulations of switchings in finite time, so-called chattering. We prove the presence of an entirely new phenomenon, namely the chaotic behaviour of bounded pieces of optimal trajectories. We find the hyperbolic domains in the neighbourhood of a homoclinic point and estimate the corresponding contraction-extension coefficients. This gives us the possibility to calculate the entropy and the Hausdorff dimension of the non-wandering set which appears to have a Cantor-like structure as in Smale's Horseshoe. The dynamics of the system is described by a topological Markov chain. In the second part it is shown that this behaviour is generic for piece-wise smooth Hamiltonian systems in the vicinity of a junction of three discontinuity hyper-surface strata.Comment: 113 pages, 22 figure

Computer Aided Verification

This open access two-volume set LNCS 11561 and 11562 constitutes the refereed proceedings of the 31st International Conference on Computer Aided Verification, CAV 2019, held in New York City, USA, in July 2019. The 52 full papers presented together with 13 tool papers and 2 case studies, were carefully reviewed and selected from 258 submissions. The papers were organized in the following topical sections: Part I: automata and timed systems; security and hyperproperties; synthesis; model checking; cyber-physical systems and machine learning; probabilistic systems, runtime techniques; dynamical, hybrid, and reactive systems; Part II: logics, decision procedures; and solvers; numerical programs; verification; distributed systems and networks; verification and invariants; and concurrency