An Introduction to Sequential Dynamical Systems

An Introduction to Sequential Dynamical Systems PDF

Author: Henning Mortveit

Publisher: Springer Science & Business Media

Published: 2007-11-27

Total Pages: 261

ISBN-13: 0387498796

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This introductory text to the class of Sequential Dynamical Systems (SDS) is the first textbook on this timely subject. Driven by numerous examples and thought-provoking problems throughout, the presentation offers good foundational material on finite discrete dynamical systems, which then leads systematically to an introduction of SDS. From a broad range of topics on structure theory - equivalence, fixed points, invertibility and other phase space properties - thereafter SDS relations to graph theory, classical dynamical systems as well as SDS applications in computer science are explored. This is a versatile interdisciplinary textbook.

Sequential Dynamical Systems with Threshold Functions

Sequential Dynamical Systems with Threshold Functions PDF

Author:

Publisher:

Published: 2001

Total Pages: 3

ISBN-13:

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A sequential dynamical system (SDS) (see [BH+01] and the references therein) consists of an undirected graph G(V, E) where each node [nu] [epsilon] V is associated with a Boolean state (s{sub [nu]}) and a symmetric Boolean function f{sub [nu]} (called the local transition function at [nu]). The inputs to f{sub {nu}} are s{sub {nu}} and the states of all the nodes adjacent to {nu}. In each step of the SDS, the nodes update their state values using their local transition functions in the order specified by a given permutation [pi] of the nodes. A configuration of the SDS is an n-tuple (b1, b2 ..., b{sub n}) where n =

On Theoretical Issues of Computer Simulations Sequential Dynamical Systems

On Theoretical Issues of Computer Simulations Sequential Dynamical Systems PDF

Author:

Publisher:

Published: 1998

Total Pages: 9

ISBN-13:

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The authors study a class of discrete dynamical systems that is motivated by the generic structure of simulations. The systems consist of the following data: (a) a finite graph Y with vertex set {l_brace}1, ..., n{r_brace} where each vertex has a binary state, (b) functions F{sub i}:F2??20--?? → F2??20--?? and (c) an update ordering?. The functions F{sub i} update the binary state of vertex i as a function of the state of vertex i and its Y-neighbors and leave the states of all other vertices fixed. The update ordering is a permutation of the Y-vertices. They derive a decomposition result, characterize invertible SDS and study fixed points. In particular they analyze how many different SDS that can be obtained by reordering a given multiset of update functions and give a criterion for when one can derive concentration results on this number. Finally, some specific SDS are investigated.

Predecessor and Permutation Existence Problems for Sequential Dynamical Systems

Predecessor and Permutation Existence Problems for Sequential Dynamical Systems PDF

Author:

Publisher:

Published: 2002

Total Pages: 25

ISBN-13:

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A class of finite discrete dynamical systems, called Sequential Dynamical Systems (SDSs), was introduced in BMR99, BR991 as a formal model for analyzing simulation systems. An SDS S is a triple (G, F, n), w here (i) G(V, E) is an undirected graph with n nodes with each node having a state, (ii) F = (fi, fi, . . ., fn), with fi denoting a function associated with node ui E V and (iii) A is a permutation of (or total order on) the nodes in V, A configuration of an SDS is an n-vector (b l, bz, . . ., bn), where bi is the value of the state of node vi. A single SDS transition from one configuration to another is obtained by updating the states of the nodes by evaluating the function associated with each of them in the order given by n. Here, we address the complexity of two basic problems and their generalizations for SDSs. Given an SDS S and a configuration C, the PREDECESSOR EXISTENCE (or PRE) problem is to determine whether there is a configuration C' such that S has a transition from C' to C. (If C has no predecessor, C is known as a garden of Eden configuration.) Our results provide separations between efficiently solvable and computationally intractable instances of the PRE problem. For example, we show that the PRE problem can be solved efficiently for SDSs with Boolean state values when the node functions are symmetric and the underlying graph is of bounded treewidth. In contrast, we show that allowing just one non-symmetric node function renders the problem NP-complete even when the underlying graph is a tree (which has a treewidth of 1). We also show that the PRE problem is efficiently solvable for SDSs whose state values are from a field and whose node functions are linear. Some of the polynomial algorithms also extend to the case where we want to find an ancestor configuration that precedes a given configuration by a logarithmic number of steps. Our results extend some of the earlier results by Sutner [Su95] and Green [@87] on the complexity of the PREDECESSOR EXISTENCE problem for 1-dimensional cellular automata. Given the underlying graph G(V, E), and two configurations C and C' of an SDS S, the PERMUTATION EXISTENCE (or PME) problem is to determine whether there is a permutation of nodes such that 8 has a transition from C' to C in one step. We show that the PME problem is NP-complete even when the function associated with each node is a simple-threshold function. We also show that a generalized version of the PME(GEN-PMEp)r oblem is NP-complete for SDSs where each node function is NOR and the underlying graph has a maximum node degree of 3. When each node computes the OR function or when each node computes the AND function, we show that the GEN-PMEpr oblem is solvable in polynomial time.

Dynamical Systems and Processes

Dynamical Systems and Processes PDF

Author: Michel Weber

Publisher: European Mathematical Society

Published: 2009

Total Pages: 778

ISBN-13: 9783037190463

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This book presents in a concise and accessible way, as well as in a common setting, various tools and methods arising from spectral theory, ergodic theory and stochastic processes theory, which form the basis of and contribute interactively a great deal to the current research on almost-everywhere convergence problems. Researchers working in dynamical systems and at the crossroads of spectral theory, ergodic theory and stochastic processes will find the tools, methods, and results presented in this book of great interest. It is written in a style accessible to graduate students.

Analysis Problems for Sequential Dynamical Systems and Communicating State Machines

Analysis Problems for Sequential Dynamical Systems and Communicating State Machines PDF

Author:

Publisher:

Published: 2001

Total Pages: 22

ISBN-13:

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A simple sequential dynamical system (SDS) is a triple (G, F, [pi]), where (i) G(V, E) is an undirected graph with n nodes with each node having a 1-bit state, (ii) F = [f1, f2 ..., f{sub n}] is a set of local transition functions with f{sub i} denoting a Boolean function associated with node Vv{sub i} and (iii) [pi] is a fixed permutation of (i.e., a total order on) the nodes in V.A single SDS transition is obtained by updating the states of the nodes in V by evaluating the function associated with each of them in the order given by [pi]. Such a (finite) SDS is a mathematical abstraction of simulation systems [BMR99, BR99]. In this paper, we characterize the computational complexity of determining several phase space properties of SDSs. The properties considered are t-REACHABILITY ('Can a given SDS starting from configuration I reach configuration B in t or fewer transitions?'), REACHABILITY('Can a given SDS starting from configuration I ever reach configuration B?') and FIXED POINT REACHABILITY ('Can a given SDS starting from configuration I ever reach configuration in which it stays for ever?'). Our main result is a sharp dichotomy between classes of SDSs whose behavior is 'easy' to predict and those whose behavior is 'hard' to predict. Specifically, we show the following. (1) The t-REACHABILITY, REACHABILITY and the FIXED POINT REACHABILITY problems for SDSs are PSPACE-complete, even when restricted to graphs of bounded bandwidth (and hence of bounded pathwidth and treewidth) and when the function associated with each node is symmetric. The result holds even for regular graphs of constant degree where all the nodes compute the same symmetric Boolean function. (2) In contrast, the t-REACHABILITYm REACHABILITY and FIXED POINT REACHABILITY problems are solvable in polynomial time for SDSs when the Boolean function associated with each node is symmetric and monotone. Two important consequences of our results are the following: (i) The close correspondence between SDSs and cellular automata (CA), in conjunctio with our lower bounds for SDSs, yields stronger lower bounds on the complexity of reachability problems for CA than known previously. (ii) REACHABILITY problems for hierarchically-specified linearly inter-connected copies of a single finite automaton are EXPSPACE-hard. The results can be combined with our related results to show hardness of a number of equivalence relations for such automata. The results can also be used to demonstrate that determining the sensitivity to initial conditions of such automata (as proposed in [Mo90, BPT91]) is computationally intractable.

Extremes and Recurrence in Dynamical Systems

Extremes and Recurrence in Dynamical Systems PDF

Author: Valerio Lucarini

Publisher: John Wiley & Sons

Published: 2016-04-04

Total Pages: 314

ISBN-13: 111863229X

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Written by a team of international experts, Extremes and Recurrence in Dynamical Systems presents a unique point of view on the mathematical theory of extremes and on its applications in the natural and social sciences. Featuring an interdisciplinary approach to new concepts in pure and applied mathematical research, the book skillfully combines the areas of statistical mechanics, probability theory, measure theory, dynamical systems, statistical inference, geophysics, and software application. Emphasizing the statistical mechanical point of view, the book introduces robust theoretical embedding for the application of extreme value theory in dynamical systems. Extremes and Recurrence in Dynamical Systems also features: • A careful examination of how a dynamical system can serve as a generator of stochastic processes • Discussions on the applications of statistical inference in the theoretical and heuristic use of extremes • Several examples of analysis of extremes in a physical and geophysical context • A final summary of the main results presented along with a guide to future research projects • An appendix with software in Matlab® programming language to help readers to develop further understanding of the presented concepts Extremes and Recurrence in Dynamical Systems is ideal for academics and practitioners in pure and applied mathematics, probability theory, statistics, chaos, theoretical and applied dynamical systems, statistical mechanics, geophysical fluid dynamics, geosciences and complexity science. VALERIO LUCARINI, PhD, is Professor of Theoretical Meteorology at the University of Hamburg, Germany and Professor of Statistical Mechanics at the University of Reading, UK. DAVIDE FARANDA, PhD, is Researcher at the Laboratoire des science du climat et de l’environnement, IPSL, CEA Saclay, Université Paris-Saclay, Gif-sur-Yvette, France. ANA CRISTINA GOMES MONTEIRO MOREIRA DE FREITAS, PhD, is Assistant Professor in the Faculty of Economics at the University of Porto, Portugal. JORGE MIGUEL MILHAZES DE FREITAS, PhD, is Assistant Professor in the Department of Mathematics of the Faculty of Sciences at the University of Porto, Portugal. MARK HOLLAND, PhD, is Senior Lecturer in Applied Mathematics in the College of Engineering, Mathematics and Physical Sciences at the University of Exeter, UK. TOBIAS KUNA, PhD, is Associate Professor in the Department of Mathematics and Statistics at the University of Reading, UK. MATTHEW NICOL, PhD, is Professor of Mathematics at the University of Houston, USA. MIKE TODD, PhD, is Lecturer in the School of Mathematics and Statistics at the University of St. Andrews, Scotland. SANDRO VAIENTI, PhD, is Professor of Mathematics at the University of Toulon and Researcher at the Centre de Physique Théorique, France.

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