A METHOD FOR FINDING AGGREGATED REPRESENTATIONS OF LINEAR DYNAMICAL SYSTEMS

2010 ◽  
Vol 13 (02) ◽  
pp. 199-215 ◽  
Author(s):  
OLOF GÖRNERUP ◽  
MARTIN NILSSON JACOBI
Keyword(s):  
Dynamical Systems ◽  
Cellular Automata ◽  
Markov Chains ◽  
Complex Systems ◽  
Diffusion Processes ◽  
Coarse Grained ◽  
Hierarchical Level ◽  
Permutation Symmetry ◽  
Linear Dynamical Systems ◽  
And Diffusion

A central problem in the study of complex systems is to identify hierarchical and intertwined dynamics. A hierarchical level is defined as an aggregation of the system's variables such that the aggregation induces its own closed dynamics. In this paper, we present an algorithm that finds aggregations of linear dynamical systems, e.g. including Markov chains and diffusion processes on weighted and directed networks. The algorithm utilizes that a valid aggregation with n states correspond to a set of n eigenvectors of the dynamics matrix such that these respect the same permutation symmetry with n orbits. We exemplify the applicability of the algorithm by employing it to identify coarse grained representations of cellular automata.

A SPECTRAL METHOD FOR AGGREGATING VARIABLES IN LINEAR DYNAMICAL SYSTEMS WITH APPLICATION TO CELLULAR AUTOMATA RENORMALIZATION

2009 ◽  
Vol 12 (02) ◽  
pp. 131-155 ◽  
Author(s):  
MARTIN NILSSON JACOBI ◽  
OLOF GÖRNERUP
Keyword(s):  
Dynamical Systems ◽  
Cellular Automata ◽  
Markov Chains ◽  
Spectral Method ◽  
Coarse Grained ◽  
Coarse Grain ◽  
Permutation Symmetry ◽  
Linear Dynamical Systems ◽  
Aggregation Of Variables ◽  
The Matrix

We present a method for identifying coarse-grained dynamics through aggregation of variables or states in linear dynamical systems. The condition for aggregation is expressed as a permutation symmetry of a set of dual eigenvectors of the matrix that defines the dynamics. The applicability of the condition is illustrated in examples from three different generic classes of reducible Markov chains: systems consisting of independent subsystems, dynamics with symmetries, and nearly decoupled Markov chains. Furthermore we show how the method can be used to coarse-grain cellular automata.

MODELING LINEAR DYNAMICAL SYSTEMS BY CONTINUOUS-VALUED CELLULAR AUTOMATA

2007 ◽  
Vol 18 (05) ◽  
pp. 833-848 ◽  
Author(s):  
JUAN CARLOS SECK TUOH MORA ◽  
MANUEL GONZALEZ HERNANDEZ ◽  
NORBERTO HERNANDEZ ROMERO ◽  
AARON RODRIGUEZ TREJO ◽  
SERGIO V. CHAPA VERGARA
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Cellular Automata ◽  
Linear Systems ◽  
Configuration Space ◽  
Integration Method ◽  
Numerical Calculations ◽  
Linear Dynamical Systems ◽  
Original Part

This paper exposes a procedure for modeling and solving linear systems using continuous-valued cellular automata. The original part of this work consists on showing how the cells in the automaton may contain both real values and operators for carrying out numerical calculations and solve a desired problem. In this sense the automaton acts as a program, where data and operators are mixed in the evolution space for obtaining the correct calculations. As an example, Euler's integration method is implemented in the configuration space in order to achieve an approximated solution for a dynamical system. Three examples showing linear behaviors are presented.

scholarly journals Investigating Transformational Complexity: Counting Functions a Region Induces on Another in Elementary Cellular Automata

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Martin Biehl ◽  
Olaf Witkowski
Keyword(s):  
Dynamical Systems ◽  
Cellular Automata ◽  
Complex Systems ◽  
Cellular Automaton ◽  
Artificial Life ◽  
Discrete Dynamical Systems ◽  
Living Systems ◽  
Finite Region ◽  
Counting Functions ◽  
Elementary Cellular Automaton

Over the years, the field of artificial life has attempted to capture significant properties of life in artificial systems. By measuring quantities within such complex systems, the hope is to capture the reasons for the explosion of complexity in living systems. A major effort has been in discrete dynamical systems such as cellular automata, where very few rules lead to high levels of complexity. In this paper, for every elementary cellular automaton, we count the number of ways a finite region can transform an enclosed finite region. We discuss the relation of this count to existing notions of controllability, physical universality, and constructor theory. Numerically, we find that particular sizes of surrounding regions have preferred sizes of enclosed regions on which they can induce more transformations. We also find three particularly powerful rules (90, 105, 150) from this perspective.

scholarly journals On the existence of solutions to the fractional derivative equations d(alpha)u/dt(alpha)+Au = f, of relevance to diffusion in complex systems

2012 ◽  
Vol 17 (2) ◽  
pp. 153-168
Author(s):  
Arnaud Heibig ◽  
Liviu Iulian Palade
Keyword(s):  
Complex Systems ◽  
Fractional Derivative ◽  
Existence Of Solutions ◽  
Diffusion Processes ◽  
Adjoint Operator ◽  
Von Neumann ◽  
Random Walker ◽  
Glassy Materials ◽  
And Diffusion ◽  
Position Probability

Fractional derivative equations account for relaxation and diffusion processes in a large variety of condensed matter systems. For instance, diffusion of position probability density displayed by a random walker in complex systems – such as glassy materials – is often modeled by fractional derivative partial differential equations. This paper deals with the existence of solutions to the general fractional derivative equation dαu/dtα+Au = f for 0 < α < 1, with A a self-adjoint operator. The results are proved using the von Neumann–Dixmier theorem.

Meridional Circulation, Turbulence, and Diffusion Processes in Stars

1976 ◽  
Vol 32 ◽  
pp. 109-116 ◽  
Author(s):  
S. Vauclair
Keyword(s):  
Convection Zone ◽  
Diffusion Processes ◽  
Meridional Circulation ◽  
Diffusion Time ◽  
Work In Progress ◽  
First Results ◽  
Stellar Envelopes ◽  
And Diffusion ◽  
Turbulence And Diffusion ◽  
Hr Diagram

This paper gives the first results of a work in progress, in collaboration with G. Michaud and G. Vauclair. It is a first attempt to compute the effects of meridional circulation and turbulence on diffusion processes in stellar envelopes. Computations have been made for a 2 Mʘstar, which lies in the Am - δ Scuti region of the HR diagram.Let us recall that in Am stars diffusion cannot occur between the two outer convection zones, contrary to what was assumed by Watson (1970, 1971) and Smith (1971), since they are linked by overshooting (Latour, 1972; Toomre et al., 1975). But diffusion may occur at the bottom of the second convection zone. According to Vauclair et al. (1974), the second convection zone, due to He II ionization, disappears after a time equal to the helium diffusion time, and then diffusion may happen at the bottom of the first convection zone, so that the arguments by Watson and Smith are preserved.

NMR-Study of Structure and Diffusion Processes in Li3N

1980 ◽  
Vol 41 (C6) ◽  
pp. C6-28-C6-31 ◽  
Author(s):  
R. Messer ◽  
H. Birli ◽  
K. Differt
Keyword(s):  
Diffusion Processes ◽  
Nmr Study ◽  
And Diffusion
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