Universality in the coarse-grained fluctuations for a class of linear dynamical systems

2018 ◽  
Vol 503 ◽  
pp. 215-220
Author(s):  
Sándor Kajántó ◽  
Zoltán Néda
Keyword(s):  
Dynamical Systems ◽  
Coarse Grained ◽  
Linear Dynamical Systems

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.

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.

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