scholarly journals On the Lyapunov Exponent of Monotone Boolean Networks †

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 1035
Author(s):  
Ilya Shmulevich
Keyword(s):  
Lyapunov Exponent ◽  
Dynamical Systems ◽  
Boolean Functions ◽  
Boolean Network ◽  
Discrete Dynamical Systems ◽  
Boolean Networks ◽  
Asymptotic Formulas ◽  
Small Perturbations ◽  
Monotone Boolean Functions ◽  
Almost All

Boolean networks are discrete dynamical systems comprised of coupled Boolean functions. An important parameter that characterizes such systems is the Lyapunov exponent, which measures the state stability of the system to small perturbations. We consider networks comprised of monotone Boolean functions and derive asymptotic formulas for the Lyapunov exponent of almost all monotone Boolean networks. The formulas are different depending on whether the number of variables of the constituent Boolean functions, or equivalently, the connectivity of the Boolean network, is even or odd.

scholarly journals The Influence of Canalization on the Robustness of Boolean Networks

2016 ◽  
Author(s):  
C. Kadelka ◽  
J. Kuipers ◽  
R. Laubenbacher
Keyword(s):  
Boolean Functions ◽  
Boolean Network ◽  
Network Models ◽  
Discrete Dynamical Systems ◽  
Boolean Networks ◽  
Molecular Networks ◽  
Weighted Sum ◽  
Computationally Efficient ◽  
Computational Tools ◽  
The Impact

AbstractTime- and state-discrete dynamical systems are frequently used to model molecular networks. This paper provides a collection of mathematical and computational tools for the study of robustness in Boolean network models. The focus is on networks governed by k-canalizing functions, a recently introduced class of Boolean functions that contains the well-studied class of nested canalizing functions. The activities and sensitivity of a function quantify the impact of input changes on the function output. This paper generalizes the latter concept to c-sensitivity and provides formulas for the activities and c-sensitivity of general k-canalizing functions as well as canalizing functions with more precisely defined structure. A popular measure for the robustness of a network, the Derrida value, can be expressed as a weighted sum of the c-sensitivities of the governing canalizing functions, and can also be calculated for a stochastic extension of Boolean networks. These findings provide a computationally efficient way to obtain Derrida values of Boolean networks, deterministic or stochastic, that does not involve simulation.

scholarly journals Chaotic Dynamical Systems Associated with Tilings of RN

2011 ◽  
pp. 19-41
Author(s):  
Lionel Rosier
Keyword(s):  
Dynamical System ◽  
Lyapunov Exponent ◽  
Dynamical Systems ◽  
Homogeneous Space ◽  
Chaos Synchronization ◽  
Discrete Dynamical Systems ◽  
Dimensional Torus ◽  
Synchronization Mechanism ◽  
Chaotic Dynamical Systems ◽  
Regular Tiling

In this chapter, we consider a class of discrete dynamical systems defined on the homogeneous space associated with a regular tiling of RN, whose most familiar example is provided by the N-dimensional torus TN. It is proved that any dynamical system in this class is chaotic in the sense of Devaney, and that it admits at least one positive Lyapunov exponent. Next, a chaos-synchronization mechanism is introduced and used for masking information in a communication setup.

Stable Index Pairs for Discrete Dynamical Systems

1997 ◽  
Vol 40 (4) ◽  
pp. 448-455 ◽  
Author(s):  
Tomasz Kaczynski ◽  
Marian Mrozek
Keyword(s):  
Dynamical Systems ◽  
Vector Fields ◽  
Discrete Dynamical Systems ◽  
Discrete Case ◽  
Smooth Vector ◽  
Small Perturbations ◽  
Index Pairs ◽  
Open Question ◽  
Stable Index

AbstractA new shorter proof of the existence of index pairs for discrete dynamical systems is given. Moreover, the index pairs defined in that proof are stable with respect to small perturbations of the generating map. The existence of stable index pairs was previously known in the case of diffeomorphisms and flows generated by smooth vector fields but it was an open question in the general discrete case.

Discrete dynamical systems on graphs and Boolean functions

2004 ◽  
Vol 66 (6) ◽  
pp. 487-497 ◽  
Author(s):  
Chris L Barrett ◽  
William Y.C Chen ◽  
Michelle J Zheng
Keyword(s):  
Dynamical Systems ◽  
Boolean Functions ◽  
Discrete Dynamical Systems

scholarly journals Parallel discrete dynamical systems on maxterm and minterm Boolean functions

2012 ◽  
Vol 55 (3-4) ◽  
pp. 666-671 ◽  
Author(s):  
J.A. Aledo ◽  
S. Martínez ◽  
F.L. Pelayo ◽  
Jose C. Valverde
Keyword(s):  
Dynamical Systems ◽  
Boolean Functions ◽  
Discrete Dynamical Systems

STABILITY AND CHAOS IN REACTION SYSTEMS

2012 ◽  
Vol 23 (05) ◽  
pp. 1173-1184 ◽  
Author(s):  
ANDRZEJ EHRENFEUCHT ◽  
MICHAEL MAIN ◽  
GRZEGORZ ROZENBERG ◽  
ALLISON THOMPSON BROWN
Keyword(s):  
Experimental Study ◽  
Dynamical Systems ◽  
Living Cell ◽  
Chaotic Behavior ◽  
Discrete Dynamical Systems ◽  
Abstract Model ◽  
Biochemical Reactions ◽  
Initial State ◽  
Small Perturbations ◽  
Reaction Systems

Reaction systems are an abstract model of biochemical reactions in the living cell within a framework of finite (though often large) discrete dynamical systems. In this setting, this paper provides an analytical and experimental study of stability. The notion of stability is defined in terms of the way in which small perturbations to the initial state of a system are likely to change the system's eventual behavior. At the stable end of the spectrum, there is likely to be no change; but at the unstable end, small perturbations take the system into a state that is probabilistically the same as a randomly selected state, similar to chaotic behavior in continuous dynamical systems.

scholarly journals Random Networks with Quantum Boolean Functions

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 792 ◽  
Author(s):  
Mario Franco ◽  
Octavio Zapata ◽  
David A. Rosenblueth ◽  
Carlos Gershenson
Keyword(s):  
Boolean Functions ◽  
Boolean Network ◽  
Complex Dynamics ◽  
Boolean Networks ◽  
Stable Complex ◽  
Complex Dynamic ◽  
Random Boolean Networks ◽  
Quantum Simulator ◽  
Novel Model ◽  
Unstable Dynamics

We propose quantum Boolean networks, which can be classified as deterministic reversible asynchronous Boolean networks. This model is based on the previously developed concept of quantum Boolean functions. A quantum Boolean network is a Boolean network where the functions associated with the nodes are quantum Boolean functions. We study some properties of this novel model and, using a quantum simulator, we study how the dynamics change in function of connectivity of the network and the set of operators we allow. For some configurations, this model resembles the behavior of reversible Boolean networks, while for other configurations a more complex dynamic can emerge. For example, cycles larger than 2N were observed. Additionally, using a scheme akin to one used previously with random Boolean networks, we computed the average entropy and complexity of the networks. As opposed to classic random Boolean networks, where “complex” dynamics are restricted mainly to a connectivity close to a phase transition, quantum Boolean networks can exhibit stable, complex, and unstable dynamics independently of their connectivity.

scholarly journals ROBUSTNESS OF TRANSCRIPTIONAL REGULATION IN YEAST-LIKE MODEL BOOLEAN NETWORKS

2010 ◽  
Vol 20 (03) ◽  
pp. 929-935 ◽  
Author(s):  
MURAT TUĞRUL ◽  
ALKAN KABAKÇIOĞLU
Keyword(s):  
Transcriptional Regulation ◽  
Boolean Functions ◽  
Boolean Network ◽  
Regulation Of Gene Expression ◽  
Boolean Networks ◽  
Transcription Level ◽  
Yeast Saccharomyces Cerevisiae ◽  
Boolean Network Model ◽  
Global Topology ◽  
Model Networks

We investigate the dynamical properties of the transcriptional regulation of gene expression in the yeast Saccharomyces Cerevisiae within the framework of a synchronously and deterministically updated Boolean network model. With a dynamically determinant subnetwork, we explore the robustness of transcriptional regulation as a function of the type of Boolean functions used in the model that mimic the influence of regulating agents on the transcription level of a gene. We compare the results obtained for the actual yeast network with those from two different model networks, one with similar in-degree distribution as the yeast and otherwise random, and another due to Balcan et al., where the global topology of the yeast network is reproduced faithfully. We, surprisingly, find that the first set of model networks better reproduce the results found with the actual yeast network, even though the Balcan et al. model networks are structurally more similar to that of yeast.

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