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.

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.

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 Facilitation in reaction systems

2020 ◽  
Vol 2 (3) ◽  
pp. 149-161
Author(s):  
Luca Manzoni ◽  
Antonio E. Porreca ◽  
Grzegorz Rozenberg
Keyword(s):  
Structural Properties ◽  
Living Cell ◽  
Formal Model ◽  
Biochemical Reactions ◽  
Reaction Systems ◽  
Model Of Computation ◽  
Dependency Graphs ◽  
Transition Graphs

Abstract Reaction systems is a formal model of computation which originated as a model of interactions between biochemical reactions in the living cell. These interactions are based on two mechanisms, facilitation and inhibition, and this is well reflected in the formulation of reaction systems. In this paper, we investigate the facilitation aspect of reaction systems, where the products of a reaction may facilitate other reactions by providing some of their reactants. This aspect is formalized through positive dependency graphs which depict explicitly such facilitating interactions. The focus of the paper is on demonstrating how structural properties of reaction systems defined through the properties of their positive dependency graphs influence the behavioural properties of (suitable subclasses of) reaction systems, which, as usual, are defined through their transition graphs.

Discrete Time Chaotic Maps With Application to Random Bit Generation

2021 ◽  
pp. 542-582
Author(s):  
Lazaros Moysis ◽  
Ahmad Taher Azar ◽  
Aleksandra Tutueva ◽  
Denis N. Butusov ◽  
Christos Volos
Keyword(s):  
Dynamical Systems ◽  
Discrete Time ◽  
Chaotic Behavior ◽  
Initial Conditions ◽  
Statistical Tests ◽  
Solution Space ◽  
Small Perturbations ◽  
Highly Sensitive ◽  
Different Shapes ◽  
Parameter Values

Chaotic behavior is a term that is attributed to dynamical systems whose solutions are highly sensitive to initial conditions. This means that small perturbations in the initial conditions can lead to completely different trajectories in the solution space. These types of chaotic dynamical systems arise in various natural or artificial systems in biology, circuits, engineering, computer science, and more. This chapter reports on some new chaotic discrete time two-dimensional maps that are derived from simple modifications to the well-known Hénon, Lozi, Sine-Sine, and Tinkerbell maps. Numerical simulations are carried out for different parameter values and initial conditions, and it is shown that the mappings either diverge to infinity or converge to attractors of many different shapes. The application to random bit generation is then considered using a collection of the proposed maps by applying a simple rule. The resulting bit generator successfully passes all statistical tests performed.

APPLICATION OF ORDER PARAMETER EQUATIONS FOR THE ANALYSIS AND THE CONTROL OF NONLINEAR TIME DISCRETE DYNAMICAL SYSTEMS

1999 ◽  
Vol 09 (08) ◽  
pp. 1618-1634 ◽  
Author(s):  
P. LEVI ◽  
M. SCHANZ ◽  
S. KORNIENKO ◽  
O. KORNIENKO
Keyword(s):  
Dynamical Systems ◽  
Order Parameter ◽  
Evolution Equations ◽  
Chaotic Behavior ◽  
Period Doubling ◽  
Original System ◽  
Discrete Dynamical Systems ◽  
Order Parameters ◽  
Control Mechanisms ◽  
Manifold Theory

This work is based on the concept of order parameters of synergetics. The order parameter equations describe the behavior of a system in the vicinity of an instability and are used here not only for the analysis but also for the control of nonlinear time discrete dynamical systems. Usually, the dimensionality of the evolution equations of the order parameters is less than the dimensionality of the original evolution equations. It is, therefore, convenient to introduce control mechanisms, first in the order parameter equations, and then to use the obtained results for the control of the original system. The aim of the control in this case is to avoid chaotic behavior of the system. This is achieved by shifting appropriate bifurcation points of a period-doubling cascade. In this work we concentrate on the shifting of only the first bifurcation point. The used control mechanisms are delayed feedback schemes. As an example the well-known Hénon map is investigated. The order parameter equation is calculated using both the adiabatic elimination procedure and the center manifold theory. Using the order parameter concept two types of control mechanisms are constructed, analyzed and compared.

New Discrete Time 2D Chaotic Maps

2017 ◽  
Vol 6 (1) ◽  
pp. 77-104 ◽  
Author(s):  
Lazaros Moysis ◽  
Ahmad Taher Azar
Keyword(s):  
Dynamical Systems ◽  
Discrete Time ◽  
Chaotic Behavior ◽  
Initial Conditions ◽  
Single Point ◽  
Solution Space ◽  
Small Perturbations ◽  
Highly Sensitive ◽  
Different Shapes ◽  
Parameter Values

Chaotic behavior is a term that is attributed to dynamical systems whose solutions are highly sensitive to initial conditions. This means that small perturbations in the initial conditions can lead to completely different trajectories in the solution space. These types of chaotic dynamical systems arise in various natural or artificial systems in biology, meteorology, economics, electrical circuits, engineering, computer science and more. Of these innumerable chaotic systems, perhaps the most interesting are those that exhibit attracting behavior. By that, the authors refer to systems whose trajectories converge with time to a set of values, called an attractor. This can be a single point, a curve or a manifold. The attractor is called strange if it is a set with fractal structure. Such systems can be both continuous and discrete. This paper reports on some new chaotic discrete time two dimensional maps that are derived from simple modifications to the well-known Hénon, Lozi, Sine-sine and Tinkerbell maps. Numerical simulations are carried out for different parameter values and initial conditions and it is shown that the mappings either diverge to infinity or converge to attractors of many different shapes.

scholarly journals Controlling nonlinear dynamical systems into arbitrary states using machine learning

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Alexander Haluszczynski ◽  
Christoph Räth
Keyword(s):  
Machine Learning ◽  
Dynamical Systems ◽  
Chaotic Systems ◽  
Nonlinear Dynamical Systems ◽  
Chaotic Behavior ◽  
Large Data ◽  
Data Sets ◽  
Initial State ◽  
Nonlinear Dynamical ◽  
Phase Space Methods

AbstractControlling nonlinear dynamical systems is a central task in many different areas of science and engineering. Chaotic systems can be stabilized (or chaotified) with small perturbations, yet existing approaches either require knowledge about the underlying system equations or large data sets as they rely on phase space methods. In this work we propose a novel and fully data driven scheme relying on machine learning (ML), which generalizes control techniques of chaotic systems without requiring a mathematical model for its dynamics. Exploiting recently developed ML-based prediction capabilities, we demonstrate that nonlinear systems can be forced to stay in arbitrary dynamical target states coming from any initial state. We outline and validate our approach using the examples of the Lorenz and the Rössler system and show how these systems can very accurately be brought not only to periodic, but even to intermittent and different chaotic behavior. Having this highly flexible control scheme with little demands on the amount of required data on hand, we briefly discuss possible applications ranging from engineering to medicine.

scholarly journals Deterministic Discrete Dynamical Systems: Advances in Regular and Chaotic Behavior with Applications

2013 ◽  
Vol 2013 ◽  
pp. 1-2
Author(s):  
Raghib Abu-Saris ◽  
Fathi Allan ◽  
Mustafa Kulenovic ◽  
Alfredo Peris
Keyword(s):  
Dynamical Systems ◽  
Chaotic Behavior ◽  
Discrete Dynamical Systems

scholarly journals On Sufficient Conditions for Chaotic Behavior of Multidimensional Discrete Time Dynamical System

2021 ◽  
Author(s):  
Nor Syahmina Kamarudin ◽  
Syahida Che Dzul-Kifli
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Discrete Time ◽  
Chaotic Behavior ◽  
Discrete Dynamical System ◽  
Sufficient Conditions ◽  
Discrete Dynamical Systems ◽  
Continuous Map ◽  
Discrete Time Dynamical System ◽  
Periodicity Property

AbstractIn this work, we look at the extension of classical discrete dynamical system to multidimensional discrete-time dynamical system by characterizing chaos notions on $${\mathbb {Z}}^d$$ Z d -action. The $${\mathbb {Z}}^d$$ Z d -action on a space X has been defined in a very general manner, and therefore we introduce a $${\mathbb {Z}}^d$$ Z d -action on X which is induced by a continuous map, $$f:{\mathbb {Z}}\times X \rightarrow X$$ f : Z × X → X and denotes it as $$T_f:{\mathbb {Z}}^d \times X \rightarrow X$$ T f : Z d × X → X . Basically, we wish to relate the behavior of origin discrete dynamical systems (X, f) and its induced multidimensional discrete-time $$(X,T_f)$$ ( X , T f ) . The chaotic behaviors that we emphasized are the transitivity and dense periodicity property. Analogues to these chaos notions, we consider k-type transitivity and k-type dense periodicity property in the multidimensional discrete-time dynamical system. In the process, we obtain some conditions on $$(X,T_f)$$ ( X , T f ) under which the chaotic behavior of $$(X,T_f)$$ ( X , T f ) is inherited from the original dynamical system (X, f). The conditions varies whenever f is open, totally transitive or mixing. Some examples are given to illustrate these conditions.

Definitions—Background

2019 ◽  
pp. 43-63
Author(s):  
Pierre-Loïc Garoche
Keyword(s):  
Dynamical System ◽  
Signal Processing ◽  
Dynamical Systems ◽  
Convex Optimization ◽  
Physical System ◽  
Discrete Dynamical Systems ◽  
Target Language ◽  
Initial State ◽  
Typical Object ◽  
Tools And Methods

This chapter presents the formalisms describing discrete dynamical systems and gives an overview on the convex optimization tools and methods used to compute the analyses. A dynamical system is a typical object used in control systems or in signal processing. In some cases, it is eventually implemented in a program to perform the desired feedback control to a cyber-physical system. Language-wise, model-based languages such as LUSTRE, ANSYS SCADE, or MATLAB Simulink provide primitives to build these dynamical systems or controllers relying on simpler constructs. In terms of programs, such dynamical systems can easily be implemented as a “while true loop” initialized by the initial state and performing the update f. The simplest systems are usually directly coded in the target language, while more advanced systems are compiled through autocoders.

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