scholarly journals Nonuniform Chaotic Dynamics and Effects of Noise in Biochemical Systems

1987 ◽  
Vol 42 (2) ◽  
pp. 136-142 ◽  
Author(s):  
H. Herzel ◽  
W. Ebeling ◽  
Th. Schulmeister
Keyword(s):  
Chaotic Dynamics ◽  
Chaotic Systems ◽  
Quantitative Description ◽  
Deterministic Chaos ◽  
Correlated Noise ◽  
Sustained Oscillations ◽  
Exponential Separation ◽  
Biochemical Systems ◽  
Relative Robustness

Biochemical models capable of sustained oscillations and deterministic chaos are investigated. Chaos is characterized by exponential separation of near-by trajectories in the long-term average. However, we observed rather large deviations from purely exponential separation termed "nonuniformity". A quantitative description and consequences of nonuniformity are discussed.Furthermore, the influence of short-correlated noise is treated using next-amplitude maps and Lyapunov exponents. Drastic amplification of fluctuations in non-chaotic systems and relative robustness of chaos were found.

DETERMINISTIC CHAOS IN AN INTERPHASE LAYER OF A LIQUID–VAPOR SYSTEM

2004 ◽  
Vol 14 (10) ◽  
pp. 3671-3678
Author(s):  
G. P. BYSTRAI ◽  
S. I. IVANOVA ◽  
S. I. STUDENOK
Keyword(s):  
Phase Transitions ◽  
Chaotic Dynamics ◽  
Deterministic Chaos ◽  
Second Order ◽  
Homogeneous Layer ◽  
Time Interval ◽  
Evaporation And Condensation ◽  
Starting Conditions ◽  
Kolmogorov's Entropy ◽  
Dimensional Mapping

A second-order nonlinear differential equation with an aftereffect for the density of a thin homogeneous layer on a liquid and vapor interface is considered. The acts of evaporation and condensation of molecules, which are regarded as periodic "impacts", excite the layer. The mentioned NDE is integrated over a finite time interval to find a 2D (two-dimensional) mapping whose numerical solution describes the chaotic dynamics of density and pressure in time. The algorithms of constructing bifurcation diagrams, Lyapunov's exponents and Kolmogorov's entropy for systems with first-order, second-order phase transitions and Van der Waals' systems were elaborated. This approach allows to associate such concepts as phase transition, deterministic chaos and nonlinear processes. It also allows to answer a question whether deterministic chaos occurs in systems with phase transitions and how fast the information about starting conditions is lost within them.

scholarly journals Long-Term Amplitude and Period Variations of δ Scuti Stars: A Sign of Chaos?

1993 ◽  
Vol 134 ◽  
pp. 173-180
Author(s):  
M. Breger
Keyword(s):  
Time Scales ◽  
Chaotic Behavior ◽  
Deterministic Chaos ◽  
Observational Evidence ◽  
Mode Interaction ◽  
Mode Switching ◽  
Completely Regular ◽  
Long Term Changes ◽  
Scuti Stars

AbstractOn short time-scales of under a year, the vast majority of δ Scuti stars studied in detail show completely regular multiperiodic pulsation. Nonradial pulsation is characterized by the excitation of a large number of modes with small amplitudes. Reports of short-term irregularity or nonperiodicity in the literature need to be examined carefully, since insufficient observational data can lead to an incorrect impression of irregularity. Some interesting cases of reported irregularities are examined.A few δ Scuti stars, such as 21 Mon, have shown stable variations with sudden mode switching to a new frequency spectrum. This situation might be an indication of deterministic chaos. However, the observational evidence for mode switching is still weak.One the other hand, the case for the existence of long-term amplitude and period changes is becoming quite convincing. Recently found examples of nonradial pulsators with long-term changes are 4 CVn, 44 Tau, τ Peg and HD 2724. (We note that other δ Scuti pulsators such as X Cae and θ2 Tau, have shown no evidence for amplitude variations over the years.) Neither the amplitude nor the period changes are periodic, although irregular cycles with time scales between a few and twenty years can be seen. While the amplitude changes can be very large, the period changes are quite small. This property is common in nonlinear systems which lead to chaotic behavior. There exists observational evidence for relatively sudden period jumps changing the period by about 10−5 and/or slow period changes near dP/dt ≤ 10−9. These period changes are an order of magnitude larger than those expected from stellar evolution.The nonperiodic long-term changes are interpreted in terms of resonances between different nonradial modes. It is shown that a large number of the nonradial acoustic modes can be in resonance with other modes once the mode interaction terms, different radial orders and rotational m-mode splitting are considered. These resonances are illustrated numerically by the use of pulsation model. Observational evidence is presented that these interaction modes exist in the low-frequency domain.

ROBUSTNESS OF ADAPTATION IN CONTROLLED SELF-ADJUSTING CHAOTIC SYSTEMS

2002 ◽  
Vol 02 (04) ◽  
pp. L285-L292 ◽  
Author(s):  
PAUL MELBY ◽  
NICHOLAS WEBER ◽  
ALFRED HÜBLER
Keyword(s):  
Chaotic Dynamics ◽  
Chaotic Systems ◽  
Logistic Map ◽  
Loop Control ◽  
Edge Of Chaos ◽  
Target Value ◽  
Chua Circuit ◽  
Modified Logistic Map ◽  
Periodic Dynamics ◽  
Strong Adaptation

It was recently shown that self-adjusting systems adapt to the edge of chaos. We study the robustness of that adaptation with respect to a controlling force. We first use numerical simulations in a modified logistic map. With these, we find that, if the controlling force has a target value of the parameter that leads to periodic dynamics, the control is successful, even for very small controlling forces. We also find, however, that if the target value for the parameter leads to chaotic dynamics, the parameter resists the control and adaptation to the edge of chaos is still observed. When the controlling force is very strong, adaptation to the edge of chaos is weaker, but still present in the system. We also perform experiments with a self-adjusting Chua circuit and find the same behavior. We quantify these results with a measurement of the robustness of the adaptation as a function of the strength of the controlling force. The control used can be expressed either as a parametric control or as an additive, closed-loop control.

Stochastic Multiresonance in an Electronic Chaotic Circuit

2014 ◽  
Vol 24 (03) ◽  
pp. 1450027
Author(s):  
Thomas Stemler ◽  
Johannes P. Werner ◽  
Hartmut Benner
Keyword(s):  
Stochastic Resonance ◽  
Linear Response ◽  
Chaotic Dynamics ◽  
Chaotic Systems ◽  
Stochastic Systems ◽  
Electronic Circuit ◽  
Chaotic Circuit ◽  
Applied Method

Methods to estimate the amplification by stochastic resonance are tested in an electronic circuit experiment showing chaotic dynamics. We demonstrate that the linear response ansatz used for the estimation in stochastic systems can be also applied to chaotic systems showing crisis induced intermittency. In addition, the applied method explains the mechanism leading to stochastic multiresonance.

Stochastic processing of chaotic dynamics outcome of biomechanical parameters in human

2015 ◽  
Vol 4 (1) ◽  
pp. 31-39
Author(s):  
Берестин ◽  
D. Berestin ◽  
Игуменов ◽  
D. Igumenov ◽  
Рассадина ◽  
...  
Keyword(s):  
Distribution Function ◽  
Stochastic Analysis ◽  
Chaotic Dynamics ◽  
Involuntary Movement ◽  
Deterministic Chaos ◽  
Special Kind ◽  
Stochastic Processing ◽  
Biomechanical Parameters ◽  
Distribution Shifts ◽  
Non Parametric

The results of the stochastic analysis of postural tremor (as alleged involuntary movement) and tapping (as supposedly voluntary movement) are considered in a comparative perspective. It is proved that the stochastic analysis of the results of chaotic dynamics in tremorogramms and tapping does not give significant differences (absence of voluntariness). Typically, all samples are significantly different and it is impossible to distinguish subjects in their tremorogramms or tapingramm. The significant differences between sites of tremorogramms in terms of a normal distribution or a non-parametric distribution are demonstrated. A continuous variation of the distribution function is observed: parametric distribution shifts to non-parametric distribution, but among themselves they (distribution function) are all different. It is well known that the unpredicta-bility and continuing changes in the state are characteristic feature of chaos. The evidence of special kind of chaos in biosystems which differs significantly from the deterministic chaos of Tom – Ar-nold is given.

scholarly journals Sources of uncertainty in deterministic dynamics: an informal overview

2011 ◽  
Vol 369 (1956) ◽  
pp. 4705-4729 ◽  
Author(s):  
Ian Stewart
Keyword(s):  
Chaotic Dynamics ◽  
Chaotic Systems ◽  
Initial Conditions ◽  
Sources Of Uncertainty ◽  
Deterministic Systems ◽  
Coin Tossing ◽  
Random Switching ◽  
Butterfly Effect ◽  
Basin Boundaries ◽  
Sensitivity To Initial Conditions

The discovery of chaotic dynamics implies that deterministic systems may not be predictable in any meaningful sense. The best-known source of unpredictability is sensitivity to initial conditions (popularly known as the butterfly effect), in which small errors or disturbances grow exponentially. However, there are many other sources of uncertainty in nonlinear dynamics. We provide an informal overview of some of these, with an emphasis on the underlying geometry in phase space. The main topics are the butterfly effect, uncertainty in initial conditions in non-chaotic systems, such as coin tossing, heteroclinic connections leading to apparently random switching between states, topological complexity of basin boundaries, bifurcations (popularly known as tipping points) and collisions of chaotic attractors. We briefly discuss possible ways to detect, exploit or mitigate these effects. The paper is intended for non-specialists.

Fundamental Sources of Economic Complexity

2016 ◽  
Vol 17 (1) ◽  
pp. 1-13 ◽  
Author(s):  
Aleksander Jakimowicz
Keyword(s):  
Deterministic Chaos ◽  
Final State ◽  
Economic Complexity ◽  
Coexistence Of Attractors ◽  
Long Term Effect ◽  
Duopoly Model ◽  
Sensitive Dependence ◽  
Business Cycle Model ◽  
The Business Cycle

AbstractThis article analyses the basic sources and types of economic complexity: chaotic attractors and repellers, complexity catastrophes, coexistence of attractors, sensitive dependence on parameters, final state sensitivity, effects of fractal basin boundaries and chaotic saddles. Four nonlinear classic models have been used for this purpose: virtual duopoly model, model of a centrally planned economy, cobweb model with adaptive expectations and the business cycle model. The issue of economic complexity has not been sufficiently dealt with in the literature. Studies of complexity in economics usually focus on identifying the conditions under which deterministic chaos emerges in models as the main form of complexity, while analyses of other forms of complexity are much less frequent. The article has two objectives: methodological and explicative, which are to shed some new light on the issue. The first objective is to make as comprehensive a catalogue of sources of economic complexity as possible; this is to be achieved by the numerical calculations presented in this article. The issue of accumulation of complexity has been emphasized, which is a type of system dynamics which has its roots in coincidence and overlapping of complexity originating in different sources. The second objective involves an explanation of the role which is played in generating complexity by classic laws of economics. It appears that there is another overarching law, which is independent of the type of system or the level of economic analysis, which states that the long-term effect of conventional economic laws is an inevitable increase in the complexity of markets and economies. Therefore, the sources of complexity discussed in this article are called fundamental ones.

CONTROL STRATEGIES FOR CHAOTIC NEUROMODULES

1996 ◽  
Vol 06 (04) ◽  
pp. 693-703 ◽  
Author(s):  
NICO STOLLENWERK ◽  
FRANK PASEMANN
Keyword(s):  
Least Squares ◽  
Chaotic Dynamics ◽  
Chaotic Systems ◽  
Least Squares Method ◽  
Control Strategies ◽  
Nonlinear Least Squares ◽  
The Neural Network ◽  
One Step ◽  
Nonlinear Least Squares Method ◽  
The One

Different strategies for control of chaotic systems are discussed: The well known Ott-Grebogi-Yorke algorithm and two alternative algorithms based on least-squares minimisation of the one step future deviation. To compare their effectiveness in the neural network context they are applied to a minimal two neuron module with discrete chaotic dynamics. The best method with respect to calculation effort, to neural implementation, and to controlling properties is the nonlinear least squares method. Furthermore, it is observed in simulations that one can stabilise a whole periodic orbit by applying the control signals only to one of its periodic points, which lies in a distinguished region of phase space.

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