scholarly journals STUDY OF SYNCHRONIZED MOTIONS IN A ONE-DIMENSIONAL ARRAY OF COUPLED CHAOTIC CIRCUITS

1999 ◽  
Vol 09 (11) ◽  
pp. 2219-2224 ◽  
Author(s):  
ZBIGNIEW GALIAS ◽  
MACIEJ J. OGORZAŁEK
Keyword(s):  
Lyapunov Exponents ◽  
Coupling Strength ◽  
Numerical Experiments ◽  
Variational Equation ◽  
Electronic Circuits ◽  
One Dimensional ◽  
Synchronous Motion ◽  
Chaotic Circuits ◽  
The Stability ◽  
Theoretical Results

We investigate the stability of synchronous motion in an array of bidirectionally coupled electronic circuits. We compute Lyapunov exponents of the generic variational equation associated with directions transversal to the synchronization subspace. Using Lyapunov exponents we derive conditions for the coupling strength for which the stable synchronous solution exists. We also find the limit on the size of the network, which can sustain stable synchronous motion. Theoretical results are compared with the results of numerical experiments.

OBSERVATIONS OF PHASE SYNCHRONIZATION PHENOMENA IN ONE-DIMENSIONAL ARRAYS OF COUPLED CHAOTIC ELECTRONIC CIRCUITS

2000 ◽  
Vol 10 (10) ◽  
pp. 2391-2398 ◽  
Author(s):  
ANDRZEJ DABROWSKI ◽  
ZBIGNIEW GALIAS ◽  
MACIEJ OGORZAŁEK
Keyword(s):  
Numerical Experiments ◽  
Phase Synchronization ◽  
Analytic Signal ◽  
Electronic Circuits ◽  
One Dimensional ◽  
Phase Plot ◽  
Chaotic Circuits ◽  
Signal Approach ◽  
Insight Into

Using numerical experiments we show that the phase synchronization concept enables better insight into the synchronization phenomena encountered in coupled nonlinear chaotic circuits. In some cases when the phase plot inspection does not allow to confirm synchrony such kind of behavior can be distinguished by inspection of the phase calculated using the analytic signal approach.

scholarly journals Hybrid Passivity Based and Fuzzy Type-2 Controller for Chaotic and Hyper-Chaotic Systems

2017 ◽  
Vol 11 (2) ◽  
pp. 96-103 ◽  
Author(s):  
Fernando Serrano ◽  
Josep M. Rossell
Keyword(s):  
Lyapunov Exponents ◽  
Chaotic System ◽  
Control Strategy ◽  
Chaotic Systems ◽  
Control Law ◽  
Four Dimensions ◽  
Design Requirements ◽  
The Stability ◽  
Theoretical Results

AbstractIn this paper a hybrid passivity based and fuzzy type-2 controller for chaotic and hyper-chaotic systems is presented. The proposed control strategy is an appropriate choice to be implemented for the stabilization of chaotic and hyper-chaotic systems due to the energy considerations of the passivity based controller and the flexibility and capability of the fuzzy type-2 controller to deal with uncertainties. As it is known, chaotic systems are those kinds of systems in which one of their Lyapunov exponents is real positive, and hyper-chaotic systems are those kinds of systems in which more than one Lyapunov exponents are real positive. In this article one chaotic Lorentz attractor and one four dimensions hyper-chaotic system are considered to be stabilized with the proposed control strategy. It is proved that both systems are stabilized by the passivity based and fuzzy type-2 controller, in which a control law is designed according to the energy considerations selecting an appropriate storage function to meet the passivity conditions. The fuzzy type-2 controller part is designed in order to behave as a state feedback controller, exploiting the flexibility and the capability to deal with uncertainties. This work begins with the stability analysis of the chaotic Lorentz attractor and a four dimensions hyper-chaotic system. The rest of the paper deals with the design of the proposed control strategy for both systems in order to design an appropriate controller that meets the design requirements. Finally, numerical simulations are done to corroborate the obtained theoretical results.

Steady-State Bifurcation for a Biological Depletion Model

2016 ◽  
Vol 26 (04) ◽  
pp. 1650066 ◽  
Author(s):  
Yan’e Wang ◽  
Jianhua Wu ◽  
Yunfeng Jia
Keyword(s):  
Steady State ◽  
Bifurcation Theory ◽  
Steady States ◽  
Two Dimensional ◽  
Implicit Function ◽  
Flux Boundary Condition ◽  
One Dimensional ◽  
Steady State Bifurcation ◽  
The Stability ◽  
Theoretical Results

A two-species biological depletion model in a bounded domain is investigated in which one species is a substrate and the other is an activator. Firstly, under the no-flux boundary condition, the asymptotic stability of constant steady-states is discussed. Secondly, by viewing the feed rate of the substrate as a parameter, the steady-state bifurcations from constant steady-states are analyzed both in one-dimensional kernel case and in two-dimensional kernel case. Finally, numerical simulations are presented to illustrate our theoretical results. The main tools adopted here include the stability theory, the bifurcation theory, the techniques of space decomposition and the implicit function theorem.

Learning Control of Time-Delay Chaotic Systems and its Applications

1998 ◽  
Vol 08 (12) ◽  
pp. 2457-2465 ◽  
Author(s):  
Keiji Konishi ◽  
Hideki Kokame
Keyword(s):  
Control System ◽  
Time Delay ◽  
Chaotic System ◽  
Numerical Experiments ◽  
Chaotic Systems ◽  
Learning Control ◽  
Uncertain Information ◽  
Stable Orbit ◽  
One Dimensional ◽  
Theoretical Results

The present paper proposes a learning control system that automatically stabilizes one-dimensional time-delayed chaotic systems. We give a systematic procedure to design the control system using a few pieces of uncertain information on the chaotic system. Furthermore, this control system can be applied to a technique that moves a stable orbit of nonchaotic systems to coexistent unstable points and stabilizes the moving orbit onto these unstable points. To check the theoretical results, we demonstrate some numerical experiments.

Analysis of Geometrically Consistent Schemes with Finite Range Interaction

2017 ◽  
Vol 22 (5) ◽  
pp. 1333-1361
Author(s):  
Hongliang Li ◽  
Pingbing Ming
Keyword(s):  
Error Estimate ◽  
Numerical Experiments ◽  
Optimal Error Estimate ◽  
Finite Range ◽  
Dimensional Problem ◽  
Optimal Error ◽  
Range Interaction ◽  
One Dimensional ◽  
Sharp Stability ◽  
Theoretical Results

AbstractWe analyze the geometrically consistent schemes proposed by E. Lu and Yang [6] for one-dimensional problem with finite range interaction. The existence of the reconstruction coefficients is proved, and optimal error estimate is derived under sharp stability condition. Numerical experiments are performed to confirm the theoretical results.

scholarly journals Synchronization of Intermittently Coupled Dynamical Networks

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Gang Zhang ◽  
Guanrong Chen
Keyword(s):  
Coupling Strength ◽  
Stability Theory ◽  
Lorenz System ◽  
Impulsive Differential Equations ◽  
Time Varying ◽  
Coupling Matrix ◽  
Dynamical Networks ◽  
Dynamical Network ◽  
The Stability ◽  
Theoretical Results

This paper investigates the synchronization phenomenon of an intermittently coupled dynamical network in which the coupling among nodes can occur only at discrete instants and the coupling configuration of the network is time varying. A model of intermittently coupled dynamical network consisting of identical nodes is introduced. Based on the stability theory for impulsive differential equations, some synchronization criteria for intermittently coupled dynamical networks are derived. The network synchronizability is shown to be related to the second largest and the smallest eigenvalues of the coupling matrix, the coupling strength, and the impulsive intervals. Using the chaotic Chua system and Lorenz system as nodes of a dynamical network for simulation, respectively, the theoretical results are verified and illustrated.

scholarly journals Stability of Exact and Discrete Energy for Non-Fickian Reaction-Diffusion Equations with a Variable Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Dongfang Li ◽  
Chao Tong ◽  
Jinming Wen
Keyword(s):  
Numerical Experiments ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Variable Delay ◽  
Discrete Energy ◽  
Continuous Problems ◽  
Rate Of Decay ◽  
The Stability ◽  
Theoretical Results

This paper is concerned with the stability of non-Fickian reaction-diffusion equations with a variable delay. It is shown that the perturbation of the energy function of the continuous problems decays exponentially, which provides a more accurate and convenient way to express the rate of decay of energy. Then, we prove that the proposed numerical methods are sufficient to preserve energy stability of the continuous problems. We end the paper with some numerical experiments on a biological model to confirm the theoretical results.

The global dynamics of a discrete nonlinear hierarchical population system

2019 ◽  
Vol 12 (02) ◽  
pp. 1950022
Author(s):  
Ze-Rong He ◽  
Huai Chen ◽  
Shu-Ping Wang
Keyword(s):  
Dynamical Systems ◽  
Lyapunov Function ◽  
Population Model ◽  
Numerical Experiments ◽  
Discrete Dynamical Systems ◽  
Reproductive Number ◽  
Global Dynamics ◽  
Population System ◽  
The Stability ◽  
Theoretical Results

This paper is concerned with the global dynamics of a hierarchical population model, in which the fertility of an individual depends on the total number of higher-ranking members. We investigate the stability of equilibria, nonexistence of periodic orbits and the persistence of the population by means of eigenvalues, Lyapunov function, and several results in discrete dynamical systems. Our work demonstrates that the reproductive number governs the evolution of the population. Besides the theoretical results, some numerical experiments are also presented.

The Mass-Preserving S-DDM Scheme for Two-Dimensional Parabolic Equations

2016 ◽  
Vol 19 (2) ◽  
pp. 411-441 ◽  
Author(s):  
Zhongguo Zhou ◽  
Dong Liang
Keyword(s):  
Domain Decomposition ◽  
Parabolic Equations ◽  
Numerical Experiments ◽  
Decomposition Method ◽  
Domain Decomposition Method ◽  
Time Step ◽  
Unconditionally Stable ◽  
One Dimensional ◽  
Splitting Technique ◽  
Theoretical Results

AbstractIn the paper, we develop and analyze a new mass-preserving splitting domain decomposition method over multiple sub-domains for solving parabolic equations. The domain is divided into non-overlapping multi-bock sub-domains. On the interfaces of sub-domains, the interface fluxes are computed by the semi-implicit (explicit) flux scheme. The solutions and fluxes in the interiors of sub-domains are computed by the splitting one-dimensional implicit solution-flux coupled scheme. The important feature is that the proposed scheme is mass conservative over multiple non-overlapping sub-domains. Analyzing the mass-preserving S-DDM scheme is difficult over non-overlapping multi-block sub-domains due to the combination of the splitting technique and the domain decomposition at each time step. We prove theoretically that our scheme satisfies conservation of mass over multi-block non-overlapping sub-domains and it is unconditionally stable. We further prove the convergence and obtain the error estimate in L2-norm. Numerical experiments confirm theoretical results.

scholarly journals On the Application of the Generalized Means to Construct Multiresolution Schemes Satisfying Certain Inequalities Proving Stability

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 533
Author(s):  
Sergio Amat ◽  
Alberto Magreñan ◽  
Juan Ruiz ◽  
Juan Carlos Trillo ◽  
Dionisio F. Yañez
Keyword(s):  
Numerical Experiments ◽  
Order Of Approximation ◽  
Two Dimensional ◽  
Nonlinear Subdivision ◽  
Proper Definition ◽  
The Stability ◽  
Definition Of ◽  
Stability Issues ◽  
Multiresolution Schemes ◽  
Theoretical Results

Multiresolution representations of data are known to be powerful tools in data analysis and processing, and they are particularly interesting for data compression. In order to obtain a proper definition of the edges, a good option is to use nonlinear reconstructions. These nonlinear reconstruction are the heart of the prediction processes which appear in the definition of the nonlinear subdivision and multiresolution schemes. We define and study some nonlinear reconstructions based on the use of nonlinear means, more in concrete the so-called Generalized means. These means have two interesting properties that will allow us to get associated reconstruction operators adapted to the presence of discontinuities, and having the maximum possible order of approximation in smooth areas. Once we have these nonlinear reconstruction operators defined, we can build the related nonlinear subdivision and multiresolution schemes and prove more accurate inequalities regarding the contractivity of the scheme for the first differences and in turn the results about stability. In this paper, we also define a new nonlinear two-dimensional multiresolution scheme as non-separable, i.e., not based on tensor product. We then present the study of the stability issues for the scheme and numerical experiments reinforcing the proven theoretical results and showing the usefulness of the algorithm.

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