Controlling Bifurcation in Electronic Implementation of Dynamical Systems

2021 ◽  
Vol 31 (06) ◽  
pp. 2150084
Author(s):  
L. R. Villa-Salas ◽  
L. J. Ontañón-García ◽  
M. T. Ramírez-Torres ◽  
J. Pena-Ramirez
Keyword(s):  
Dynamical Systems ◽  
Chaotic Systems ◽  
Experimental Studies ◽  
Bifurcation Parameter ◽  
Electronic Components ◽  
Rossler System ◽  
Arduino Uno ◽  
Parameter Values ◽  
Electronic Implementation ◽  
Topological Changes

In the theoretical and experimental studies of bifurcations in dynamical systems, the adjustments of the parameter values play a key role. The reason is because small variations in these values may result in topological changes in the behavior of the flow of the system. Taking this into account, in this paper, a new design for controlling bifurcation, suitable for electronic implementations of chaotic systems, is presented. The variation of the bifurcation parameter is performed by means of an Arduino UNO micro-controller and a digital controlled potentiometer. In this way, the variation of the electronic components is performed in an automated manner, avoiding the intrinsic problems of a manual variation of the circuit parameters. As a particular example, a scaled Rössler system is considered. One of the advantages of the controlled automated bifurcation is that it is useful for analyzing the robustness of the different limiting behaviors of the system against parameter mismatches.

scholarly journals Finite Cascades of Pitchfork Bifurcations and Multistability in Generalized Lorenz-96 Models

2020 ◽  
Vol 25 (4) ◽  
pp. 78
Author(s):  
Anouk F. G. Pelzer ◽  
Alef E. Sterk
Keyword(s):  
Dynamical Systems ◽  
Scaling Law ◽  
Chaotic Attractors ◽  
Bifurcation Parameter ◽  
Coexisting Attractors ◽  
Pitchfork Bifurcations ◽  
Primary Focus ◽  
Divisibility Properties ◽  
Lorenz 96 ◽  
Parameter Values

In this paper, we study a family of dynamical systems with circulant symmetry, which are obtained from the Lorenz-96 model by modifying its nonlinear terms. For each member of this family, the dimension n can be arbitrarily chosen and a forcing parameter F acts as a bifurcation parameter. The primary focus in this paper is on the occurrence of finite cascades of pitchfork bifurcations, where the length of such a cascade depends on the divisibility properties of the dimension n. A particularly intriguing aspect of this phenomenon is that the parameter values F of the pitchfork bifurcations seem to satisfy the Feigenbaum scaling law. Further bifurcations can lead to the coexistence of periodic or chaotic attractors. We also describe scenarios in which the number of coexisting attractors can be reduced through collisions with an equilibrium.

scholarly journals A New Chaotic Mixer Design Based on the Delta Robot and Its Experimental Studies

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Onur Kalayci ◽  
Ihsan Pehlivan ◽  
Akif Akgul ◽  
Selcuk Coskun ◽  
Ersin Kurt
Keyword(s):  
Distribution Ratio ◽  
Chaotic Systems ◽  
Liquid Mixture ◽  
Dynamic Properties ◽  
Mixing Time ◽  
Experimental Studies ◽  
Chaotic Mixing ◽  
Position Information ◽  
Delta Robot ◽  
Arduino Uno

In this study, a new chaotic mixer based on the Delta robot was designed and produced which had been controlled with Arduino Uno card and MATLAB. First of all, chaotic mixing systems with different dynamic properties were chosen for the chaotic mixing process. Then, by solving the chaotic systems selected in the MATLAB with the Runge Kutta 45 (RK45) numerical solution algorithm, the results in the integer format were obtained. The obtained chaotic time-series results were transformed into 3-dimensional position information for the servomotors used in the mixer with the algorithm developed in MATLAB. The supervision was provided to ensure that the newly designed chaotic mixer was pacing chaotically in x, y, and z coordinates by transferring the chaotic position information to the Arduino Uno R3 card via USB 2.0. With the software developed in MATLAB, the performances of 7 diversified chaotic systems’ trajectories and circular motion trajectories were compared over the numerical simulation orbital distribution ratio (ODR). In the final stage, in a solid-liquid mixture type, at the selected constant mixing time, experimental studies were performed where homogeneity and orbital distribution ratio (ODR) parameters were compared by using 7 diversified chaotic systems. The designed and produced chaotic mixer can also be used in experimental studies of certain liquid-liquid mixture types. It is thought that this prototype presented in the article will serve the aim of developing new chaotic mixer systems and algorithms to derive more homogeneous mixtures in a shorter time.

Non Linear Dynamical Systems and Chaos Synchronization

2010 ◽  
Vol 1 (2) ◽  
pp. 43-57
Author(s):  
Ayub Khan ◽  
Prempal Singh
Keyword(s):  
Dynamical Systems ◽  
Chaos Synchronization ◽  
Chaotic Systems ◽  
Three Dimensional ◽  
Linear Feedback ◽  
Algebraic Form ◽  
Linear Dynamical Systems ◽  
Non Linear ◽  
Linear State ◽  
Parameter Values

In this paper, the authors study chaos synchronization of chaotic systems, which can exhibit a two scroll attractor for different parameter values via linear feedback control. First, chaos synchronization of three dimensional systems is studied and ‘generalized non-linear dynamical systems’ are analyzed. The considered synchronization criterion consists of identical drive and response systems coupled with linear state error variables. As a consequence, the authors have proposed some theorems for synchronization. This paper features sufficient synchronization criteria for the linear coupled generalized non-linear dynamical systems obtained in an explicit algebraic form and the new synchronization criteria for some typical chaotic systems. Finally, the optimized criteria are applied to explain the Rossler system.

scholarly journals Differential Transform Method as an Effective Tool for Investigating Fractional Dynamical Systems

2021 ◽  
Vol 11 (15) ◽  
pp. 6955
Author(s):  
Andrzej Rysak ◽  
Magdalena Gregorczyk
Keyword(s):  
Optimal Parameter ◽  
Differential Transform Method ◽  
Bifurcation Diagrams ◽  
Transform Method ◽  
Rossler System ◽  
Differential Transform ◽  
Rössler System ◽  
Fractional System ◽  
Parameter Values

This study investigates the use of the differential transform method (DTM) for integrating the Rössler system of the fractional order. Preliminary studies of the integer-order Rössler system, with reference to other well-established integration methods, made it possible to assess the quality of the method and to determine optimal parameter values that should be used when integrating a system with different dynamic characteristics. Bifurcation diagrams obtained for the Rössler fractional system show that, compared to the RK4 scheme-based integration, the DTM results are more resistant to changes in the fractionality of the system.

A UNIFIED FRAMEWORK FOR SYNCHRONIZATION AND CONTROL OF DYNAMICAL SYSTEMS

1994 ◽  
Vol 04 (04) ◽  
pp. 979-998 ◽  
Author(s):  
CHAI WAH WU ◽  
LEON O. CHUA
Keyword(s):  
Dynamical Systems ◽  
Chaotic Systems ◽  
Main Tool ◽  
Lyapunov Stability Theory ◽  
Asymptotical Stability ◽  
Unified Framework ◽  
Partial Synchronization ◽  
Chua’S Oscillator ◽  
And Control

In this paper, we give a framework for synchronization of dynamical systems which unifies many results in synchronization and control of dynamical systems, in particular chaotic systems. We define concepts such as asymptotical synchronization, partial synchronization and synchronization error bounds. We show how asymptotical synchronization is related to asymptotical stability. The main tool we use to prove asymptotical stability and synchronization is Lyapunov stability theory. We illustrate how many previous results on synchronization and control of chaotic systems can be derived from this framework. We will also give a characterization of robustness of synchronization and show that master-slave asymptotical synchronization in Chua’s oscillator is robust.

Noninvertible Piecewise Linear Maps Applied to Chaos Synchronization and Secure Communications

1997 ◽  
Vol 07 (07) ◽  
pp. 1617-1634 ◽  
Author(s):  
G. Millerioux ◽  
C. Mira
Keyword(s):  
Dynamical Systems ◽  
Chaos Synchronization ◽  
Chaotic Systems ◽  
Piecewise Linear ◽  
A Priori ◽  
Secure Communications ◽  
Technological Parameters ◽  
Linear Maps ◽  
Piecewise Linear Maps ◽  
Necessary And Sufficient

Recently, it was demonstrated that two chaotic dynamical systems can synchronize each other, leading to interesting applications as secure communications. We propose in this paper a special class of dynamical systems, noninvertible discrete piecewise linear, emphasizing on interesting advantages they present compared with continuous and differentiable nonlinear ones. The generic aspect of such systems, the simplicity of numerical implementation, and the robustness to mismatch of technological parameters make them good candidates. The classical concept of controllability in the control theory is presented and used in order to choose and predict the number of appropriate variables to be transmitted for synchronization. A necessary and sufficient condition of chaotic synchronization is established without computing numerical quantities, introducing a state affinity structure of chaotic systems which provides an a priori establishment of synchronization.

scholarly journals Numerical detection of unstable periodic orbits in continuous-time dynamical systems with chaotic behaviors

2007 ◽  
Vol 14 (5) ◽  
pp. 615-620 ◽  
Author(s):  
Y. Saiki
Keyword(s):  
Dynamical Systems ◽  
Periodic Orbits ◽  
Infinite Number ◽  
Continuous Time ◽  
Chaotic Systems ◽  
Lorenz System ◽  
Complex Phenomenon ◽  
Unstable Periodic Orbits ◽  
Newton Raphson ◽  
The Lorenz System

Abstract. An infinite number of unstable periodic orbits (UPOs) are embedded in a chaotic system which models some complex phenomenon. Several algorithms which extract UPOs numerically from continuous-time chaotic systems have been proposed. In this article the damped Newton-Raphson-Mees algorithm is reviewed, and some important techniques and remarks concerning the practical numerical computations are exemplified by employing the Lorenz system.

Modal Interactions in Dynamical and Structural Systems

1989 ◽  
Vol 42 (11S) ◽  
pp. S175-S201 ◽  
Author(s):  
A. H. Nayfeh ◽  
B. Balachandran
Keyword(s):  
Dynamical Systems ◽  
Surface Waves ◽  
Nonlinear Response ◽  
Experimental Studies ◽  
Structural Systems ◽  
Qualitative Behavior ◽  
Modal Interactions ◽  
Chaotic Motions ◽  
Beam Structures ◽  
Quadratic Nonlinearities

We review theoretical and experimental studies of the influence of modal interactions on the nonlinear response of harmonically excited structural and dynamical systems. In particular, we discuss the response of pendulums, ships, rings, shells, arches, beam structures, surface waves, and the similarities in the qualitative behavior of these systems. The systems are characterized by quadratic nonlinearities which may lead to two-to-one and combination autoparametric resonances. These resonances give rise to a coupling between the modes involved in the resonance leading to nonlinear periodic, quasi-periodic, and chaotic motions.

ENERGY-LIKE FUNCTIONS FOR SOME DISSIPATIVE CHAOTIC SYSTEMS

2005 ◽  
Vol 15 (08) ◽  
pp. 2507-2521 ◽  
Author(s):  
C. SARASOLA ◽  
A. D'ANJOU ◽  
F. J. TORREALDEA ◽  
A. MOUJAHID
Keyword(s):  
Phase Space ◽  
Dynamic Behavior ◽  
Fourth Order ◽  
Chaotic Systems ◽  
Quantitative Measure ◽  
Energy Functions ◽  
Dissipation Of Energy ◽  
Rossler System ◽  
Order Polynomial ◽  
Fourth Order Polynomial

Functions of the phase space variables that can considered as possible energy functions for a given family of dissipative chaotic systems are discussed. This kind of functions are interesting due to their use as an energy-like quantitative measure to characterize different aspects of dynamic behavior of associated chaotic systems. We have calculated quadratic energy-like functions for the cases of Lorenz, Chen, Lü–Chen and Chua, and show the patterns of dissipation of energy on their respective attractors. We also show that in the case of the Rössler system at least a fourth-order polynomial is required to properly represent its energy.

A Consistent Approach for the Development of a Comprehensive Data Base of Time-Dependent Parameters for Concrete Engineered Barriers

2013 ◽  
Author(s):  
Suresh C. Seetharam ◽  
Dirk Mallants ◽  
Janez Perko ◽  
Diederik Jacques
Keyword(s):  
Data Base ◽  
Experimental Studies ◽  
Cementitious Materials ◽  
Time Dependent ◽  
Extensive Literature ◽  
Near Surface ◽  
Comprehensive Data ◽  
Engineered Barriers ◽  
Parameter Values ◽  
Consistent Approach

This paper presents a consistent approach for the development of a comprehensive data base of time-dependent hydraulic and transport parameters for concrete engineered barriers of the future Dessel near surface repository for low level waste. The parameter derivation is based on integration of selected data obtained through an extensive literature review, data from experimental studies on cementitious materials specific for the Dessel repository and numerical modelling using physically-based models of water and mass transport. Best estimate parameter values for assessment calculations are derived, together with source and expert range and their probability density function wherever the data was sufficient. We further discuss a numerical method for upscaling laboratory derived parameter values to the repository scale; the resulting large-scale effective parameters are commensurate with numerical grids used in models for radionuclide migration. To accommodate different levels of conservatism in the various assessment calculations defined by ONDRAF/NIRAS, several sets of parameter values have been derived based on assumptions that introduce different degrees of conservatism. For pertinent parameters, the time evolution of such properties due to the long-term concrete degradation is also addressed. The implementation of the consistent approach is demonstrated by considering the pore water diffusion coefficient as an example.

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