Theory Of Chaotic Systems Research Articles

Qualitative Properties of a Physically Extended Six-Dimensional Lorenz System

In this paper, the qualitative properties of a physically extended six-dimensional Lorenz system, with additional physical terms describing rotation and density, which was proposed in [Moon et al., 2019] have been investigated. The dissipation, invariance, Lyapunov exponents, Kaplan–Yorke dimension, ultimate boundedness and global attractivity of this six-dimensional Lorenz system have been discussed in detail according to the chaotic systems theory. We find that this system exhibits chaos phenomena for a new set of parameters. It is well known that the general method for studying the bounds of a chaotic system is to construct a suitable Lyapunov-like function (or the generalized positive definite and radically unbounded Lyapunov function). However, the higher the dimension of a chaotic system, the more difficult it is to construct the Lyapunov-like function. The innovation of this paper is that we first construct the suitable Lyapunov-like function for this six-dimensional Lorenz system, and then we prove that this system is not only globally bounded for varying parameters, but it also gives a collection of global absorbing sets for this system with respect to all parameters of this system according to Lyapunov’s direct method and the optimization method. Furthermore, we obtain the rate of the trajectories going from the exterior to the global absorbing set. Some numerical simulations are presented to validate our research results. Finally, we give a direct application of the results obtained in this paper. According to the results of this paper, we can conclude that the equilibrium point [Formula: see text] of this system is globally exponentially stable.

Chaotic control of the dynamical behavior of COVID-19 through the electromagnetic fields

Investigating the dynamical behavior of a system is an effective method to predict and control its future behavior. Studying the dynamic behavior of a virus can prevent the pathogenicity of a virus and save human lives during the disease epidemic. If the transmission of information from the virus genome to the environment is locked, the pathogenicity of the virus stops. Information transmission can be checked via the investigation of the spin information transport. In the current study, we have characterized the dynamical behavior of the virus by studying the spin transport through its RNA chain to estimate the information transfer path in the system. A voltage generator with adjustable frequency as a control system has been designed using the control theory of chaotic systems. Our aim is disturbing and reduces the transmitted information from viruses to the environment. The external stimuli can propel the system to the locked information transfer situation. Applying an external field noise with a specific frequency range (200–500 GHz) controlled through the external controller system can destroy the information transmitted by the virus to the environment. Disturbance intensity as a control parameter adjusts the external field frequency to push the system to the chaotic behavior which will be able to lock the information transfer and then prevent the spread of the epidemic.

Visualization of chaotic attractors in 3D as motivating tool for introductory physics course

It is well known that simple nonlinear systems can produce inherently unpredictable behaviour, which is called chaotic motion. Interest in the theory behind it has risen rapidly and much effort has been invested in integrating it into graduate and undergraduate curricula. However, the concepts and ideas that are appropriate for the description of these motions are very abstract and intricate. Human beings are visual creatures in nature; therefore all aids and procedures which give a visible demonstration of an abstract idea are of great value from a didactic point of view. Our paper offers the teachers a method for the demonstration of strange attractors which play a central role in the theory of chaotic systems. The images of the 2D projection of the strange attractors are well known and common, however their 3D visualisation gives a real sensation for the human brain so they are much more suitable for capturing the students’ interest. With the help of our downloadable MAPLE program the DXF files of strange attractors can be easily generated and their images can be engraved in glass. E-materials belonging to the method can be downloaded from our webpage.

Dynamical Temperature from the Phase Space Trajectory

Abstract In classical statistical mechanics the trajectory in phase space represents the propagation of a classical Hamiltonian system. While trajectories play a key role in chaotic system theory, exploitation of a single trajectory has yet to be considered. This work shows that for ergodic dynamical systems the dynamical temperature can be derived using phase space trajectories.

Dual Combination Synchronization of Six Chaotic Systems

Based on combination synchronization of three chaotic systems and combination–combination synchronization of four chaotic systems, a novel scheme of dual combination synchronization is investigated for six chaotic systems in the paper. Using combined adaptive control and Lyapunov stability theory of chaotic systems, some sufficient conditions are attained to realize dual combination synchronization of six chaotic systems. The corresponding theoretical proofs and numerical simulations are presented to demonstrate the effectiveness and correctness of the dual combination synchronization. Due to the complexity of dual combination synchronization, it will be more secure and interesting to transmit and receive signals in application of communication.

CHAOS AND WEATHER FORECASTING: THE ROLE OF THE UNSTABLE SUBSPACE IN PREDICTABILITY AND STATE ESTIMATION PROBLEMS

In the first part of this paper, we review some important results on atmospheric predictability, from the pioneering work of Lorenz to recent results with operational forecasting models. Particular relevance is given to the connection between atmospheric predictability and the theory of Lyapunov exponents and vectors. In the second part, we briefly review the foundations of data assimilation methods and then we discuss recent results regarding the application of the tools typical of chaotic systems theory described in the first part to well established data assimilation algorithms, the Extended Kalman Filter (EKF) and Four Dimensional Variational Assimilation (4DVar). In particular, the Assimilation in the Unstable Space (AUS), specifically developed for application to chaotic systems, is described in detail.

Research on Railway Engineering Cost Prediction Model Based on Chaotic Neural Networks and CS

It is always hard to draw on the experience of completed projects to predict engineering cost, and the nonlinear characteristic of the influence factors of engineering cost increases the difficulty of prediction. Less efforts and higher accuracy are the objects pursued by related researchers. In this paper, the Cost Significant theorem is applied to simplify computing and the chaotic neural network is used to improve accuracy. The prediction model is rooted from the nonlinear dynamic chaotic system theory and two techniques employed are phase space reconstruction and chaotic neural network construction. The experiment results indicate that the model is suitable for estimating short-term engineering investment and the prediction accuracy is improved.

A self-adaptive embedded chaotic particle swarm optimization for parameters selection of Wv-SVM

Particle swarm optimization (PSO) is a population-based swarm intelligence algorithm driven by the simulation of a social psychological metaphor instead of the survival of the fittest individual. Based on the chaotic system theory, this paper proposes new PSO method that uses chaotic mappings for parameter adaptation of Wavelet v-support vector machine (Wv-SVM). Since chaotic mapping enjoys certainty, ergodicity and the stochastic property, the proposed PSO introduces chaos mapping using logistic mapping sequences which increases its convergence rate and resulting precision. The simulation results show the parameter selection of Wv-SVM model can be solved with high search efficiency and solution accuracy under the proposed PSO method.

The Study of the Chaotic Behavior in Retailer's Demand Model

Based on the work of domestic and foreign scholars and the application of chaotic systems theory, this paper presents an investigation simulation of retailer's demand and stock. In simulation of the interaction, the behavior of the system exhibits deterministic chaos with consideration of system constraints. By the method of space's reconstruction, the maximal Lyapunov exponent of retailer's demand model was calculated. The result shows the model is chaotic. By the results of bifurcation diagram of model parameters , and changing initial condition, the system can be led to chaos.

A novel chaotic particle swarm optimization approach using Hénon map and implicit filtering local search for economic load dispatch

Particle swarm optimization (PSO) is a population-based swarm intelligence algorithm driven by the simulation of a social psychological metaphor instead of the survival of the fittest individual. Based on the chaotic systems theory, this paper proposed a novel chaotic PSO combined with an implicit filtering (IF) local search method to solve economic dispatch problems. Since chaotic mapping enjoys certainty, ergodicity and the stochastic property, the proposed PSO introduces chaos mapping using Hénon map sequences which increases its convergence rate and resulting precision. The chaotic PSO approach is used to produce good potential solutions, and the IF is used to fine-tune of final solution of PSO. The hybrid methodology is validated for a test system consisting of 13 thermal units whose incremental fuel cost function takes into account the valve-point loading effects. Simulation results are promising and show the effectiveness of the proposed approach.

Theoretical time series analysis from electric field oscillations generated by rate equations of generation–recombination processes in n-type semiconductors

We have analyzed the nonlinear electric field time series theoretically generated by a set of three coupled first-order differential equations in order to interpret the chaotic behavior present in measurements of low-frequency current oscillations in semiconductor samples. In the present work, we carry out a direct investigation of the theoretical time series applying techniques of experimental nonlinear time series analysis. We have reconstructed the phase space of the system and calculated some parameters as the correlation dimension of the reconstructed attractors, the Lyapunov exponent's spectrum and the Kaplan–Yorke dimension. The conclusions are consistent with the theory of chaotic systems and with the available experimental results for semiconductors.

Predicting chaotic time series of PM10 concentration using artificial neural network

Respirable particulate matter (PM10) concentration at one residential site in Delhi, India was predicted using the neural network approach. The concepts of chaotic systems theory were utilized to build the neural network model. The embedding dimension was estimated to provide the inputs to the neural network. The model evaluation results indicated the importance of noise reduction before selecting the embedding dimension of the time series. The selection of a proper embedding dimension is considered to be essential for obtaining reliable predictions. The model’s performance shows the capability of neural networks in modelling the chaotic time series.

Organizational Values as 'Attractors of Chaos': An Emerging Supply Chain Management Change Designed to Overcome Organizational Complexity

This paper considers supply chain management organizations with the aid of 'chaotic systems theory' developed originally in physics and mathematics. Since innovations in supply chain management are vital for organizational survival 'complex systems theory' may assist in fine-tuning managerial philosophies that provide stability in supply chain management because it is on the boundary of chaos that the greatest innovational creativity occurs. Neither 'management by rigid objectives' (MBO) nor 'management by instruction' (MBI) will be suitable for the information society of the twenty-first century because chaotic social systems will no longer be effectively managed. However, the capacity for self-organization will be derived essentially from how supply chain management members accept a shared set of values or principles for action-'management by values' (MBV). Complex systems theory deals with systems that show complex structures in time or space, often hiding simple deterministic rules. This theory holds that once these rules are found, it is possible to make effective predictions and even to control the apparent complexity. The state of chaos that self-organizes is attributable to the appearance of the 'strange attractor' and provides the ideal basis for creativity and innovation in the twenty-first century. In the self-organized state of chaos, social members are not confined to narrow roles and gradually develop their capacity for differentiation and relationships, growing progressively towards their maximum potential contribution to the efficiency of the organization. In this meaning, values act as organizers of 'attractors' of disorder, which, in the theory of chaos, are systems represented by usually regular geometric configurations that predict the long-term behavior of complex systems. In supply chain management organizations (as in all kinds of social systems) the initial principles end up as the final principles in the long term. An attractor is a model representation of the behavioral results of the system. The attractor is not a force of attraction or a goal-oriented presence in the system; it simply depicts where the system is heading based on its rules of motion. In supply chain management organizations that cultivate or shares values of autonomy, responsibility, independence, innovation, creativity, and proactivity, the risk of short-term chaos is mitigated by external complexities that organizations are currently confronting. The strategy is to alter the supply chain management's surroundings so that they can benefit from management by values (MBV).

Classical invariants and the quantization of chaotic systems

Due to their exponential proliferation, long periodic orbits constitute a serious drawback in Gutzwiller's theory of chaotic systems. Therefore, it would be desirable that other classical invariants, not suffering from the same problem, could be used in alternative semiclassical quantization schemes. In this Rapid Communication, we demonstrate how a suitable dynamical analysis of chaotic quantum spectra unveils the role played, in this respect, by classical invariant areas related to the stable and unstable manifolds of short periodic orbits.

Geometry of gene expression dynamics

A gene expression trajectory moves through a high dimensional space where each axis represents the mRNA abundance of a different gene. Genome wide gene expression has a dynamic structure, especially in studies of development and temporal response. Both visualization and analyses of such data require an explicit attention to the temporal structure. Using three cell cycle trajectories from Saccharomyces cerevisiae to illustrate, we present several techniques which reveal the geometry of the data. We import phase-delay time plots from chaotic systems theory as a dynamic data visualization device and show how these plots capture important aspects of the trajectories. We construct an objective function to find an optimal two-dimensional projection of the cell cycle, demonstrate that the system returns to this plane after three different initial perturbations, and explore the conditions under which this geometric approach outperforms standard approaches such as singular value decomposition and Fourier analysis. Finally, we show how a geometric analysis can isolate distinct parts of the trajectories, in this case the initial perturbation versus the cell cycle. junhyong.kim@yale.edu

Inequivalence of weak localization and coherent backscattering.

We define a current-conserving approximation for the local conductivity tensor of a disordered system which includes the effects of weak localization. Using this approximation we show that the weak localization effect in conductance is not obtained simply from the diagram corresponding to the coherent backscattering peak observed in optical experiments. Other diagrams contribute to the effect at the same order and decrease its value. These diagrams appear to have no semiclassical analogues, a fact which may have implications for the semiclassical theory of chaotic systems. The effects of discrete symmetries on weak localization in disordered conductors are evaluated and compared to results from chaotic scatterers.

Communicating with chaos.

The use of chaos to transmit information is described. Chaotic dynamical systems, such as electrical oscillators with very simple structures, naturally produce complex wave forms. We show that the sym- bolic dynamics of a chaotic oscillator can be made to follow a desired symbol sequence by using small perturbations, thus allowing us to encode a message in the wave form. We illustrate this using a simple numerical electrical oscillator model. PACS numbers: 05.45.+b Much of the fundamental understanding of chaotic dy- namics involves concepts from information theory, a field developed primarily in the context of practical communi- cation. Concepts from information theory used in chaos include metric entropy, topological entropy, Markov par- titions, and symbolic dynamics (1). On the other hand, because of their exponential sensitivity, chaotic systems are often said to evolve randomly. This terminology is partially justified if one regards the information obtained by detailed observation of the chaotic orbit as being less significant than the statistical properties of the orbits. The object of this Letter is to show that we can use the close connection between the theory of chaotic systems and information theory in a way that is more than purely formal. In particular, we show that the recent realization that chaos can be controlled with small perturbations (2) can be utilized to cause the symbolic dynamics of a chaotic system to track a prescribed symbol sequence, thus allowing us to encode any desired message in the sig- nal from a chaotic oscillator. The natural complexity of chaos thus provides a vehicle for information transmission in the usual sense. Furthermore, we argue that this method of communication will often have technological advantages. Specifically, assume that there is an electrical oscillator producing a large amplitude chaotic signal that one wishes to use for communication. The so-called double

Chaotic mixing of a viscous fluid in a screw impeller mixer

Several studies on model systems1,2 have established the potential utility of the theory of chaotic systems for the analysis of mixing in laminar flow fields. Here we present an application of the theory to the mixing, under creeping flow conditions, in a screw impeller mixer, which is used in a number of industrial mixing operations. The flow in the system was found to comprise five main components: (i) unidirectional flow in the annulus; (ii) turning slow flow in the lower compartment; (iii) double helical motion in the screw element; (iv) helical motion in the upper part of the draft tube; and (v) turning flow in the upper compartment. Under creeping flow conditions, the flow is independent of the rotational speed of the impeller, thus mixing can be altered only by changing the system geometry. The flow is time independent with respect to a frame of reference fixed on the impeller, and can be considered to be a more complicated version of that in the partitioned pipe mixer.3 Streak lines of the flow are invariant to 2π rotations of the impeller, and illustrate the stretching and folding of material lines by the flow. The development of a streak line starting in the annulus was studied experimentally using a fluorescent tracer dye. Each streak line was found to be segmented into an infinite number of loops each starting at a point on the impeller blade. Each loop in turn was found to be wrapped toroidally around the draft tube. Poincaré sections were obtained by noting the points of intersection of a tracer particle trajectory with a horizontal plane. The coordinate system was fixed either on the tank or on the impeller. Experimentally obtained Poincaré sections were chaotic but showed no structure in either case indicating a well mixed-system to the resolution of the measurements. Studies are currently underway with impellers of varying pitch and length to study their effect on the quality of mixing in terms of Poincaré sections and the extent of stretching of a streak line loop in each rotation of the impeller.