Chaotic Behavior Of System Research Articles

Measuring Out-of-Time-Order Correlators on a Nuclear Magnetic Resonance Quantum Simulator

The idea of the out-of-time-order correlator (OTOC) has recently emerged in the study of both condensed matter systems and gravitational systems. It not only plays a key role in investigating the holographic duality between a strongly interacting quantum system and a gravitational system, but also diagnoses the chaotic behavior of many-body quantum systems and characterizes the information scrambling. Based on the OTOCs, three different concepts -- quantum chaos, holographic duality, and information scrambling -- are found to be intimately related to each other. Despite of its theoretical importance, the experimental measurement of the OTOC is quite challenging and so far there is no experimental measurement of the OTOC for local operators. Here we report the measurement of OTOCs of local operators for an Ising spin chain on a nuclear magnetic resonance quantum simulator. We observe that the OTOC behaves differently in the integrable and non-integrable cases. Based on the recent discovered relationship between OTOCs and the growth of entanglement entropy in the many-body system, we extract the entanglement entropy from the measured OTOCs, which clearly shows that the information entropy oscillates in time for integrable models and scrambles for non-intgrable models. With the measured OTOCs, we also obtain the experimental result of the butterfly velocity, which measures the speed of correlation propagation. Our experiment paves a way for experimentally studying quantum chaos, holographic duality, and information scrambling in many-body quantum systems with quantum simulators.

On disappearance of chaos in fractional systems

In a seminal paper, Grigorenko and Grigorenko [15], numerically studied fractional order dynamical systems (FODS) of the form Dαixi=fi(x1,x2,x3),0≤αi≤1,(i=1,2,3); and showed the existence of chaos in case of fractional Lorenz system when Σ=α1+α2+α3≤3. Since then voluminous numerical work has been done to explore various FODS, in this regard. It is now an established fact that Σ acts as a chaos controlling parameter.In the present article we take a survey of present literature on chaotic behavior of fractional order dynamical systems. Further we numerically explore fractional Chen, Rössler, Bhalekar–Gejji, Lorenz and Liu systems and observe that chaos always disappear if Σ ≤ 2. The existing examples in the literature along with the systems that we have analyzed lead us to conjecture non-existence of chaos if Σ ≤ 2; which in some sense is a generalization of classical Poincaré–Bendixon theorem for FODS.

Chaotic behavior of discrete-time linear inclusion dynamical systems

Given any finite family of real d-by-d nonsingular matrices {S1,…,Sl}, by extending the well-known Li–Yorke chaos of a deterministic nonlinear dynamical system to a discrete-time linear inclusion or hybrid or switched system:xn∈{Skxn−1;1≤k≤l},x0∈Rdandn≥1,we study the chaotic dynamics of the state trajectory (xn(x0, σ))n ≥ 1 with initial state x0∈Rd, governed by a switching law σ:N→{1,…,l}. Two sufficient conditions are given so that for a “large” set of switching laws σ, there exhibits the scrambled dynamics as follows: for all x0,y0∈Rd,x0≠y0,lim infn→+∞∥xn(x0,σ)−xn(y0,σ)∥=0andlim supn→+∞∥xn(x0,σ)−xn(y0,σ)∥=∞.This implies that there coexist positive, zero and negative Lyapunov exponents and that the trajectories (xn(x0, σ))n ≥ 1 are extremely sensitive to the initial states x0∈Rd. We also show that a periodically stable linear inclusion system, which may be product unbounded, does not exhibit any such chaotic behavior. An explicit simple example shows the discontinuity of Lyapunov exponents with respect to the switching laws.

Nonlinear dynamics as an engine of computation.

Control of chaos teaches that control theory can tame the complex, random-like behaviour of chaotic systems. This alliance between control methods and physics-cybernetical physics-opens the door to many applications, including dynamics-based computing. In this article, we introduce nonlinear dynamics and its rich, sometimes chaotic behaviour as an engine of computation. We review our work that has demonstrated how to compute using nonlinear dynamics. Furthermore, we investigate the interrelationship between invariant measures of a dynamical system and its computing power to strengthen the bridge between physics and computation.This article is part of the themed issue 'Horizons of cybernetical physics'.

Incremental response sensitivity approach for parameter identification of chaotic and hyperchaotic systems

This paper develops an incremental response sensitivity approach for parameter identification of chaotic and hyperchaotic systems. In doing so, there hinge on two points. The first point is the establishment of the incremental objective function; this origins from the idea that long-term behavior of (hyper-)chaotic systems is often sensitive to the change of system parameters, but the short-term behavior is not. The advantage over the general objective function resides in that the incremental objective function is often well-defined, that is, having only one local minimal in the parametric space. The second is the utilization of the response sensitivity approach to get the minimal or identified parameters of the incremental objective function. In order to guarantee the performance, the trust-region constraint is additionally enforced to enhance the response sensitivity approach and it is practically dealt with by the Tikhonov regularization. Numerical tests are performed to verify the proposed approach.

NON‐PDC Observer‐Based T‐S Fuzzy Tracking Controller Design and its Application in CHAOS Control

AbstractIn many mechanical devices with chaotic behavior, stabilizing unstable periodic orbits (UPOs) of the system has positive effects in the lifetime and effectiveness of these devices. In this study, a new non‐parallel distributed compensation (non‐PDC) observer‐based tracking controller is presented for Takagi–Sugeno fuzzy systems to control the chaotic behavior of such systems. Asymptotic stability synthesis of the closed‐loop system is investigated using a fuzzy Lyapunov function to derive less conservative conditions than common quadratic Lyapunov function‐based approaches. To tackle the main drawback of the fuzzy Lyapunov‐based approaches, which assume some upper bounds on the derivatives of the fuzzy grade functions, we propose a new procedure by considering a constraint on the control signal. The new design conditions are given in the form of linear matrix inequalities (LMIs). The proposed control structure is applied to spinning disks in which chaos phenomena appear in lateral vibration. Simulation results are given to show the applicability of the proposed tracker to the UPO problem.

Electrokinetically Control Flow Mixing in Microfluidic Chip Utilizing Microelectrode Based on Indium Tin Oxide

In order to blend the liquid in the microfluidic chip adequately, this paper presented a novel mixing scheme in which the species streams was mixed via the application of Lorenz chaotic system algorithm to three pair of parallel electrodes mounted on the surfaces of the mixing chamber to produces chaotic electric fields. To select the better microelectrode arrays, the paper also discussed the advantages of the indium tin oxide (ITO) materials and used ITO to design the microelectrode array. Finally, simulations of the mixing process under the control of the proposed chaotic electric fields are performed. Analysis result shows that the correlation coefficient between the largest Lyapunov exponents on Lorenz chaotic system and chaotic flow behavior is 0.72 which shows a better synchronism, and it also proved that ITO is a better material to fabricate the microelectrode arrays.

Design of Initial Value Control for Modified Lorenz-Stenflo System

For the sake of complexity, unpredictability, and exceeding sensitivity to initial conditions in the chaotic systems, there were many studies for information encryption of chaotic systems in recent years. Enhancing the security in information encryption of chaotic systems, an initial value control circuit for chaotic systems is proposed in this paper. By way of changing the initial value, we can change the behavior of chaotic systems and also change the key of information encryption. An analog circuit is implemented to verify the initial value control circuit design.

A new wind power prediction method based on chaotic theory and Bernstein Neural Network

The accuracy of wind power prediction is important for assessing the security and economy of the system operation when wind power connects to the grids. However, multiple factors cause a long delay and large errors in wind power prediction. Hence, efficient wind power forecasting approaches are still required for practical applications. In this paper, a new wind power forecasting method based on Chaos Theory and Bernstein Neural Network (BNN) is proposed. Firstly, the largest Lyapunov exponent as a judgment for wind power system's chaotic behavior is made. Secondly, Phase Space Reconstruction (PSR) is used to reconstruct the wind power series' phase space. Thirdly, the prediction model is constructed using the Bernstein polynomial and neural network. Finally, the weights and thresholds of the model are optimized by Primal Dual State Transition Algorithm (PDSTA). The practical hourly data of wind power generation in Xinjiang is used to test this forecaster. The proposed forecaster is compared with several current prominent research findings. Analytical results indicate that the forecasting error of PDSTA + BNN is 3.893% for 24 look-ahead hours, and has lower errors obtained compared with the other forecast methods discussed in this paper. The results of all cases studying confirm the validity of the new forecast method.

The Control of The Lorenz Chaotic System Blended With The Noise

In this study, the Lorenz chaotic system behavior is investigated according to the change in the noise of the system and measurement. The noise was emerged as an important impact on the Lorenz chaotic system. The discrete-time LQR ( Linear Quadratic Regulator) control method and discrete-time PI controller method have been used to the noisy system controls. Noise was accepted as a destructor that changes the output behavior of the chaotic system. Noisy system outputs were enforced to move to the equilibrium points by the support of controllers. Optimization of the controller parameters for noiseless system models was performed using a genetic algorithm method. Then, both control system algorithms determined as successful in transporting to the equilibrium point for state variables which are blended with the noise. Changes, associated with the noise conditions are given in the graphics and tables. Besides, the error performance criteria and the control performance criteria were obtained to evaluate the results. This study was performed by MATLAB-Simulink simulation program.

The Routh–Hurwitz conditions of fractional type in stability analysis of the Lorenz dynamical system

This paper discusses stability conditions and a chaotic behavior of the Lorenz dynamical system involving the Caputo fractional derivative of orders between 0 and 1. Contrary to some existing results on the topic, we study these problems with respect to a general (not specified) value of the Rayleigh number as a varying control parameter. Such a bifurcation analysis is known for the classical Lorenz system; we show that analysis of its fractional extension can yield different conclusions. In particular, we theoretically derive (and numerically illustrate) that nontrivial equilibria of the fractional Lorenz system become locally asymptotically stable for all values of the Rayleigh number large enough, which contradicts the behavior known from the classical case. As a main proof tool, we derive the optimal Routh–Hurwitz conditions of fractional type, i.e., necessary and sufficient conditions guaranteeing that all zeros of the corresponding characteristic polynomial are located inside the Matignon stability sector. Beside it, we perform other bifurcation investigations of the fractional Lorenz system, especially those documenting its transition from stability to chaotic behavior.

The Influence of Road Bumps Characteristics on the Chaotic Vibration of a Nonlinear Full-Vehicle Model with Driver

In this paper, the chaotic behavior of a ground vehicle system with driver subjected to road disturbances is studied and the relationship between the nonlinear vibration of the vehicle and ride comfort is evaluated. The vehicle system is modeled as fully nonlinear with seven degrees of freedom and an additional degree of freedom for driver (8-DOF). The excitation force is the road irregularities that are assumed as road speed control bumps. The sinusoidal, consecutive half-sine and dented-rectangular waveforms are considered to simulate the road speed control bumps. The nonlinearities of the system are due to the nonlinear springs and dampers that are used in the suspension system and tires. The governing differential equations are extracted under Newton–Euler laws and solved via numerical methods. The chaotic behaviors were studied in more detail with special techniques such as bifurcation diagrams, phase plane portrait, Poincaré map and Lyapunov exponents. The ride comfort was evaluated as the RMS value of the vertical displacement of the vehicle body and driver. Firstly, the effect of amplitude (height) and frequency (vehicle’s speed) of these speed control bumps on chaotic vibrations of vehicle are studied. The obtained results show that various forms of vibrations, such as periodic, subharmonic and chaotic vibrations, can be detected in the system behavior with the change of the height and frequency of speed control bumps and present different types of strange attractors in the vehicle with and without driver. Then, the influence of nonlinear vibration on ride comfort and the relationship between chaotic vibrations of the vehicle and driving comfort are investigated. The results of analyzing the RMS diagrams reveal that the chaotic behaviors can directly affect the driving comfort and lead to the driver’s comfort being reduced. The obtained results can be used in the design of vehicle and road bumps pavement.

A NOVEL APPLICATION OF CHAOTIC PROPERTIES IN WATER TREATMENT PLANT

This paper aims at presenting a new optimization proposal to enhance the flocculation process in Water Treatment (WT) plant using a better flash mixing, located at KELAVERAPALLY, in Krishnagiri district, Tamil Nadu, India. Further, Sludge removal is done efficiently which decreases the water wastage as well as improvement in output water quality. Though WT plants are already equipped with systematic and sequential physicochemical processes, still they need to be optimized to obtain a better treated drinking water to maintain the quality standards as prescribed by World Health Organization. Chaotic behavior in chemical systems has been used to optimize the performance of WT plant. Measurement systems implemented in WT plant yield several chaotic based measurement parameters which are used to control the system operations to maintain the target water quality. This intelligible data extraction through the proposed measurement  systems in a short span of time improves the plant performance without adding any costly systems except few changes in the existing plant setup.  Chaotic behavior is ensured through Lyapunov Exponents and Kolmogorov-Sinai Entropies. Both, water quality improvement and water wastage reduction is achieved simultaneously in the proposed work when a dosage prediction is done using Feed Forward Neural Networks. The treatment plant investigated has a maximum capacity of 14 MLD (Million litres per day) using two parallel streams with 7 MLD each

The chaotic behaviour of cardiovascular system during open heart surgery

The chaotic behaviour of cardiovascular system during open heart surgery

Missing chaos in global climate change data interpreting?

The main problem of ecological data modeling is their interpretation and its correct understanding. This problem cannot be solved solely by a big data collection. To sufficiently understand ecosystems we need to know how these processes behave and how they respond to internal and external factors. Similarly, we need to know the behavior of processes that are involved in the climate system and the biosphere of the earth. In order to characterize precisely the behavior of individual elements and ecosystems we need to use deterministic, stochastic and chaotic behavior. Unfortunately, the chaotic part of systems is typically completely ignored in almost all approaches. Ignoring of chaotical part leads to many biased outcomes. To overcome this gap we model chaotic system behavior by random iterated function system which provides a generic guideline for such data management. This also allows to replicate a complexity and chaos of ecosystem.

About existence of stationary points for the Arnold–Beltrami–Childress (ABC) flow

The existence of stationary points for the dynamical system of ABC-flow is considered.The ABC-flow, a three-parameter velocity field that provides a simple stationary solution of Euler's equations in three dimensions for incompressible, inviscid fluid flows, is the prototype for the study of turbulence (it provides a simple example of dynamical chaos).But, nevertheless, between the chaotic trajectories of the appropriate solutions of such a system we can reveal the stationary points, the deterministic basis among the chaotic behaviour of ABC-flow dynamical system. It has been proved the existence of 1 point for two partial cases of parameters {A, B, C}: (1) A = B = 1; (2) C = 1 (A² + B² = 1). Moreover, dynamical system of ABC-flow allows 3 points of such a type, depending on the meanings of parameters {A, B, C}.

Effective Control and Bifurcation Analysis in a Chaotic System with Distributed Delay Feedback

We discuss the dynamic behavior of a new Lorenz-like chaotic system with distributed delayed feedback by the qualitative analysis and numerical simulations. It is verified that the equilibria are locally asymptotically stable whenα∈(0,α0)and unstable whenα∈(α0,∞); Hopf bifurcation occurs whenαcrosses a critical valueα0by choosingαas a bifurcation parameter. Meanwhile, the explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is derived by normal form theorem and center manifold argument. Furthermore, regardingαas a bifurcation parameter, we explore variation tendency of the dynamics behavior of a chaotic system with the increase of the parameter valueα.

Efficient method to generate time evolution of the Wigner function for open quantum systems

The Wigner function is a useful tool for exploring the transition between quantum and classical dynamics, as well as the behavior of quantum chaotic systems. Evolving the Wigner function for open systems has proved challenging however; a variety of methods have been devised but suffer from being cumbersome and resource intensive. Here we present an efficient fast-Fourier method for evolving the Wigner function, that has a complexity of $O(N\log N)$ where $N$ is the size of the array storing the Wigner function. The efficiency, stability, and simplicity of this method allows us to simulate open system dynamics previously thought to be prohibitively expensive. As a demonstration we simulate the dynamics of both one-particle and two-particle systems under various environmental interactions. For a single particle we also compare the resulting evolution with that of the classical Fokker-Planck and Koopman-von Neumann equations, and show that the environmental interactions induce the quantum-to-classical transition as expected. In the case of two interacting particles we show that an environment interacting with one of the particles leads to the loss of coherence of the other.

A novel hybrid color image encryption algorithm using two complex chaotic systems

Based on complex Chen and complex Lorenz systems, a novel color image encryption algorithm is proposed. The larger chaotic ranges and more complex behaviors of complex chaotic systems, which compared with real chaotic systems could additionally enhance the security and enlarge key space of color image encryption. The encryption algorithm is comprised of three step processes. In the permutation process, the pixels of plain image are scrambled via two-dimensional and one-dimensional permutation processes among RGB channels individually. In the diffusion process, the exclusive-or (XOR for short) operation is employed to conceal pixels information. Finally, the mixing RGB channels are used to achieve a multilevel encryption. The security analysis and experimental simulations demonstrate that the proposed algorithm is large enough to resist the brute-force attack and has excellent encryption performance.

Designing fuzzy logic controller based on combination of terminal sliding mode and state feedback controllers for stabilizing chaotic behaviour in rod-type plasma torch system

In this paper, a fuzzy logic controller is designed for controlling non-linear behaviours in a rod-type plasma torch system. This system may behave chaotically when its parameters pass through critical values, and gives rise to undesirable oscillations, which degrade the performance of the system. In order to avoid such behaviours, the state feedback controller and the terminal sliding mode controller are combined into a fuzzy system to stabilize the equilibrium point of the system. The fuzzy rules include consequent parts in which state feedback control or terminal sliding mode control is utilized for different conditions. The proposed scheme provides the advantages of both methods, i.e. a decreased settling time is achieved using terminal sliding mode control, and using state feedback control, the chattering phenomenon is removed. The proposed method provides robustness against the external disturbance and uncertainty. Numerical simulations are given to verify that the proposed fuzzy control scheme can effectively control chaotic behaviour of the rod-type plasma torch system and the method provides the best performance among the other methods discussed in the paper.