Chaos Attractors Research Articles

Nonlinear Dynamics of an Unsymmetric Cross-Ply Square Composite Laminated Plate for Vibration Energy Harvesting

The nonlinear behaviors and energy harvesting of an unsymmetric cross-ply square composite laminated plate with a piezoelectric patch is presented. The unsymmetric cross-ply square composite laminated plate has two stable equilibrium positions by applying thermal stress, thus having snap-through with larger amplitude between the two stable equilibrium positions relative to the general laminated plate. Based on the von-Karman large deformation theory, the nonlinear electromechanical coupling equations of motion of the unsymmetric composite laminated plate with a piezoelectric patch are derived by using Hamilton’s principle. The influence of the base excitation amplitude on nonlinear behaviors and energy harvesting are investigated. For different base excitation amplitudes, the motions of the system demonstrate periodic motion, quasi-periodic motion, chaotic motion and snap-through, and two single-well chaotic attractors and a two-well chaos attractor coexist. Moreover, the power generation efficiency is optimal when the excitation amplitude is in a certain range due to its own unique nonlinear characteristics. The unsymmetric cross-ply square composite laminated plate subjected to thermal stress can actually be called a kind of bistable composite shell structure that has a broad application prospect in combination with morphing aircraft, large deployable antenna and solar panel, which are very likely to have nonlinear vibration.

Effect of Different Forms of Periodic Piecewise Linear Forces on Duffing Oscillator

The paper highlights the effect of different forms of periodic piecewise linear forces in the ubiquitous Duffing oscillator equation. The external periodic piecewise linear forces considered are Triangular, Hat, Trapezium, Quadratic and Rectangular. With the aid of some numerical simulation tools such as bifurcation diagram, phase portrait and Poincare´ map, the different routes to chaos and various strange attractors are found to occur due to the applied forces. The effect of an ε-parametric control force in the Duffing system is also analyzed. To characterize the regular and chaotic behaviours of this system, the maximal Lyapunov exponent is employed.

On scenarios of the onset of homoclinic attractors in three-dimensional non-orientable maps

We study scenarios of the appearance of strange homoclinic attractors (which contain only one fixed point of saddle type) for one-parameter families of three-dimensional non-orientable maps. We describe several types of such scenarios that lead to the appearance of discrete homoclinic attractors including Lorenz-like and figure-8 attractors (which contain a saddle fixed point) as well as two types of attractors of spiral chaos (which contain saddle-focus fixed points with the one-dimensional and two-dimensional unstable manifolds, respectively). We also emphasize peculiarities of the scenarios and compare them with the known scenarios in the orientable case. Examples of the implementation of the non-orientable scenarios are given in the case of three-dimensional non-orientable generalized Hénon maps.

Coexisting Multiple Attractors in a Fourth Order Chua’s Circuit With Experimental Verifications by Analog and Digital Circuits

Exploring coexistence of multiple attractors brought by the multistability for circuits and systems has a significant meaning in the theoretical researches and practical applications for chaos. In this article, a succinct fourth order Chua’s circuit is proposed by replacing the negative resistance with an ordinary positive resistance in a traditional fourth order one. The two-dimensional stability analysis for equilibrium points shows that this circuit possesses one unstable saddle-focus point with index 1 and two stable node-focus points. Coexisting bifurcation models, multiple attractors and the corresponding attraction basins are revealed by a series of numerical simulations. The clear crisis scenario of the coexisting limit cycles of period-3 bridging the coexisting single-scroll attractors of chaos and the double-scroll one is observed by the bifurcation analyses. The dual-mode experimental verifications by the analog and digital circuits are carried out on the self-made printed circuit boards, which validate the simulated dynamical behaviors with the combination of physics and engineering.

Stochastic Bifurcations, Chaos and Phantom Attractors in the Langford System with Tori

The variability of stochastic dynamics for a three-dimensional dynamic model in a parametric zone with 2-tori is investigated. It is shown how weak Gaussian noise transforms deterministic quasiperiodic oscillations into noisy bursting. The phenomenon of stochastic generation of a phantom attractor and its shift with noise amplification is revealed. This phenomenon, accompanied by order-chaos transitions, is studied in terms of stochastic [Formula: see text]- and [Formula: see text]-bifurcations.

The Reliability of IoT Networks With Characteristics of Abnormal Induced Signals

The abnormal nodes of the Internet of Things (IoT) are closely related to the reliability of the network, but it is always a difficulty to predict the characteristics of the induced signals that threaten reliability. In this article, a feature analysis method for the abnormal signal inducing the unreliable anomaly in IoT based on chaos attractor is proposed to make phase space reconstruction of the abnormal signal that the nodes of IoT can induce unreliability. The phase space structure is constructed by the sequence of abnormal signals. The chaotic attractor is used to analyze the phase space characteristics and reflect the phase space characteristics of unreliable signals induced by the network from the geometric point of view. The mutual information method is used to solve the time delay of the unreliable abnormal signal of IoT, calculate the embedding dimension, find out the chaotic attractors under different characteristics, and judge the reliability change characteristics by analyzing the chaotic attractors. The experimental results show that the method has a better time slot consumption, throughput, and idle energy indexes. Furthermore, our experiments show that the chaos attractor method can effectively predict the reliability trend of IoT, and better predict the reliability anomalies of IoT.

A Novel Hybrid Secure Image Encryption Based on the Shuffle Algorithm and the Hidden Attractor Chaos System.

Aiming at the problems of small key space, low security of encryption structure, and easy to crack existing image encryption algorithms combining chaotic system and DNA sequence, this paper proposes an image encryption algorithm based on a hidden attractor chaotic system and shuffling algorithm. Firstly, the chaotic sequence generated by the hidden attractor chaotic system is used to encrypt the image. The shuffling algorithm is used to scramble the image, and finally, the DNA sequence operation is used to diffuse the pixel value of the image. Experimental results show that the key space of the scheme reaches 2327 and is very sensitive to keys. The histogram of encrypted images is evenly distributed. The correlation coefficient of adjacent pixels is close to 0. The entropy values of encrypted images are all close to eight and the unified average change intensity (UACI) value and number of pixel changing rate (NPCR) value are close to ideal values. All-white and all-black image experiments meet the requirements. Experimental results show that the encryption scheme in this paper can effectively resist exhaustive attacks, statistical attacks, differential cryptanalysis, known plaintext and selected plaintext attacks, and noise attacks. The above research results show that the system has better encryption performance, and the proposed scheme is useful and practical in communication and can be applied to the field of image encryption.

Autonomous chaos of exciton-polariton condensates

We study the formation of chaos and strange attractors in the order parameter space of a system of two coupled, non-resonantly driven exciton-polariton condensates. The typical scenario of bifurcations experienced by the system with increasing external pumping consists of (i) formation of $\pi$-synchronized condensates at low pumping, (ii) symmetry breaking pitchfork bifurcation leading to unequal occupations of the condensates with non-trivial phase difference between them, (iii) loss of the stability of all fixed points in the system resulting in chaotic dynamics, (iv) limit cycle dynamics of the order parameter, which ends up in (v) in-phase synchronized condensates via the Hopf bifurcation from a limit cycle. The chaotic dynamics of the order parameter is evidenced by calculating the maximal Lyapunov exponent. The presence of a chaotic domain is studied as a function of polariton-polariton interaction and the Josephson coupling between the condensates. At some values of the parameters the bifurcation route is more complex and the strange attractor can coexist with the stable fixed-point lasing. We also investigate how the chaotic dynamics is reflected in the light emission spectrum from the microcavity.

An image encryption algorithm based on a hidden attractor chaos system and the Knuth–Durstenfeld algorithm

Chaotic systems have been widely applied in digital image encryption due to their complex properties such as ergodicity, pseudo randomness and extreme sensitivity to their initial values and parameters. An image encryption algorithm based on a hidden attractor chaos system and Knuth–Durstenfeld algorithm is proposed. First, a hidden attractor chaos system is used to encrypt digital image. Compared to a self-excited attractor, the hidden attractor's attracting basin does not intersect with any small neighbourhoods of the equilibria. It is difficult for attackers to reconstruct the attractor by finding equilibrium points. Therefore, the hidden attractor chaotic system is difficult to decrypt. Meanwhile, the hidden attractor chaos system is very sensitive to initial values and parameters. Second, the Knuth–Durstenfeld algorithm has good randomness. In addition, the Knuth–Durstenfeld algorithm can reduce the time complexity and the space complexity of the permutation while achieving good permutation effects. Thus, Knuth–Durstenfeld algorithm is used to permutate the digital image. Finally, DNA sequence operations are used to diffuse image pixels values. Some experimental analyses have been applied to measure the new scheme, and the experimental results illustrate the scheme possesses better encryption performances. This method can be applied in secure image communication fields.

Antimonotonicity, Chaos and Multidirectional Scroll Attractor in Autonomous ODEs Chaotic System

Three-dimensional autonomous ordinary differential equations (ODEs) are the simplest and most important chaotic systems in nonlinear dynamics. In fact, they have been applied in many fields. In this paper, a systematic methodology for analyzing complex behavior of the ODEs chaotic system, as one of the ODEs chaotic systems, the improved TCS which satisfies the condition $a_{12}a_{21} = 0$ , is proposed. It is dissipative, chaos, symmetric, antimonotonicity and can generate multiple directional ( $M\times N\times L$ ) scroll attractors. Then, bifurcation diagrams, Lyapunov exponents, time series, Poincare sections, and Hausdroff dimensions are analyzed by setting the parameters and initial value. More interestingly, antimonotonicity (named reverse period-doubling bifurcation) and coexisting bifurcations are also reported. Finally, the results of theoretical analyses may be verified by electric experimental.

Strange nonchaos in self-excited singing flames

We report the first experimental evidence of a strange nonchaotic attractor (SNA) in the natural dynamics of a self-excited laboratory-scale system. In the previous experimental studies, the birth of a SNA was observed in quasiperiodically forced systems; however, such evidence of a SNA in an autonomous laboratory system is yet to be reported. We discover the presence of a SNA between the attractors of quasiperiodicity and chaos through a fractalization route in a laboratory thermoacoustic system. The observed dynamical transitions from order to chaos via a SNA is confirmed through various nonlinear characterization methods prescribed for the detection of a SNA.

Short-term metro passenger flow forecasting using ensemble-chaos support vector regression

ABSTRACTWith reliable and accurate predictions of short-term passenger flow, metro agencies can assign proper trains and crews, optimize schedules, and operate efficiently. This paper presents a hybrid model, which is intended for forecasting short-term passenger flow. The forecasting model aggregates SVR-based sub-models developed with subsets extracted by combining bootstrap sampling with dimensions subsampling. The sub-models consider not only the data from the reconstructed chaos attractor but also the periodic historical data. Passenger flow data collected from four different metro stations in Nanjing are used for model implementation and performance evaluation. Results confirm that the proposed EICSVR model was able to significantly improve predictive performance and generalization. The excellent accuracy and stability obtained in the empirical study indicate that the proposed model has good development potential for forecasting short-term passenger flow.

A novel chaotic modulation approach of packaged antenna for secured wireless medical sensor network in E‐healthcare applications

AbstractThis article first time reports the chaotic modulation approach toward RF signal processing for secured wireless medical sensor network (WMSN) in E‐healthcare applications. A Lorenz based chaotic modulation approach is implemented which provides lowest bit error rate (BER). The definite analytical expressions for BER in a differential chaos‐shift keying (DCSK) modulation scheme is derived and it predicted good correlation between simulated and theoretical. It is observed that proposed Lorenz chaos‐based DCSK modulation scheme is a potential candidate to provide high security in the patient data for WMSN. An off‐body UWB slotted antenna is designed which could avoid limitation of short‐range distance like implanted ones. The entire work includes numerical, simulated and experimental data in three phases. In first phase, Lorenz chaotic oscillator with electronics compatibility is executed which acts as data acquisition unit and demonstrates two‐dimensional and three‐dimensional chaos attractors. While in the second phase, analysis of BER achieves value of less than 10−4 by providing pseudorandom bit sequence at 5 Gb/s. A chaos modulated envelope using Lorenz based DCSK modulation is obtained by delay element γ. Finally, the third phase is designed on‐wafer off‐body antenna and demonstrates 3.1 to 10.6 GHz UWB toward RF signal processing in E‐healthcare applications.

Chaos and multiple attractors in a fractal–fractional Shinriki’s oscillator model

In this paper, we obtain novel chaotic behaviors using differential and integral operators with power-law, exponential-decay and Mittag-Leffler law for the Shinriki’s oscillator model. We studied the uniqueness and existence of the solutions employing the fixed point postulate. Also, we consider fractal–fractional operators to capture self-similarities for this chaotic attractor. These novel operators predict chaotic behaviors involving the fractal derivative in convolution with power-law, exponential decay law and the Mittag-Leffler function. The generalized model with power-law and Mittag-Leffler kernel was solved numerically via the Adams–Bashforth–Moulton and Adams–Moulton scheme, respectively. For another cases, the numerical schemes are based on the fundamental theorem of fractional calculus and the Lagrange polynomial interpolation. Numerical simulations for the symmetric and asymmetric cases are obtained to show the applicability and computational efficiency of these methods.

The Stability and Complexity Analysis of a Low-Carbon Supply Chain Considering Fairness Concern Behavior and Sales Service.

This paper studies a low-carbon dual-channel supply chain in which a manufacturer sells products through the direct channel and traditional channel, and a retailer sells products through the traditional channel. The manufacturer considers carbon emission reduction and has fairness concern behavior. The retailer provides sales service in the traditional channel and considers fairness concern behavior. The objective of this paper is to analyze the effects of different parameter values on the price stability and utility of the supply chain system emphatically using 2D bifurcation diagram, parameter plot basin, the basins of attraction, chaos attractor and sensitivity to the initial value, etc. The results find that the retailer’s fairness concern behavior shrinks the stability of the supply chain system more than that of the manufacturer’s fairness concern behavior. The system stability region decreases with the increase of carbon emission reduction level and the retailer’s fairness concern. The customers’ preference for the direct channel decreases the stable range of the direct channel, while it enlarges the stable range of the traditional channel. The supply chain system enters into chaos through flip bifurcation with the increase of price adjustment speed. In a stable state, the manufacture improving customer’s preference for the direct channel and the retailer choosing the appropriate fairness concern level can achieve the maximum utility separately. In a chaotic state, the average utilities of the manufacturer and retailer all decline, while that of the retailer declines even more. By selecting appropriate control parameter, the low-carbon dual-channel supply chain system can return to a stable state from chaos again. The research of this paper is of great significance to price decisions of participants and supply chain operation management.

Chaos in a duopoly model of technological innovation with bounded rationality based on constant conjectural variation

Abstract In this paper, we construct a duopoly model of technological innovation based on constant conjectural variation under the assumption of bounded rationality. The equilibrium points of constant conjectural variation and their local stability are investigated. By the theoretical proof, we prove that the model displays Li-Yorke chaos and distributional chaos in the ranges of the output adjusted coefficients by employing the snap-back repeller theory. The technological innovation is the main source of the core competitiveness of a firm, and technological content is an important factor of technological innovation. We find that the firms have the optimal technological contents while they could have the maximal profits in the two different situations, non-cooperative technological innovation and cooperative technological innovation. It’s proved that the cooperation of technological innovation can increase the marginal technological content, and benefit to technological progress. Then Numerical simulations are used to illustrate the dynamic analysis of the model under the situations with different technological innovation and conjectural variation. When the output adjusted coefficient changes, a series of complex phenomena including bifurcation, chaos and strange attractor can be observed in our model. And when the technological content is larger, the chaos of the system finally appears.

The Complexity Analysis for Price Game Model of Risk-Averse Supply Chain Considering Fairness Concern

This paper, considering risk aversion and fair concern, establishes a dynamic price game model of a dual-channel supply chain in which dual-channel retailer sells products through traditional channel and online channel and the online retailer only sells products through online channel. The stability of the system and the influences of different parameter values on utilities are analyzed emphatically using game theory and nonlinear dynamic theory, such as 2D and 3D bifurcation diagram, parameter plot basin, chaos attractor, and sensitivity to initial value. The results find that the system is more likely to lose stability and fall into chaos with the customer demand fluctuating greatly. The system enters into chaos through flip bifurcation with the increase of the price adjustment speed; adjusting the risk-aversion levels or the fairness concern levels of the two retailers can make the system be in a stable state or delay the occurrence of system instability. When the system is in chaos, the average utility of the online retailer will decrease and one of the dual-channel retailers will increase. Using the state feedback control method, the system can return to a stable state from chaos by selecting appropriate control parameters. The research of this paper is of great significance to the decision-makers’ price decision and supply chain operation management.

Homoclinic-doubling and homoclinic-gluing bifurcations in the Takens-Bogdanov normal form with symmetry.

In this paper, we investigate the dynamics of a fourth-order normal form near a double Takens-Bogdanov bifurcation. The reduced system of this normal form possesses eight pairs of homoclinic orbits for certain parameter values. The nonlinear time transformation method is applied to obtain an analytical approximation of the homoclinic orbit in the perturbed system and to construct the homoclinic bifurcation curve as well. Using numerical continuation, period-doubling and homoclinic-doubling cascades emanating from a codimension-2 bifurcation point are found. A codimension-2 homoclinic-gluing bifurcation point at which several homoclinic orbits concerning the origin glue together to form a new homoclinic orbit is also obtained. It is shown that in the vicinity of these bifurcation points, the system may exhibit chaos and chaotic attractors.

Application of chaos in a recurrent neural network to control in ill-posed problems: a novel autonomous robot arm.

Inspired by a viewpoint that complex/chaotic dynamics would play an important role in biological systems including the brain, chaotic dynamics introduced in a recurrent neural network was applied to robot control in ill-posed situations. By computer experiments we show that a model robot arm without an advanced visual processing function can catch a target object and bring it to a set position under ill-posed situations (e.g., in the presence of unknown obstacles). The key idea in these works is adaptive switching of a system parameter (connectivity) between a chaos regime and attractor regime in a neural network model, which generates, depending on environmental circumstances, either chaotic motions or definite motions corresponding to embedded attractors. The adaptive switching results in useful functional motions of the robot arm. These successful experiments indicate that chaotic dynamics is potentially useful for practical engineering control applications. In addition, this novel autonomous arm system is implemented in a hardware robot arm that can avoid obstacles and reach for a target in a situation where the robot can get only rough target information, including uncertainty, by means of a few sensors, as indicated in the appendix, A1 and A2.

Dynamical analysis of a novel single Opamp-based autonomous LC oscillator: antimonotonicity, chaos, and multiple attractors

A novel autonomous RLCC-Diodes-Opamp chaotic oscillator with a pair of antiparallel semiconductor diodes implementing hyperbolic sine nonlinearity is introduced. Basic dynamic properties of the new system are categorized numerically with respect to its parameters by exploiting standard nonlinear analysis tools such as time series, bifurcation diagrams, plots of largest Lyapunov exponent, phase portraits, Poincare sections, and basins of attraction. Some striking phenomena are reported including antimonotonicity, period doubling, crises, chaos, hysteresis, and coexisting bifurcations. More importantly, one of the most interesting results is the finding of various regions in the parameters’ space in which the proposed oscillator develops the phenomenon of multiple attractors characterized by the coexistence of up to four disconnected periodic and chaotic attractors for the same values of parameters. Laboratory measurements are consistent with the theoretical results.