Chaotic Dynamic Behavior Research Articles

Unstable periodic orbits analysis in the Qi system

We use the variational method to extract the short periodic orbits of the Qi system within a certain topological length. The chaotic dynamical behaviors of the Qi system with five equilibria are analyzed by the means of phase portraits, Lyapunov exponents, and Poincaré maps. Based on several periodic orbits with different sizes and shapes, they are encoded systematically with two letters or four letters for two different sets of parameters. The periodic orbits outside the attractor with complex topology are discovered by accident. In addition, the bifurcations of cycles and the bifurcations of equilibria in the Qi system are explored by different methods respectively. In this process, the rule of orbital period changing with parameters is also investigated. The calculation and classification method of periodic orbits in this study can be widely used in other similar low-dimensional dissipative systems.

Design and Implementation of Simplified Symmetry Chaotic Circuit

In order to reduce the circuit cost and improve the stability and flexibility of the circuit, a simplified symmetry chaotic circuit was designed and implemented by using an inverse integration circuit and a voltage follower as isolators. The change of different symmetry chaotic dynamic behaviors caused by the change of parameters can be realized by adjusting the time constant. The behavior coexistence characteristics and amplitude control characteristics under different initial conditions were verified. The results of circuit experiments are in good agreement with those of numerical simulation and theoretical analysis. This method is effective and feasible.

On Petrenko's deviations and the Julia limiting directions of solutions of complex differential equations

The Julia set J(f) of a transcendental meromorphic function f has chaotic behavior in the complex dynamics. The ray arg⁡z=θ∈[0,2π) is said to be a limiting direction of J(f) if there is an unbounded sequence {zn}⊆J(f) such that limn→∞⁡arg⁡zn=θ. In this paper, we mainly investigated the Julia limiting directions of entire solutions of complex differential equations, of which coefficients are associated with Petrenko's deviations. In fact, we proved the lower bounds of the sets of Julia limiting directions of these solutions have closed relations with Petrenko's deviations of the coefficients.

Dynamical analysis of a new chaotic system and its application in ADC

The improvement of Analog-to-Digital Converter (ADC) performance is crucial to improve the data quality of the signal acquisition system. Therefore, based on the dithering technology, combined with the chaotic system, the study designs and analyses a novel method to improve the Spurious-Free dynamic range (SFDR) of the ADC. First, a new five-dimensional chaotic system is proposed in this paper. The chaotic characteristics and dynamic behavior of the new system have been deeply analysed. The analysis methods mainly consist of the Lyapunov exponent, bifurcation diagram, power spectrum and sample entropy. The corresponding simulation model has been developed in Simulink, and the simulation results prove the reality and feasibility of the new chaotic system. Second, the random number homogenization universal algorithm is used to modify the amplitude of the chaotic sequences generated by the new chaotic system. The uniformity and correlation analysis prove that the Dither signal has strong uniformity and randomness. Third, a behaviour-level simulation model of the pipeline ADC is built; then, the narrowband, broadband and different amplitude Dither signals are added to verify the improvement effect for the dynamic performance of the ADC. The experimental results demonstrate that the designed method can improve the SFDR of the ADC by 10.37 dBc, which effectively eliminates the harmonic energy and improves the dynamic performance of the ADC.

A New Robust Method to Investigate Dynamic Instability of FTV for the Double Tripod Industrial Driveshafts in the Principal Parametric Resonance Region

The present work aims to design a robust method to detect and certify the deterministic chaos or ergodic process for the forced torsional vibrations (FTV) of a double tripod industrial driveshaft (DTID) in transition through the principal parametric resonance region (PPRR) which is considered by the researchers in the field as one of the most important resonance regions for the systems having parametric excitations. The DTID’s model for FTV considers the following effects: nonuniformities of inertial characteristics of the DTID’s elements, the harmonic torque excitation induced by the asynchronous electrical motor used for a heavy-duty grain mill, and the harmonic reaction torque generated by different granulation of the substance needed to be milled. Based on these aspects, a model of the FTV for the DTID was designed which was a modified, physically consistent model already used by the authors to investigate the FTV of automotive driveshafts (homokinetic transmission). For the DTID elements, the dynamic instability for nonstationary FTV in the PPRR using time–history analysis (THA) was analyzed—THA represents the phase portraits. Time–history analysis is a detection method for possible chaotic dynamic behavior for the nonstationary FTV (NFTV) in transition through PPRR. If this dynamic behavior was seen, a new robust method LEA–PM was created to certify and confirm the deterministic chaos for the NFTV of DTID. The new method, LEA–PM, is composed of the Lyapunov exponent’s approach (LEA) coupled with the Poincaré Map (PM) applied to the global system of differential equations that describe the FTV of DTID in the PPRR. This new robust method, which embeds LEA and PM, LEA–PM, establishes if the mechanical system has a deterministic chaotic dynamic behavior (strange attractor) or an ergodic dynamic process in this resonant region. LEA represents a new method that includes not only the maximal Lyapunov exponent method (MLEM) but also new mathematical criteria that is “the sum of all Lyapunov exponents has to be negative” which, coupled with MLEM, indicates the presence of deterministic chaos (strange attractors). THA–LEA–PM had been used for the NFTV of DTID computing the phase portraits, the Lyapunov exponents, and representing the Poincaré Maps of the NFTV for the DTID’s elements in transition through PPRR, founding deterministic chaos or ergodic dynamic behavior. Based on the obtained results, numerical simulations revealed the pitting manifestations of the DTID’s elements, typical for the geared systems transmission, mentioned recently in experimental data research for the homokinetic transmissions. Using the new robust method, THA–LEA–PM (time–history analysis coupled with LEA–PM) can be used in future research for chaotic dynamic analysis of DTID’s NFTV transition through superharmonic resonances, subharmonic resonances, combination resonances, and internal resonances. Time–history analysis as a detection method for chaos and LEA–PM as a certifying method for deterministic chaos can be integrated as a design tool for DTID’s FTV control of the homokinetic transmission.

Predicting the dynamical behaviors for chaotic semiconductor lasers by reservoir computing.

We demonstrate the successful prediction of the continuous intensity time series and reproduction of the underlying dynamical behaviors for a chaotic semiconductor laser by reservoir computing. The laser subject to continuous-wave optical injection is considered using the rate-equation model. A reservoir network is constructed and trained using over 2 × 104 data points sampled every 1.19 ps from the simulated chaotic intensity time series. Upon careful optimization of the reservoir parameters, the future evolution of the continuous intensity time series can be accurately predicted for a time duration of longer than 0.6 ns, which is six times the reciprocal of the relaxation resonance frequency of the laser. Moreover, we demonstrate for the first time, to the best of our knowledge, that the predicted intensity time series allows for accurate reproduction of the chaotic dynamical behaviors, including the microwave power spectrum, probability density function, and the chaotic attractor. In general, the demonstrated approach offers a relatively high flexibility in the choice of reservoir parameters according to the simulation results, and it provides new insights into the learning and prediction of semiconductor laser dynamics based on measured intensity time series.

Passive chaos suppression for the planar slider-crank mechanism with a clearance joint by attached vibro-impact oscillator

Chaotic dynamic behavior in clearance joints is the prominent nonlinear factor for multibody systems. This study presents a passive chaos suppression method to stabilize the planar slider-crank mechanism with a clearance joint. The strategy resorts to a vibro-impact oscillator attached to the slider. Primarily, the impact principle is employed to characterize the dynamics in clearance joints. The mathematical model of the integral system is conducted subsequently. The chaotic system behavior is evaluated by bifurcation analysis varying clearance size and crank speed. When the controller is activated, parametric optimizations of the stiffness, clearance coefficient, and particle mass are conducted to achieve optimum chaos elimination. Phase portraits, the largest Lyapunov exponent, and time-frequency distribution are evaluated. Numerical results illuminate that an adjusted oscillator can suppress the chaotic high-frequency composition of system response and help to maintain continuous contact between the joint elements. The results suggest applying the identical clearance size in the revolute joint and vibro-impact oscillator for fast optimization. Moreover, the method is still technology valid when considering different contact principles. Ultimately, a test platform is established to achieve some experimental validations.

Qualitative geometric analysis of traveling wave solutions of the modified equal width Burgers equation

This paper devotes to the qualitative geometric analysis of the traveling wave solutions of MEW‐Burgers wave equation. Firstly, MEW‐Burgers equation is transformed into an equivalent planar dynamical system by using traveling wave transformation. Then the global structure of the planar system is presented, and solitary waves, kink waves (anti‐kink waves), and periodic waves are found. Secondly, Jacobi stability for the planar system is studied based on KCC theory, and Jacobi stability of any point on the trajectory of system is dicussed. The dynamical behavior of the deviation vector near the equilibrium points is analyzed, and the numerical simulation is consistent with the theoretical analysis. Lyapunov stability and Jacobi stability of the equilibrium points are also compared and analyzed. The obtaining results show that Lyapunov stability of the equilibrium points of the system is not exactly consistent with Jacobi stability. Finally, the planar system with periodic disturbance is transformed into a six dimensional nonlinear system, and the periodic, quasi‐periodic, and chaotic dynamical behaviors of the system are numerically simulated.

Chaotic Dynamics in the 2D System of Nonsmooth Ordinary Differential Equations

Over the last decade, the chaotic behaviors of dynamical systems have been extensively explored. Recently, discovering or developing a 2D system of ordinary differential equations (ODEs) capable of exhibiting chaotic dynamical behaviors is an attractive research topic. In this study, a chaotic system with a 2D system of nonsmooth ODEs has been developed. This system is can exhibit chaotic dynamical behaviors. Its main dynamical behaviors, including time-series trajectories, phase portraits of attractors, and equilibria and their stability, have been investigated. The developed system has been verified by an excessive variety of fascinating chaotic behaviors, such as chaotic attractor, symmetry, sensitivity to initial conditions (ICs), fractal dimension, autocorrelation, power spectrum, Lyapunov exponent, and bifurcation diagram. Analytical and numerical simulations are used to study the dynamical behaviors of such a system. The developed system has extreme sensitivity to ICs, a fractal dimension of more than 1.8 and less than 2.05, an autocorrelation fluctuating randomly about an average of zero, a broadband power spectrum, and one positive Lyapunov exponent. The obtained numerical simulation results have proven the capability of the developed 2D system for exciting chaotic dynamical behaviors

A Novel Discrete‐Time Leslie–Gower Model with the Impact of Allee Effect in Predator Population

The discrete‐time system has more complex and chaotic dynamical behaviors as compared to the continuous‐time system. This paper extends a discrete Leslie–Gower predator‐prey system with the Allee effect in the predator’s population, whose dynamics are analyzed and explored. We have determined the equilibrium points and studied their local stability properties. We find that the system undergoes flip bifurcation and Neimark–Sacker bifurcation around the interior equilibrium point by choosing the Allee parameter as a bifurcation parameter. We discuss the stability and direction of both bifurcations with the help of the normal form theory and center manifold theorem. The flip bifurcation and Neimark–Sacker bifurcation are the most common routes to the chaotic orbit in the discrete system. Moreover, we utilize state feedback, pole placement, and hybrid control methods to control the chaos in the system. The work is complete with the numerical simulations to confirm the analytical findings.

Performance of the 2D Coupled Map Lattice Model and Its Application in Image Encryption

The two‐dimensional coupled map lattice (2D CML) model has been extensively employed as the basis component for designing various schemes in the cryptography system due to its complicated chaotic dynamic behavior. In this study, we analyze the chaotic characteristics of the 2D CML model, such as the Lyapunov exponent (LE), synchronization stability, bifurcation, and ergodicity. We then show that the chaotic sequences generated by the 2D CML model are random according to the NIST testing. Furthermore, we propose an image encryption scheme based on the 2D CML model and Singular Value Decomposition (SVD). In our scheme, the SVD method is used to reduce the image storage, and the Red, Green, and Blue channels of a color image will be encrypted through confusion and diffusion. The simulation results, as well as the results of the comparison with other schemes, demonstrate that our scheme possesses outstanding statistics, excellent encryption performance, and high security. It has great potential for ensuring the security of digital images in real applications.

CDSK Modulator and Demodulator System Based on FPAA Technology

Experts have been concerned about the security of digital communication systems and the reduction of interception in recent years. Because of its inherent characteristics and high efficiency in encoding information, chaotic signals can play a significant role in this respect. A dynamically reconfigurable FPAA AN231E04 from Anadigm is used to implement of Correlation Delay Shift Keying (CDSK) communication system. The FPAA chip can be used in almost any electronic hardware for analog system implementations. It permits the experimental selection or amendment of the system structure to achieve its specific properties. The performance of the Rossler oscillator was evaluated via MATLAB Simulink environment. The practical implementation demonstrates and validates chaotic dynamic behavior. The effect of Additive White Gaussian Noise (AWGN) on the performance of the proposed CDSK systems was studied. Under the impact of the AWGN channel, the performance efficiency of the suggested CDSK and noncoherent differential chaos shift keying (NC-DCSK) systems were examined. The CDSK bit transfer technique is safer than the NC-DCSK bit transfer technology. The power consumption of the proposed system is (only 763±18 mW) according to performance studies.

Chaotic dynamic behavior of switched reluctance motor taking into account of magnetic saturation effects

Chaotic dynamic behavior of switched reluctance motor taking into account of magnetic saturation effects

Biomimetic Chaotic Sensor for Moderate Static Magnetic Field.

The effects of a static magnetic field on systems with chaotic dynamical behavior have attracted little attention so far. Here, Chua’s electronic circuit with an inductor placed in a static uniform magnetic field operating in a chaotic double-scroll regime is studied experimentally. The effect of the magnetic field on the duty cycle factor and the spike count rate, with spikes defined by crossings between the scrolls of the double-scroll attractor, is described. A slow monotonic variation in the duty cycle factor and constant spike count rate is observed for magnetic field intensities up to the threshold, where both these metrics change severely; the dynamic trajectory remains on one scroll and spikes disappear. The dependence of the static magnetic field intensity on Chua’s circuit resistivity at the threshold is given. Two biomimetic magnetic chaotic sensors are proposed: one based on one Chua’s circuit and another that can have various transfer functions and is composed of several independent Chua’s circuits.

The full phase space dynamics of a magnetically levitated electromagnetic vibration harvester

We consider the motion of an electromagnetic vibrational energy harvester (EMVEH) as function of the initial position and velocity and show that this displays a classical chaotic dynamical behavior. The EMVEH considered consists of three coaxial cylindrical permanent magnets and two coaxial coils. The polarities of the three magnets are chosen in such a way that the central magnet floats, with its lateral motion being prevented by enclosion in a hollow plastic tube. The motion of the floating magnet, caused by e.g. environmental vibrations, induces a current in the coils allowing electrical energy to be harvested. We analyze the behavior of the system using a numerical model employing experimentally verified expressions of the force between the magnets and the damping force between the floating magnet and the coils. We map out the phase space of the motion of the system with and without gravity, and show that this displays a fractal-like behavior and that certain driving frequencies and initial conditions allow a large power to be harvested, and that more stable states than two exists. Finally, we show that at leasts fifth order polynomial approximation is necessary to approximate the magnet-magnet force and correctly predict the system behavior.

Chaotic Dynamical Behavior of Coupled One-Dimensional Wave Equations

In this paper, we consider the chaotic oscillation of coupled one-dimensional wave equations. The symmetric nonlinearities of van der Pol type are proposed at the two boundary endpoints, which can cause the energy of the system to rise and fall within certain bounds. At the interconnected point of the wave equations, the energy is injected into the system through an anti-damping velocity feedback. We prove the existence of the snapback repeller when the parameters enter a certain regime, which causes the system to be chaotic. Numerical simulations are presented to illustrate our theoretical results.

Combination Forecast of Economic Chaos Based on Improved Genetic Algorithm

The deterministic economic system will also produce chaotic dynamic behaviour, so economic chaos is getting more and more attention, and the research of economic chaos forecasting methods has become an important topic at present. The traditional economic chaos forecasting models are mostly based on large samples, but in actual production activities, there are a large number of small-sample economic chaos problems, and there is still no effective solution. This paper proposes a combined forecasting model based on the traditional economic chaos forecasting method. First of all, through the decision tree classification, priority selection of features, rough prediction is achieved. Secondly, we use BP neural network to make secondary prediction. Because the initial weight is randomly selected, it is easy to fall into the defect of local minimum. This paper optimizes the BP neural network. Finally, the decision tree model and the BP neural network model optimized by the improved genetic algorithm are combined, and the combined model is optimized by the improved GA. This method can take advantage of many prediction models and combine the prediction information of multiple different prediction models to effectively improve the fitting ability of the model and improve the prediction accuracy.

Single Neuronal Dynamical System in Self-Feedbacked Hopfield Networks and Its Application in Image Encryption.

Image encryption is a confidential strategy to keep the information in digital images from being leaked. Due to excellent chaotic dynamic behavior, self-feedbacked Hopfield networks have been used to design image ciphers. However, Self-feedbacked Hopfield networks have complex structures, large computational amount and fixed parameters; these properties limit the application of them. In this paper, a single neuronal dynamical system in self-feedbacked Hopfield network is unveiled. The discrete form of single neuronal dynamical system is derived from a self-feedbacked Hopfield network. Chaotic performance evaluation indicates that the system has good complexity, high sensitivity, and a large chaotic parameter range. The system is also incorporated into a framework to improve its chaotic performance. The result shows the system is well adapted to this type of framework, which means that there is a lot of room for improvement in the system. To investigate its applications in image encryption, an image encryption scheme is then designed. Simulation results and security analysis indicate that the proposed scheme is highly resistant to various attacks and competitive with some exiting schemes.

An Energy-conserving Integrator for Conservative Hamiltonian Systems with Ten-dimensional Phase Space

Abstract In this paper, an implicit nonsymplectic exact energy-preserving integrator is specifically designed for a ten-dimensional phase-space conservative Hamiltonian system with five degrees of freedom. It is based on a suitable discretization-averaging of the Hamiltonian gradient, with a second-order accuracy to numerical solutions. A one-dimensional disordered discrete nonlinear Schrödinger equation and a post-Newtonian Hamiltonian system of spinning compact binaries are taken as our two examples. We demonstrate numerically that the proposed algorithm exhibits good long-term performance in the preservation of energy, if roundoff errors are neglected. This result is independent of time steps, initial orbital eccentricities, and regular and chaotic orbital dynamical behavior. In particular, the application of appropriately large time steps to the new algorithm is helpful in reducing time-consuming and roundoff errors. This new method, combined with fast Lyapunov indicators, is well suited to studying the influence of some parameters or initial conditions related to chaos in the two example problems. It is found that chaos in the former system is mainly responsible for one of the parameters. In the latter problem, a combination of small initial separations and high initial eccentricities can easily induce chaos.

Analysis and control of a 4D hyperchaotic system with passive control

In this study, the dynamical behavior of the four-dimensional (4D) hyperchaotic system is analyzed. Its chaotic dynamical behaviors and basic dynamical properties are presented by Lyapunov exponents, stability analysis, and Kaplan-Yorke dimension. Then, the control of 4D hyperchaotic system is implemented by using passive control. The global asymptotic stability of the system is guaranteed by using Lyapunov function. Simulation results are shown to validate all theoretical analysis and demonstrate the effectiveness of the proposed control method. By applying the passive controllers, the system under chaotic behavior converges to the equilibrium point at origin asymptotically.