Duffing Chaotic System Research Articles

Soliton and lump and travelling wave solutions of the (3 + 1) dimensional KPB like equation with analysis of chaotic behaviors

Nonlinear evolution equations (NLEEs) have a wide range of applications in various fields, including physics, engineering, biology, economics, and more. These equations are used to describe complex systems that change over time, where the state or behavior of the system not only depends on the current situation, but may also be related to the past. A deep understanding of the solutions to NLEEs is of great significance for predicting and controlling the behavior of complex systems. The (3 + 1)-dimensional Kadomtsev-Petviashvili-Boussinesq-like (KPB-like) equation represents a partial differential equation that describes nonlinear wave phenomena in multi-dimensional spaces. This model is commonly employed in fields such as fluid dynamics, plasma physics, and nonlinear optics to study wave behaviors. This paper investigates the KPB-like equations, which have meaningful physical implications in describing nonlinear wave phenomena. Hirota bilinear method is employed to derive the bilinear form for the KPB-like equation and 1-soliton, 2-soliton, lump, and travelling wave solutions are studied to this kind of nonlinear model. In addition, the chaotic behaviors of soliton and lump solutions are explored via applying the Duffing chaotic system. To have a better understanding of dynamic structures, 3D, line, density and contour map plots of begotten results are displayed. Our results indicate the directness and effectiveness of the applied method for analyzing high-dimensional differential equations that arise in nonlinear science.

Soliton, breather, lump, interaction solutions and chaotic behavior for the (2+1)-dimensional KPSKR equation

Nonlinear evolution equations are widely applied in various fields, and understanding their solutions is crucial for predicting and controlling the behavior of complex systems. In this paper, the (2+1)-dimensional Kadomtsev–Petviashvili–Sawada–Kotera–Ramani (KPSKR) equation is investigated, which has rich physical significance in nonlinear waves. Making use of Hirota’s bilinear form, soliton, breather and lump solutions of the (2+1)-dimensional KPSKR equation are derived. The interactions between lump solutions and exponential function, as well as between lump solutions and hyperbolic cosine function, are explored. Furthermore, the chaotic behavior of 1-soliton, 2-soliton, lump and interaction solutions are studied via applying the Duffing chaotic system. The physical structure and characteristics of begotten results are illustrated through 3D plots and corresponding two-dimensional profiles. These results indicate that the strategies utilized are more direct and effective, enriching the study of dynamics in high-dimensional nonlinear differential equations.

Adaptive fuzzy finite-time PID backstepping control for chaotic systems with full states constraints and unmodeled dynamics

To solve the problem that the controlled object and system model cannot be fully mastered or the system parameters cannot be well obtained through conventional measurement methods, in this paper, a novel adaptive fuzzy finite-time PID backstepping control method is developed for a class of nonlinear chaotic systems with full state constraints and unmodeled dynamics. The unmodeled dynamics are handled by employing a dynamic signal. To avoid the violations of full state constraints (FSCs), a barrier Lyapunov function is used to implement a PID backstepping framework, whose merit consists in that three control gain constants KP, KI and KD with corresponding proportional relationships are designed to further improve control precision. Besides, the PID control method also embodies three dynamic gain components ΔKP, ΔKI and ΔKD, all of which are free functions that can be self-tuning online, and have a significant link with the adaptive law and modify accordingly. Finally, theoretical analysis confirms that all signals are ultimately uniformly bounded, the error signal converges to a nearby area near the origin in a finite-time, and the FSCs are not smashed. The practicality of the proposed method is confirmed by two simulation studies using the second-order Duffing chaotic system and third-order Chua's circuit.

Research on the Threshold Determination Method of the Duffing Chaotic System Based on Improved Permutation Entropy and Poincaré Mapping.

The transition from a chaotic to a periodic state in the Duffing chaotic oscillator detection system is crucial in detecting weak signals. However, accurately determining the critical threshold for this transition remains a challenging problem. Traditional methods such as Melnikov theory, the Poincaré section quantitative discrimination method, and experimental analyses based on phase diagram segmentation have limitations in accuracy and efficiency. In addition, they require large computational data and complex algorithms while having slow convergence. Improved permutation entropy incorporates signal amplitude information on the basis of permutation entropy and has better noise resistance. According to the characteristics of improved permutation entropy, a threshold determination method for the Duffing chaotic oscillator detection system based on improved permutation entropy (IPE) and Poincaré mapping (PM) is proposed. This new metric is called Poincaré mapping improved permutation entropy (PMIPE). The simulation results and the verification results of real underwater acoustic signals indicate that our proposed method outperforms traditional methods in terms of accuracy, simplicity, and stability.

Fractional order PI observer for a class of fractional order systems

Currently the study of fractional order systems has become of great research interest, in particular the state estimation stands out within the lines of studies for this type of systems. Different methodologies have been proposed in order to solve this problem, however most of the techniques involve complete information of the system. Thus, this work presents a methodology for state estimation in a class of fractional order systems based on a fractional order observer which is constructed through an algebraic technique. This observer presents some significant properties, for instance, only variables of interest are estimated, i.e., it is a non redundant observer, it does not need complete information of the system and the initial conditions for the observer can be freely assigned. The stability of this observer is analyzed with the global Mittag-Leffler boundedness approach. Finally, the effectiveness and accuracy of the proposed methodology is verified through state estimation of a fractional-order Duffing chaotic system.

Studying the Effect of Initial Conditions and System Parameters on the Behavior of a Chaotic Duffing System

This work presents a five-period chaotic system called the Duffing system, in which the effect of changing the initial conditions and system parameters d, g and w, on the behavior of the chaotic system, is studied. This work provides a complete analysis of system properties such as time series, attractors, and Fast Fourier Transformation Spectrum (FFT). The system shows periodic behavior when the initial conditions xi and yi equal 0.8 and 0, respectively, then the system becomes quasi-chaotic when the initial conditions xi and yi equal 0 and 0, and when the system parameters d, g and w equal 0.02, 8 and 0.09. Finally, the system exhibits hyperchaotic behavior at the first two conditions, 0 and 0, and the bandwidth of the chaotic signal becomes wider (2 a.u.) than in the first case. So, this system can be used in many physical applications, such as encrypting confidential information.

Mechanical Chaotic Duffing System with Magnetic Springs

Mechanical systems with inherent chaotic behavior are of notable practical interest due to their applicability in many fields of technology, from industrial mills and concrete mixers to microscopic micromechanical random bit generators. One of the most generic mathematical models for designing chaotic mechanical systems is the Duffing oscillator, which demonstrates chaotic motion under periodic excitation. The mechanical implementation of Duffing oscillator requires nonlinear springs, which can be implemented using different physical principles. In the current study, we propose the mechanical Duffing oscillator with magnetic springs as a low-wear, robust and easy-to-implement solution. We show by simulation and experimentation that the developed mechanical system performs chaotic oscillations in a wide range of parameters. The proposed design can be revised in a problem-specific manner and achieve many practical applications.

Preface

The 2022 6th International Conference on Mechanics, Mathematics and Applied Physics (ICMMAP 2022) was successfully held in Qingdao, China on August 19-21, 2022. As an annual conference, ICMMAP 2022 provided a platform for experts and scholars from home and abroad to share scientific research results and cutting-edge technologies, and strengthen academic research and discussion, promoting cooperation in the industrialization of academic results.In the conference, we were honored to invite four professors to serve as our Conference General Chairs. Respectively, they are Prof. Harith Bin Ahmad from University of Malaya, Malaysia, Prof. Ooi Chong Heng from University of Malaya, Malaysia, Prof. Rozaini Roslan FIMA from Universiti Tun Hussein Onn Malaysia, Malaysia and Prof. Mário Fernando Santos from University of Aveiro, Portugal. 170 individuals attended the conference covering keynote speeches and oral presentations. Firstly, the keynote speakers were each allocated 30-45 minutes to address their speeches. Then in the oral presentations, the excellent papers we had selected were presented by their authors by turns.Five distinguished professors were invited to hold keynote speeches during the conference. Among them, Prof. Weiwei Zhang from Dongguan University of Technology, China performed a thought-provoking speech on the title Studies on Chaos System Inspection of Weak Ultrasonic Guided Wave. To improve the detection efficiency of small defects in the pipeline, a weak guided wave identification method based on the parameter sensitivity of the Duffing chaotic system was proposed in this research. Then, two chaotic systems were proposed to identify the amplitude identification and arriving time of the defect echo, and the size and location of pipeline defects were identified based on them. Moreover, A. Prof. Fengming Du from Dalian Maritime University, China made a report on: Research on Thermal and Mechanical Behavior of Continuous Casting Mold. This paper focused on the heat transfer, solidification, deformation and friction in continuous casting mold and slab, and is significant for improving slab quality and optimizing process parameters. Their insightful speeches had triggered heated discussion. And every participant praised this conference for disseminating useful and insightful knowledge.After months of hard work and preparation, we selected a bunch of high-quality papers from the submissions and compiled them into the proceedings after rigorously reviewing them. These papers feature but are not limited to the following topics: Applied Mechanics, Aeroelasticity, Combinatorial Mathematics, Electromagnetism, etc. All the papers have been checked through rigorous review and processes to meet the requirements of publication.We would like to acknowledge all of those who supported ICMMAP 2022 and made it a great success. Particularly, we would like to thank the Journal of Physics: Conference Series, for all the efforts of its colleagues in publishing this paper volume. We sincerely hope that the ICMMAP 2022 turned out to be a forum for excellent discussions that enable new ideas to come about, promoting collaborative research.The Committee of ICMMAP 2022List of Committee member is available in this Pdf.

An Improved Linear State Error Feedback Synchronization Control Criteria for a Six-Axis Duffing Chaotic System

By the employment of an improved linear state error feedback method, the synchronization control of a six-axis Duffing chaotic system was studied. Compared with previous methods, it has two advantages: the nonlinear term of the Duffing chaotic system is reserved in the synchronization error system, and the trajectory bound of the response system is predicted in advance to deduce the synchronization criteria. First, a typical ship parametric excitation roll chaotic system with parametric and forced excitation is taken as the control object to realize chaos synchronization control. Then, the control method is applied to the conventional six-axis Duffing oscillatory chaotic system and the four-axis Duffing oscillatory chaotic system. Finally, three simulation cases are presented to illustrate the validity of the synchronization criteria.

Human-autonomous devices for weak signal detection method based on multimedia chaos theory

The detection of weak signals has been widely used in communication, radar and other fields. The detection of weak signals in the background of strong noise is an important research hotspot of modern information theory, and prompts people to explore and study new theories and new methods of weak signal detection. The device takes Holmes-type Duffing mapping as the research object, and uses the Lyapunov exponent as the criterion for chaos identification. The chaotic critical value of the equation is changed from chaotic state to periodic state. Whether it contains the detection algorithm of the target signal, the signal detection for the unknown frequency and the Holmes-type Duffing system is improved by the sliding mode variable structure control method in the control theory. The simulation results show that the improved chaotic Duffing system can effectively suppress the noise and detect the frequency of the weak signal through the power spectrum of the system.

Threshold determination method of Duffing chaotic system based on multi-scale entropy

Entropy is one of the parameters describing the state of matter in thermodynamics, it can be used to measure the degree of confusion. The entropy of the signal can be used to express the complexity of the signal. The threshold for the transition of the Duffing chaotic system from the critical chaotic state to the large-scale periodic state is called the transition threshold. It is an important parameter for the analysis of chaotic systems, and its solution method is one of the problems urgently to be solved in chaos theory. If the jump threshold is smaller than the real threshold of the system, it will affect its detection signal-to-noise ratio. If the jump threshold is larger than the real threshold, it will cause incorrect detection results, so it is very important to accurately determine the jump threshold. In this study, we found that the multiscale sample entropy value of the Duffing system is significantly different when the system is in the chaotic state and the periodic state, when the system is in a chaotic state, the entropy value is larger, when the system is in a periodic state, the entropy value is smaller, and when the system enters the periodic state, the multiscale entropy value tends to be stable, this paper proposes to use this phenomenon to determine the transition threshold by analyzing the relationship between the entropy of the system and the amplitude of the driving force. When the entropy value is obviously smaller and tends to be stable, the corresponding driving force amplitude is the jump threshold. using this method, the jump threshold of the sinusoidal signal and square wave signal detection system is calculated, the results show that the method is fast, accurate and simple to calculate. However, this method may have a problem that the calculated threshold value is smaller than the real threshold value, our analysis is that the random selection of the subsequence used for calculation causes the calculation threshold value to be too small, so the method is improved in conjunction with genetic algorithm, using genetic algorithm to find the most complicated subsequence in the whole sequence, then this subsequence is used to solve the threshold, Through a large number of calculations and analysis, it can be seen that the problem of a small threshold is no longer present, and the improved method can obtain the jump threshold of the Duffing system very accurately.

Onset of stochastic synchronization induced by diffusion processes in a generalized Duffing system

In this study, two generalized Duffing systems, which are not directly coupled but subjected to a common stochastic excitation, are studied to address the problem of chaotic synchronization. Three types of stochastic diffusion processes generated by the first-order filter are taken as driving excitations and employed to survey their effects on the onset of synchronization with identical spectral densities but different probability density functions (PDFs). By applying the mean largest Lyapunov exponent, we observe that synchronization can indeed occur in the chaotic Duffing systems when the excitation amplitudes are sufficiently larger than the thresholds needed of the synchronization response. Meanwhile, we show the effects for the three diffusion processes on the onset of synchronization are similar. Above all, the most noteworthy result of this work is that the synchronization thresholds induced by the three diffusion processes are almost identical regardless of the selections of different PDFs. In this sense, the PDFs have almost no influence on the onset of synchronization, while, the threshold amplitudes are merely determined by the spectral densities. This can be comprehended from the view of “energy” since the identical spectral densities of the three diffussion processes embody the same total power.

Dynamic Analysis of Modified Duffing System via Intermittent External Force and Its Application

Over the past century, a tremendous amount of work on the Duffing system has been done with continuous external force, including analytical and numerical solution methods, and the dynamic behavior of physical systems. However, hows does the Duffing oscillator behave if the external force is intermittent? This paper investigates the Duffing oscillator with intermittent external force, and a modified Duffing chaotic system is proposed. Different from the continuous-control method, an intermittent external force of cosine function was designed to control the Duffing oscillator, such that the modified Duffing (MD) system could behave chaotically. The dynamic characteristics of MD system, such as the strange attractors, Lyapunov exponent spectra, and bifurcation diagram spectra were outlined with numerical simulations. Numerical results showed that there existed a positive Lyapunov exponent in some parameter intervals. Furthermore, by combining it with chaos scrambling and chaos XOR encryption, a chaos-based encryption algorithm was designed via the pseudorandom sequence generated from the MD. Finally, feasibility and validity were verified by simulation experiments of image encryption.

Chaos-based signal detection with discrete-time processing of the Duffing attractor

The results of detection of periodic signals using the chaos theory based on discrete processing of the Duffing attractor in the Poincare section were considered.A chaotic Duffing system characterized by high sensitivity to periodic signals and a possibility of implementation by means of a relatively simple circuit was chosen for the study.Response of the Duffing system to the periodic influence was analyzed. It was shown that when amplitude of periodic components of the input signal grows at a frequency of driving oscillations, there is a shift of the phase trajectory along the Poincare section which is characterized by fractal geometry. Types of the Duffing attractor changes that result from the influence of a periodic input signal were determined. Control regions for recording types of the phase trajectory dynamics were identified in the phase plane formed by the output signal and its derivative. In accordance with the characteristics of the obtained phase trajectories, a truth table was constructed. It enables estimation of influence of the periodic component with a sufficiently large time sampling increment which is important for ensuring speed of the signal processing devices. Transforms were obtained that describe the process of detecting periodic signals by discrete processing of the Duffing attractor in the Poincare section.Based on the formulated transforms and the truth table, a block diagram of a device for detecting periodic signals in noise was proposed. The proposed device can be used as an input unit to implement the Duffing system based on an analog electric circuit.Values of discrete estimates of amplitude of the periodic component of the input signal according to the shift of the phase trajectory of the Duffing system with respect to the attractor in the Poincare section were obtained. According to the modeling results, the proposed circuit makes it possible to detect periodic signals at low values of the signal-to-noise ratio.

An adaptive data detection algorithm based on intermittent chaos with strong noise background

In order to realize the signal detection under the condition of lower SNR, this paper introduced the adaptive phase length based on the Duffing chaotic system and verified the measured signal at the optimal excitation frequency. The existence of the target signal was judged by observing whether there are two consecutive intermittent chaos in the time domain. Then the envelope of the intermittent chaos was obtained by Hilbert transform. Finally, the exact value of envelope spectrum was obtained by using the one-and-half-dimension spectrum, which can calculate the precise value of the frequency of the signal to be measured. The experimental results showed that the proposed algorithm can achieve a lower SNR than the conventional detection. Compared with the general chaotic detection, this algorithm can realize smart self-adaptation. It is unnecessary to specify different excitation frequencies and chaotic thresholds for different frequencies to be measured. In addition to the existence of the target signal judgment, the algorithm can also achieve accurate calculation of the frequency of the signal to be measured.

Robust backstepping control for uncertain chaotic multi-inputs multi-outputs systems using type 2 fuzzy systems

In this paper, a new backstepping type 2 fuzzy control is developed for a class of uncertain multi-inputs multi-outputs (MIMO) chaotic systems, with a complicate of uncertainties and external disturbances. The system consists of interconnected subsystems that are duffing equation. In the control design, type 2 fuzzy logic systems are used to approximate the unknown functions. Hybrid adaptive robust tracking control schemes that are based upon a combination of bounds of type 2 fuzzy approximation parameters and the backstepping design are developed such that all the states and signals are bounded and the proposed approach alleviate the online computation burden and improves the robustness to dynamic uncertainties and external disturbances. Finally, the control of chaotic duffing systems with unknown parameters is given as example to verify the effectiveness of the proposed approach.

Defect detection of pipes using Lyapunov dimension of Duffing oscillator based on ultrasonic guided waves

This study proposes a novel small defect detection approach for steel pipes using the Lyapunov dimension (D) of the Duffing chaotic system based on ultrasonic guided waves. In this paper, inspection model is constructed by inputting the measured guided wave signal into the Duffing equation as the external turbulent driving force term and thenDis calculated. The properties of the Duffing system's noise immunity are first demonstrated theoretically based on the Lyapunov exponents. By comparingDof the Duffing inspection system between the conditions of the inputted pure noise and the guided wave signal, the amplitude of the periodic force (F), the important parameter of the Duffing inspection system, could be determined. The values of other parameters of the Duffing inspection system are subsequently determined according to the numerical investigation. Furthermore, a time-moving window function is constructed to scan along the measured signal to locate the defect. And the small defect echo signal polluted by the noise is illustrated to prove the availability of the proposed method. Both numerical and experimental results show that the proposed approach can be used to improve the sensitivity of small defect detection and locate the small defect in pipes.

Mechanism for the coexistence phenomenon of random phase suppressing chaos and stochastic resonance in Duffing system

Noise, which is ubiquitous in real systems, has been the subject of various and extensive studies in nonlinear dynamical systems. In general, noise is regarded as an obstacle. However, counterintuitive effects of noise on nonlinear systems have recently been recognized, such as noise suppressing chaos and stochastic resonance. Although the noise suppressing chaos and stochastic resonance have been studied extensively, little is reported about their relation under coexistent condition. In this paper by using Lyapunov exponent, Poincaré section, time history and power spectrum, the effect of random phase on chaotic Duffing system is investigated. It is found that as the intensity of random phase increases the chaotic behavior is suppressed and the power response amplitude passes through a maximum at an optimal noise intensity, which implies that the coexistence phenomenon of noise suppressing chaos and stochastic resonance occurs. Furthermore, an interesting phenomenon is that the optimal noise intensity at the SR curve is just the critical point from chaos to non-chaos. The average effect analysis of harmonic excitation with random phase and the system’s bifurcation diagram shows that the increasing of random phase intensity is in general equivalent to the decreasing of harmonic excitation amplitude of the original deterministic system. So there exists the critical noise intensity where the chaotic motion of large range disintegrates and non-chaotic motion of small scope appears, which implies the enhancing of the regularity of system motion and the increasing of the response amplitude at the input signal frequency. After that, the excess noise will not change the stability of the system any more, but will increase the degree of random fluctuation near the stable motion, resulting in the decreasing of the response amplitude. Therefore, the formation of stochastic resonance is due to the dynamical mechanism of random phase suppressing chaos.

Application of the Rise Feedback Control in Chaotic Systems

In this paper a new RISE controller is gained to control chaos in a tracking task. The technique copes with the chattering phenomenon whilst works for different classes of nonlinear systems incorporating different relative degrees. This control strategy will be primarily implemented on a Duffing chaotic system. In order to assess performance of the controller, the technique will be implemented on a more complex system, so called Genesio-Tesi dynamic. The result will be finally compared with an optimal controller. The capability of the proposed feedback technique to control the chaos is verified through simulation study with respect to similar classic approaches.

State Observer Synchronization Used in the Three-Dimensional Duffing System

Synchronization of chaotic systems has attracted extensive concern in the past few years. In this study, we investigate a new structure of Duffing system by the variable decomposition method. Then, we analyze the state observer synchronization based on the new Duffing system. It is proved theoretically that the designed observer can keep synchronization with Duffing chaotic system in transmitter. The design is presented reasonably with the conditional Lyapunov exponents, and its effectiveness is clearly shown in simulation results.