Search Results (60)

This study investigates the dynamics of dust acoustic periodic waves in a three-component, unmagnetized dusty plasma system using generalized $(r^{\star },q^{\star })$ distributions. First, boundary conditions are applied to reduce the model to a second-order nonlinear ordinary differential equation. The Galilean transformation is subsequently applied to reformulate the second-order ordinary differential equation into an unperturbed dynamical system. Next, phase portraits of the system are examined under all possible conditions of the discriminant of the associated cubic polynomial, identifying regions of stability and instability. The Runge–Kutta method is employed to construct the phase portraits of the system. The Hamiltonian function of the unperturbed system is subsequently derived and used to analyze energy levels and verify the phase portraits. Under the influence of an external periodic perturbation, the quasi-periodic and chaotic dynamics of dust ion acoustic waves are explored. Chaos detection tools confirm the presence of quasi-periodic and chaotic patterns using Basin of attraction, Lyapunov exponents, Fractal Dimension, Bifurcation diagram, Poincaré map, Time analysis, Multi-stability analysis, Chaotic attractor, Return map, Power spectrum, and 3D and 2D phase portraits. In addition, the model’s response to different initial conditions was examined through sensitivity analysis. Full article

In this article, we investigate the fractional soliton solutions as well as the chaotic analysis of the fourth-order nonlinear Ablowitz–Kaup–Newell–Segur wave equation. This model is considered an intriguing high-order nonlinear partial differential equation that integrates additional spatial and dispersive effects to extend the concepts to more intricate wave dynamics, relevant in engineering and science for understanding complex phenomena. To examine the solitary wave solutions of the proposed model, we employ sophisticated analytical techniques, including the generalized projective Riccati equation method, the new improved generalized exponential rational function method, and the modified F-expansion method, along with mathematical simulations, to obtain a deeper insight into wave propagation. To explore desirable soliton solutions, the nonlinear partial differential equation is converted into its respective ordinary differential equations by wave transforms utilizing $\beta$-fractional derivatives. Further, the solutions in the forms of bright, dark, singular, combined, and complex solitons are secured. Various physical parameter values and arrangements are employed to investigate the soliton solutions of the system. Variations in parameter values result in specific behaviors of the solutions, which we illustrate via various types of visualizations. Additionally, a key aspect of this research involves analyzing the chaotic behavior of the governing model. A perturbed version of the system is derived and then analyzed using chaos detection techniques such as power spectrum analysis, Poincaré return maps, and basin attractor visualization. The study of nonlinear dynamics reveals the system’s sensitivity to initial conditions and its dependence on time-decay effects. This indicates that the system exhibits chaotic behavior under perturbations, where even minor variations in the starting conditions can lead to drastically different outcomes as time progresses. Such behavior underscores the complexity and unpredictability inherent in the system, highlighting the importance of understanding its chaotic dynamics. This study evaluates the effectiveness of currently employed methodologies and elucidates the specific behaviors of the system’s nonlinear dynamics, thus providing new insights into the field of high-dimensional nonlinear scientific wave phenomena. The results demonstrate the effectiveness and versatility of the approach used to address complex nonlinear partial differential equations. Full article

This study explores analytical soliton solutions for the cubic–quintic time-fractional nonlinear non-paraxial pulse transmission model. This versatile model finds numerous uses in fiber optic communication, nonlinear optics, and optical signal processing. The strength of the quintic and cubic nonlinear components plays a crucial role in nonlinear processes, such as self-phase modulation, self-focusing, and wave combining. The fractional nonlinear Schrödinger equation (FNLSE) facilitates precise control over the dynamic properties of optical solitons. Exact and methodical solutions include those involving trigonometric functions, Jacobian elliptical functions (JEFs), and the transformation of JEFs into solitary wave (SW) solutions. This study reveals that various soliton solutions, such as periodic, rational, kink, and SW solitons, are identified using the complete discrimination polynomial methods (CDSPM). The concepts of chaos and bifurcation serve as the framework for investigating the system qualitatively. We explore various techniques for detecting chaos, including three-dimensional and two-dimensional graphs, time-series analysis, and Poincarè maps. A sensitivity analysis is performed utilizing a variety of initial conditions. Full article

The dynamic response of symmetrical press mechanisms is severely affected by revolute clearance, translational clearance, and the elasticity of the components. Therefore, the coupling effects of disturbance factors were studied in this paper, including revolute clearance, translational clearance, and component elastic deformation; the influence of their coupling effects on the dynamic chaos characteristic are also discussed. A dynamic model of a rigid–flexible coupling mechanism with revolute clearance and translational clearance was established. Using MATLAB R2024a to solve the model, chaos identification was researched through phase diagrams, Poincaré maps, and maximum Lyapunov exponents. Under the parameters studied in this paper, the maximum Lyapunov exponents at revolute clearance A (X direction and Y direction) and translational clearance B (X direction and Y direction) were 0.0521, 0.0573, 0.3915, and −0.0287, respectively. The motion state of revolute pair A (X direction and Y direction) and translational pair B (X direction) were more prone to chaotic states; translational pair B (Y direction) was more prone to periodic motion. The influence of various factors on the dynamic response were analyzed. With the increase in driving speed and clearance value, as well as the decrease in friction coefficient, the stability of the mechanism weakened, and the vibration of the mechanism’s dynamic response intensified. This paper provides theoretical support for the establishment of precise dynamic models for multi-link symmetrical structure press mechanisms. Full article

There are three principal novelties in the present investigation. It is the first time Cohen–Grossberg-type neural networks are considered with the most general delay and advanced piecewise constant arguments. The model is alpha unpredictable in the sense of electrical inputs and is researched under the conditions of alpha unpredictable and Poisson stable outputs. Thus, the phenomenon of ultra Poincaré chaos, which can be indicated through the analysis of a single motion, is now confirmed for a most sophisticated neural network. Moreover, finally, the approach of pseudo-quasilinear reduction, in its most effective form is now expanded for strong nonlinearities with time switching. The complexity of the discussed model makes it universal and useful for various specific cases. Appropriate examples with simulations that support the theoretical results are provided. Full article

As an inherent property of polyvinyl chloride (PVC) gel material, viscoelasticity is closely related to the deformation of the material, which will affect its dynamic behavior. However, the existing theoretical model does not consider the influence of time-varying damping on its nonlinear vibration, which leads to the unclear nonlinear dynamic behavior of the material under the dual influence of viscoelasticity and electromechanical parameters and limits the further application of the material. Therefore, in this study, the standard linear solid (SLS) model was used to describe the time-varying dynamic change of viscoelasticity of PVC gel, and the electromechanical coupling second-order nonlinear constitutive equation of PVC gel actuator was established by combining the Gent free energy theory model. The harmonic resonance, stability and periodicity of PVC gel actuator under different loading conditions were investigated by using dynamic analysis methods such as phase path, Poincaré map, bifurcation diagram, and Lyapunov exponent. Through the systematic research in this study, the deformation law of PVC gel with time-varying damping under different electromechanical parameters was revealed, and the parameter control strategy of deformation stability and chaos was obtained, which provided the design method and theoretical basis for the further application of the material. Full article

While billiard systems of various shapes have been used as paradigmatic model systems in the fields of nonlinear dynamics and quantum chaos, few studies have investigated anisotropic billiards. Motivated by the tremendous advances in using and controlling electronic and optical mesoscopic systems with bilayer graphene (BLG), representing an easily accessible anisotropic material for electrons when trigonal warping is present, we investigate billiards of various anisotropies and geometries using a trajectory-tracing approach founded on the concept of ray–wave correspondence. We find that the presence of anisotropy can change the billiards’ dynamics dramatically from its isotropic counterpart. It may induce chaotic and mixed dynamics in otherwise integrable systems and may stabilize originally unstable trajectories. We characterize the dynamics of anisotropic billiards in real and phase space using Lyapunov exponents and the Poincaré surface of section as phase space representation. Full article

The isolated boost full-bridge converter (IBFBC) is a DC–DC conversion topology that achieves a high boost ratio and provides electrical isolation, making it suitable for applications requiring both. Its operational dynamics are often intricate due to its inherent characteristics and the prevalent usage of nonlinear switching elements, leading to bifurcation and chaos. Chaos theory was employed to investigate the impact of changes in the voltage feedback coefficient K and input voltage E on the dynamic behavior of the IBFBC when operating in discontinuous conduction mode (DCM). Based on an analysis of its operating principles, a discrete iterative mapping model of the system in DCM is constructed using the stroboscopic mapping method. The effects of two control parameters, namely the proportional coefficient and input voltage, on system performance are studied using bifurcation diagrams, fold diagrams, Poincaré sections, and Lyapunov exponents. Simulation experiments are conducted using time-domain and waveform diagrams to validate the discrete mapping model and confirm the correctness of the theoretical analysis. The results indicate that when the IBFBC operates in DCM, its operating state is influenced by the voltage feedback coefficient $K$ and input voltage $E$. Under varying values of $K$ and $E$, the system may operate in a single-period stable state, multi-period bifurcation, or chaotic state, exhibiting typical nonlinear behavior. Full article

The switched arrival system is a typical hybrid system that is commonly used to simulate industrial control systems. The corresponding mathematical model and switching time are described. In order to be closer to the actual industrial control systems, the switching time is changed from 0 to greater than 0. In this case, the system not only generates chaos but also system losses. For this purpose, firstly, the causes of system losses are analyzed. Secondly, the Poincare section is selected to define the control target—periodic orbits. And then, the delayed impulsive feedback method is improved for the system at a switching time greater than 0, and extended to each boundary in order to enhance the control effect. This not only controls chaos in the system but also reduces the loss ratio and detects periodic orbits. Finally, numerical simulations of the system orbits and loss ratio with and without implementing control are compared. The possible intervals for the optimal control coefficient under the same initial conditions are detected. Period-1 orbits are detected at switching times greater than 0, and the stability of system operation is verified. Full article

In the present paper, we study both classical and quantum Hénon–Heiles systems. In particular, we make a comparison between the classical and quantum trajectories of integrable and nonintegrable Hénon–Heiles Hamiltonians. From a classical standpoint, we study both theoretically and numerically the form of invariant curves in the Poincaré surfaces of section for several values of the coupling parameter in the integrable case and compare them with those in the nonintegrable case. Then, we examine the corresponding Bohmian trajectories, and we find that they are chaotic in both cases, but with chaos emerging at different times. Full article

The conventional winding number method is extended in this research to characterize the nonlinear dynamical systems, especially in differentiating partially predictable chaos from strong chaos. On modern robotics’ challenges with increased degrees of freedom, traditional methods like the Lyapunov exponent are insufficient for distinguishing between strong and partially predictable chaos. The proposed methods examine the winding number’s sensitivity with respect to the center and its standard deviations across time sequences to assess predictability and differentiate between different motion types. The Duffing–Van der Pol system is used to show the effectiveness in identifying different chaotic behaviours, offering significant implications for the control of complex robotic systems. Full article

This paper investigates the explicit, accurate soliton and dynamic strategies in the resolution of the Wazwaz–Benjamin–Bona–Mahony (WBBM) equations. By exploiting the ensuing wave events, these equations find applications in fluid dynamics, ocean engineering, water wave mechanics, and scientific inquiry. The two main goals of the study are as follows: Firstly, using the dynamic perspective, examine the chaos, bifurcation, Lyapunov spectrum, Poincaré section, return map, power spectrum, sensitivity, fractal dimension, and other properties of the governing equation. Secondly, we use a generalized rational exponential function (GREF) technique to provide a large number of analytical solutions to nonlinear partial differential equations (NLPDEs) that have periodic, trigonometric, and hyperbolic properties. We examining the wave phenomena using 2D and 3D diagrams along with a projection of contour plots. Through the use of the computational program Mathematica, the research confirms the computed solutions to the WBBM equations. Full article

In chaos control, one usually seeks to stabilize the unstable periodic orbits (UPOs) that densely inhabit the attractors of many chaotic dynamical systems. These orbits collectively play a significant role in determining the dynamics and properties of chaotic systems and are said to form the skeleton of the associated attractors. While UPOs are insightful tools for analysis, they are naturally unstable and, as such, are difficult to find and computationally expensive to stabilize. An alternative to using UPOs is to approximate them using cupolets. Cupolets, a name derived from chaotic, unstable, periodic, orbit-lets, are a relatively new class of waveforms that represent highly accurate approximations to the UPOs of chaotic systems, but which are generated via a particular control scheme that applies tiny perturbations along Poincaré sections. Originally discovered in an application of secure chaotic communications, cupolets have since gone on to play pivotal roles in a number of theoretical and practical applications. These developments include using cupolets as wavelets for image compression, targeting in dynamical systems, a chaotic analog to quantum entanglement, an abstract reducibility classification, a basis for audio and video compression, and, most recently, their detection in a chaotic neuron model. This review will detail the historical development of cupolets, how they are generated, and their successful integration into theoretical and computational science and will also identify some unanswered questions and future directions for this work. Full article

When confronted with the imminent threat of predation, the prey instinctively employ strategies to avoid being consumed. These anti-predator tactics involve individuals acting collectively to intimidate predators and reduce potential harm during an attack. In the present work, we propose a state-dependent feedback control predator-prey model that incorporates a nonmonotonic functional response, taking into account the anti-predator behavior observed in pest-natural enemy ecosystems within the agricultural context. The qualitative analysis of this model is presented utilizing the principles of impulsive semi-dynamical systems. Firstly, the stability conditions of the equilibria are derived by employing pertinent properties of planar systems. The precise domain of the impulsive set and phase set is determined by considering the phase portrait of the system. Secondly, a Poincaré map is constructed by utilizing the sequence of impulsive points within the phase set. The stability of the order-1 periodic solution at the boundary is subsequently analyzed by an analog of the Poincaré criterion. Additionally, this article presents various threshold conditions that determine both the existence and stability of an order-1 periodic solution. Furthermore, it investigates the existence of order-k ($k\ge 2$) periodic solutions. Finally, the article explores the complex dynamics of the model, encompassing multiple bifurcation phenomena and chaos, through computational simulations. Full article

Rotate vector (RV) reducers have widely been used in high-performance precision drives for industrial robots. However, the current nonlinear dynamic studies on RV reducers are not extensive and require a deeper focus. To bridge this gap, a translational–torsional nonlinear dynamic model for an RV reducer transmission system is proposed. The gear backlash, time-varying mesh stiffness, and comprehensive meshing errors are taken into account in the model. The dimensionless vibration differential equations of the system were derived and solved numerically. By means of bifurcation diagrams, phase trajectories, Poincaré sections, and the power spectrum, the motion state of the system was studied with the bifurcation parameters’ variation, including excitation frequency and meshing damping. The results demonstrate that this system presents enriched nonlinear dynamic characteristics under different parameter combinations. The motion state of the system is more susceptible to change at lower frequencies. Increasing the meshing damping coefficient proves effective in suppressing the occurrence of chaos and reducing vibration amplitudes, significantly enhancing the stability of the transmission system. Full article