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This study delves into the exploration of the dynamics of (3 + 1) -dimensional Painlevé integrable generalized model from different perspectives, which delineates the evolution of nonlinear phenomena in three spatial dimensions and one temporal dimension, displaying the remarkable Painlevé integrability property. The ϕ 6 -expansion technique is applied to extract traveling wave solutions in the form of Jacobi elliptic functions. To give physical insights of obtained solutions, we present these solutions through graphs such as 2D, 3D and contour plots. Further, to understand the planar dynamical system, we employ the concepts of bifurcation, chaos theory and sensitivity analysis. Bifurcation analysis reveals the dependence on the solution of a planar dynamical system at critical points. Additionally, the detection of chaotic movements in the perturbed dynamical system is achieved by detecting tools. Also the sensitivity analysis of the model is investigate by three distinct initial conditions. The findings are innovative, valuable and captivating for the readers in exploring this model. [ABSTRACT FROM AUTHOR]

This study delves into the exploration of the dynamics of (3+1)-dimensional Painlevé integrable generalized model from different perspectives, which delineates the evolution of nonlinear phenomena in three spatial dimensions and one temporal dimension, displaying the remarkable Painlevé integrability property. The ϕ6-expansion technique is applied to extract traveling wave solutions in the form of Jacobi elliptic functions. To give physical insights of obtained solutions, we present these solutions through graphs such as 2D, 3D and contour plots. Further, to understand the planar dynamical system, we employ the concepts of bifurcation, chaos theory and sensitivity analysis. Bifurcation analysis reveals the dependence on the solution of a planar dynamical system at critical points. Additionally, the detection of chaotic movements in the perturbed dynamical system is achieved by detecting tools. Also the sensitivity analysis of the model is investigate by three distinct initial conditions. The findings are innovative, valuable and captivating for the readers in exploring this model.

Microcavity photon dynamics in curved space is an emerging interesting area at the crossing point of nanophotonics, chaotic science, and non-Euclidean geometry. We report the sharp difference between the regular and chaotic motions of cavity photons subjected to the varying space curvature. While the island modes of regular motion rise in the phase diagram in the curved space, the chaotic modes show special mechanisms to adapt to the space curvature, including the fast diffusion of ray dynamics, and the localization and hybridization of the Husimi wavepackets among different periodic orbits. These observations are unique effects enabled by the combination of the chaotic trajectory, the wave nature of light, and the non-Euclidean orbital motion, and therefore make the system a versatile optical simulator for chaotic science under quantum mechanics in curved space-time. [ABSTRACT FROM AUTHOR]

This paper investigates the dynamic behavior of a tethered satellite system in an elliptical orbit as the tethered subsatellite is deployed from a mother satellite at an out-of-plane angle while maintaining the tether at a constant in-plane pitch angle by a predefined control force acting on the subsatellite. A variable-length pendulum model is employed to describe the purely out-of-plane tether deployment dynamics, from which an analytical tether deployment control law is derived. The stability condition of the analytical control law is examined by Floquet theory. The corresponding analytical deployment law is asymptotically stable, and the requirement for a positive tension control input is guaranteed to ensure the controllability of the tether system. Following deployment, the critical conditions for the out-of-plane libration of the tethered subsatellite to become chaotic during the station-keeping phase are examined. Finally, simulation results reveal that the proposed approach can effectively and rapidly identify periodic, quasiperiodic, and chaotic motions using bifurcation diagrams, which provides valuable insights for enhancing the stability and reliability of such tethered satellite systems. • Proposed an analytical deployment law for a purely out-of-plane tethered satellite. • Revealed nonlinear phenomena of the system during the station-keeping phase. • Explored bifurcation characteristic and chaotic windows of the system. [ABSTRACT FROM AUTHOR]

This study explores the intrinsic characteristics of the (2+1)-dimensional Schwarz-Korteweg-de Vries equation used to describe shallow water waves. The multiple solitons are successfully constructed using a technique involving multiple exponential functions. Graphical representations of the results are provided in 3D, 2D, and density plots to assess the compatibility of the solutions. Also, the dynamic nature of studied equation is examined based on bifurcation and chaos theory for nonlinear systems. Bifurcation signifies how our dynamical system is affected by physical parameters in planar dynamical system. After that, we apply the external force on planar dynamical system to show the chaotic like behavior of the studied model. Such behavior is confirmed by utilizing different chaos detecting tools. The obtained results serve to explain the effectiveness and applicability of the utilized methodologies in comprehend the exact solution and qualitative behavior of nonlinear physical models.

In order to describe and solve chaotic yawing in straight-line navigation, taking large tanker Davis Sea as an example, the yawing model and corresponding high-gain adaptive controller is established. Chaotic yawing is proved to exist. Mathematical analysis indicates the error of the controlled system is globally ultimately bounded. Simulation results show the proposed control law is effective and robust to model perturbation. Compared with sliding mode control, the accuracy of proposed method is 69.9% better and the control energy consumption is 32.6% smaller. [ABSTRACT FROM AUTHOR]

This paper researches the nonlinear behavior of a nonlinear torsional vibration isolator (NTVI) which is composed of flexible rod element and electromagnetic element. And the NTVI has a quasi-zero stiffness (QZS) characteristic. Firstly, a statics model of the QZS-NTVI and the parameter relationship between the nonlinear electromagnetic element and the flexible rod are established; the effects of geometric parameters on the QZS behavior and restoring torque are described. Then, the statics model approximated by the Taylor series is incorporated into the dynamic model of QZS-NTVI. The stability of the harmonic solution is analyzed, and the amplitude–frequency response and the torque transmissibility curves under different geometric parameters are derived according to the harmonic balance method. Furthermore, numerical analysis effort is performed to study the nonlinear behavior of QZS-NTVI. The results show that the QZS-NTVI exhibits super-harmonic and sub-harmonic resonances and undergoes the periodic and chaotic motions alternatively with the change in the excitation amplitude and the angular frequency. Finally, both the simulation and experimental efforts are performed to validate the statics model of the flexible rod and to study the vibration isolation performance and nonlinear behavior of QZS-NTVI. The results demonstrate that the vibration isolation performance of QZS-NTVI notably outperforms the linear system, as well as nonlinear behavior with frequency jump. [ABSTRACT FROM AUTHOR]

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The two-bodies problem can be fully solved, and was solved by Kepler (1609) and Newton (1687). The general three-body problem is often given as an example of a mathematical problem that ‘can’t be solved’. So, there is no general analytical solution. This problem can be significant and a special case of this problem is the Circular Restricted Three-Body Problem (CRTBP), which can be applied to the Earth-Moon system with a spacecraft, the Sun-Earth system with an asteroid, etc. In this paper, let’s focus on the motion of a test particle near the triangular Lagrange points L4 and L5 in the Earth-Moon and the Sun-Earth systems. Studying the movement of an object around these points is especially important for space mission design. To generate a trajectory around these points, the non-linear equations of motion for the circular restricted three-body problem were numerically integrated into MATLAB®2023 software and the results are presented in the plane (x, y) and the phase plane (x, vx) and (y, vy). By numerical orbit integration, it is possible to investigate what happens when the displacement is relatively large or short from the Lagrange points. Then the small astronomical body may vibrate around these points. The results in this paper are shown in the rotating and inertia axes. Various initial positions near the Lagrange points and velocities are used to produce various paths the test particle can take. The same examples of numerical studies of trajectories associated with Lagrange points are shown in the inertial and the rotating coordinates system and are discussed. From the results of the numerical tests performed in MATLAB®2023, it is possible to saw that there are different types of periodic, quasi-periodic, and chaotic orbits. [ABSTRACT FROM AUTHOR]

The nonlinear problem of the Watt governor dynamics is investigated. It is assumed to be installed on a machine that performs specified harmonic vibrations of small amplitude along the vertical. Viscous friction forces are assumed to arise in regulator hinges of the, and these forces are small. In the main operating mode of the regulator, its rods, carrying massive weights, are deflected from the downward vertical by a constant acute angle. If friction and vertical vibrations of the machine are neglected, an approximate problem is obtained in which the regulator dynamics is described by an autonomous Hamiltonian system with one degree of freedom. On the phase portrait of the approximate problem, the operating mode corresponds to a singular point of the center type. The trajectories encircling this point lie inside the separatrix, which is a homoclinic doubly asymptotic trajectory that passes through the equilibrium position corresponding to the vertical position of the rods with weights. In the phase portrait, this position corresponds to a saddle singular point. The Melnikov method is used to obtain the splitting condition for the unperturbed separatrix in the complete perturbed problem, taking into account dissipation in the hinges and vertical vibrations of the machine. [ABSTRACT FROM AUTHOR]

This study explores the modified Oskolkov equation, which depicts the behavior of the incompressible viscoelastic Kelvin–Voigt fluid. The primary focus of this research lies in several key areas. Firstly, the Lie symmetries of the considered equation are identified. These symmetries are utilized to transform the discussed model into an ordinary differential equation. Analytical solutions are subsequently derived using the new auxiliary equation technique. Next, a comprehensive analysis of the equation's dynamic nature is undertaken from multiple aspects. Bifurcation is carried out at fixed points within the system, and chaotic behavior is unveiled by introducing an external force to the dynamic system. Various tools, including 3D and 2D phase plots, time series, Poincaré maps, and multistability analysis, are employed to identify the chaotic nature of the system. Furthermore, the sensitivity of the model is explored across diverse initial conditions. In general, comprehending the dynamic characteristics of systems holds immense significance in forecasting outcomes and innovating new technologies. [ABSTRACT FROM AUTHOR]

The active material of liquid crystal elastomer (LCE) is capable of harvesting energy directly from the surroundings and sustaining continuous motion in the presence of light and heat. It is extensively employed in active machinery, soft robotics, biomedicine, and other fields. To date, there is barely any research on the sustained chaotic motion system of LCE pendulum. The main objective of this paper is to put forward a sustained motion system of a simple pendulum comprising photosensitive LCE. In accordance with the LCE dynamic model, a nonlinear dynamic model of the LCE simple pendulum is established, and its motion behavior characteristics under periodic illumination are examined. The numerical outcomes demonstrate that apart from the in-situ vibration mode, the LCE pendulum experiences two types of displacement motion modes as well, namely periodic oscillation mode and chaotic motion mode. The mechanism underlying the sustained periodic oscillation and chaotic motion is revealed by compensating for the damping dissipation with work done by the LCE contraction. Moreover, the discussion also covers how the system parameters affect the motion modes of the LCE simple pendulum. Through altering the parameters such as illumination period, contraction coefficient, light intensity, damping coefficient and gravitational acceleration, it is possible to realize the distinct motion modes of the LCE simple pendulum. This research may deepen the comprehension on the motion behavior of the simple pendulum, and provide scientific guidance for the design and exploration of chaotic systems based on active materials.

In this study, (4+1)-dimensional Davey Srewartson Kadomtsev Petviashvili equation is investigated, which can be employed to depict internal wave interactions encompassing both elastic and non-elastic behaviors. Initially, the translational symmetries of the equation are identified by employing Lie symmetry method. Then, by utilizing these symmetries, the discussed model is translated into an ordinary differential equation. The new auxiliary equation approach is then employed to obtain precise solutions, and the results are visually represented. Afterward, the dynamical behavior of the equation is analyzed from multiple perspectives, including bifurcation, chaos, and sensitivity analysis. Bifurcation is conducted at fixed points, while an external force is applied to the dynamical system, leading to the discovery of chaotic behavior using various tools such as 3D and 2D phase plots, time series plots, Poincaré maps, and multistability analysis. Furthermore, sensitive nature of the model is examined at different initial values, revealing the significant sensitivity of the proposed model, where even small changes in the initial conditions result in notable variations. At the end, conclusion is presented.

• Multi-segmented constraints effects on the dynamics of pipe conveying fluid are investigated. • Dependence of pipeline's sticking to the constraint is explored. • Asymmetry of the multi-segment constraint is studied. • System's response is characterized using bifurcation diagrams, Poincare maps, power spectra. Previous studies of the cantilevered pipeline conveying fluid system have included motion-limiting constraints in the form of trilinear springs. While this is desirable in experimental scenarios, it may not be representative of real-world applications. Therefore, this study focuses on multi-segmented motion-limiting constraints. As this type of motion-limiting constraint has not been investigated with a cantilevered pipeline system, a wide variety of outer and inner constraint stiffness and constraint gap sizes are investigated in this study to gain a comprehensive understanding of how the multi-segmented constraints affect the dynamics of the cantilevered pipeline. In this effort, bifurcation diagrams, phase portraits, Poincare maps, time histories, and power spectra are used to investigate the dynamics of the system, and the fluid flow speeds where dynamic characteristics are considered. In general, it is found that critical flow speeds like when the pipe sticks in the constraints are reduced as the constraint stiffnesses are increased. Additionally, the sticking flow speed occurred at lower flow speeds as the gap sizes of the inner and outer constraints decrease, and a larger constraint offset results in a smaller inner gap size leading to critical behaviors occurring at earlier flow speeds. [ABSTRACT FROM AUTHOR]

For the chaotic motion control of a single-degree-of-freedom vibro-impact system with soft constraint, two-parameter cooperative intelligent optimal control method based on LRS-QPSO (i.e., Local Rotational Search Quantum Particle Swarm Optimization) algorithm optimization SVM (i.e., Support Vector Machines) was proposed. Firstly, in the parameter plane (ω,ζ), the motion distribution of the system and the change of the maximum Lyapunov exponent is obtained by numerical simulation, and the correlation between parameters change and the transition from chaotic motion to periodic motion is analyzed. Secondly, by introducing ω and ζ into the construction of the input variables of the SVM controller, a chaotic motion two-parameter cooperative intelligent optimal control strategy is proposed to meet the requirements of expected control objectives. Therefore, by the virtue of nonlinear mapping characteristics of SVM, the combined effects of ω and ζ on the dynamic state transition of the system can be reflected in the output variables of the controller. Finally, with single parameter ω and single parameter ζ as the research object, the bifurcation diagram, Lyapunov exponential spectrum, and controlled periodic motion orbit diagram are analyzed, and the advantages of two-parameter cooperative intelligent optimal control are compared and analyzed. [ABSTRACT FROM AUTHOR]

In this paper, the abundant nonlinear dynamical behaviors of a fractional-order time-delayed Duffing system under harmonic excitation are studied. By constructing Melnikov function, the necessary conditions of chaotic motion in horseshoe shape are detected, and the chaos threshold curve is obtained by comparing the results obtained through the Melnikov theory and numerical iterative algorithm. The results show that the trend of change is the same, which confirms the accuracy of the chaos threshold curve. It could be found that when the excitation frequency ω is larger than a certain value, the Melnikov theory is not valid for these values. Furthermore, by numerical simulation, some numerical results are obtained, including phase portraits, the largest Lyapunov exponents, and the bifurcation diagrams, Poincare maps, time histories, and frequency spectrograms at some typical points. These numerical simulation results show that the system exhibits some new complex dynamical behaviors, including entry into the state of chaotic motion from single period to period-doubling bifurcation and chaotic motion and periodic motion alternating under the necessary condition of chaotic occurrence. In addition, the effects of time delay, fractional-order coefficient, fractional order, linear viscous damping coefficient, and linear stiffness coefficient on the chaotic threshold curve are discussed, respectively. Those results reveal that there exist abundant nonlinear dynamic behaviors in this fractional-order system, and by adjusting these parameters reasonably, the system could be transformed from chaotic motion to non-chaotic motion. [ABSTRACT FROM AUTHOR]

Power reflux hydraulic transmission system (PRHTS), which is a recently introduced continuously variable transmission system, enables the improvement of fuel economy of construction vehicles. For investigating the nonlinear dynamic characteristics of PRHTS, its nonlinear dynamic model is established by merging the dynamic models of a planetary gear train and torque converter. A dynamical model of the planetary gear train reveals the parameters of mesh damping, time-varying mesh stiffness, and transmission error. The nonlinear dynamic equations of the PRHTS are solved using the fourth-order Runge-Kutta method. The dynamic orbits of the system are observed through bifurcation diagrams, which use the internal excitation frequency and meshing damping ratios, both of which are dimensionless, as control parameters. Numerical examples show the dynamic evolution mechanism involving one-period motion, multi- periodic motion, quasi-periodic motion, and chaotic motion. The onset of chaotic motion is identified from bifurcation diagrams, dynamic trajectories, phase plane diagram, and Poincare maps of the PRHTS. The simulation results provide an understanding of the operating conditions under which undesirable dynamic motion occurs in PRHTS and serve as invaluable information for effective dynamic design of PRHTS. [ABSTRACT FROM AUTHOR]

Vibration is used to compact the concrete mixture in the manufacture of ferroconcrete parts without molds in long configurations. The transfer of vibrations from a cylindrical punch to filler particles in a viscous concrete mixture is studied. A mathematical description is developed for the motion of a spherical filler particle in a viscous medium set vibrating by a cylindrical punch, and its mean velocity and path in chaotic motion are determined. The range of velocity and path values for punch vibration with industrial frequencies and amplitudes is established. Mathematical modeling is employed. The possibility of varying the mean velocity and path in a viscous medium by adjusting punch vibration is established for particles of different size. [ABSTRACT FROM AUTHOR]

Slight variations induced by thermal effects may bring unexpected discrepancies in both the system's linear and non-linear responses. The present study investigates the temperature effects on the non-linear coupled motions of suspended cables subject to one-to-one internal resonances between the in-plane and out-of-plane modes. The classical non-linear flexible system is excited by a uniform distributed harmonic excitation with the primary resonance. Introducing a two-mode expansion and applying the multiple scale method, the polar and Cartesian forms of modulation equations are obtained. Several parametric investigations have highlighted the qualitative and quantitative discrepancies induced by temperature through the curves of force/frequency-response amplitude, time history diagrams, phase portraits, frequency spectrum, and Poincaré sections. Based on the bifurcation and stability analyses, temperature effects on the multiple steady-state solutions, as well as static and dynamic bifurcations, it is observed that the periodic motions may be bifurcated into the chaotic motions in thermal environments. The saddle-node, pitch-fork, and Hopf bifurcations are sensitive to temperature changes. Finally, our perturbation solutions are confirmed by directly integrating the governing differential equations, which yield excellent agreement with our results and validate our approach. [ABSTRACT FROM AUTHOR]

The nonlinear dynamic behaviors of viscoelastic axially functionally graded material (AFG) pipes conveying pulsating internal flow are very complex. And the dynamic behavior will induce the failure of the pipes, and research of vibration and stability of pipes becomes a major concern. Considering that the elastic modulus, density, and coefficient of viscoelastic damping of the pipe material vary along the axial direction, the transverse vibration equation of the viscoelastic AFG pipe conveying pulsating fluid is established based on the Euler-Bernoulli beam theory. The generalized integral transform technique (GITT) is used to transform the governing fourth-order partial differential equation into a nonlinear system of fourth-order ordinary differential equations in time. The time domain diagram, phase portraits, Poincaré map and power spectra diagram at different dimensionless pulsation frequencies, are discussed in detail, showing the characteristics of chaotic, periodic, and quasi-periodic motion. The results show that the distributions of the elastic modulus, density, and coefficient of viscoelastic damping have significant effects on the nonlinear dynamic behavior of the viscoelastic AFG pipes. With the increase of the material property coefficient k, the transition between chaotic, periodic, and quasi-periodic motion occurs, especially in the high-frequency region of the flow pulsation. [ABSTRACT FROM AUTHOR]

The effectiveness of continuous vibro-impact forcing representations for the cantilevered pipe that conveys fluid is explored and analyzed. The previously accepted forcing model utilizing a smoothened trilinear spring is estimated using three continuous forcing representations, namely, polynomial, rational polynomial, and hyperbolic tangent. The accuracy of the estimated forcing functions is investigated and analyzed by calculating the root mean square error, and bifurcation diagrams are generated and compared to the nominal system. Additionally, the dynamic response of the system is further characterized using Poincare maps, power spectra, and basins of attraction. Once all continuous forcing representations are analyzed and compared to the nominal system, the computational cost of each method is examined, and further limitations of the hyperbolic tangent method are discovered. It is proved that the hyperbolic tangent forcing representation most accurately captures the dynamic response of the pipeline, and the least accurate representation is the rational polynomial representation. Additionally, considerable computational cost is saved when employing the hyperbolic tangent representation compared to the discontinuous representation. [ABSTRACT FROM AUTHOR]

The main idea of this paper is to study the chaotic behavior of Zakharov–Kuznetsov equation with perturbation. By taking the traveling wave transformation, we transform the perturbed Zakharov–Kuznetsov equation with dual-power law and triple-power law nonlinearity into planar dynamic systems, and then analyze how the external perturbed terms affect the chaotic behavior. We emphasize here that there is no chaotic phenomenon for the non-perturbed ZK equation, thus it is only caused by the external perturbed terms.

In this study, nonlinear vibration and stability of a pipe conveying fluid, with axially moving supports, immersed in an external fluid is studied. The equation of motion is derived with the aid of the Newtonian method via employing the Euler–Bernoulli beam theory and plug flow assumption. Considering the stretching effect, the nonlinear equation of motion is obtained, and it is discretized via the Galerkin method. Afterward, the stability of the system is investigated using two different approaches, and their results are compared. Numerical simulations show the system has complex dynamic behavior. To unveil the dynamics of the system, various analysis techniques such as bifurcation diagram of Poincaré map, phase portrait, power spectrum, and time trajectories are used. Limit cycle, period-n, quasi-periodic, and chaotic regime are observed. It is found that pipe can flutter around the trivial solution or post-buckling equilibrium points. Moreover, the influences of some system parameters on the chaotic behavior of the pipe are discussed. [ABSTRACT FROM AUTHOR]

Solitary waves in hyperelastic structures propagate stably in absence of external forces. However, as the external forces increase, the stability of solitary waves may lose and then chaotic behaviors may appear. In this paper, solitary waves and chaotic motions of an infinitely long cylinder with an internal heat source are investigated. Moreover, the cylinder is composed of nearly compressible thermo-hyperelastic neo-Hookean materials. Based on the variational principle, in a non-uniform steady-state temperature field and under a uniformly distributed radial periodic load, a mathematical model describing motions of the infinite cylinder is established. The governing equations are analyzed by qualitative and bifurcation theorems. The effects of structure boundary temperature on the qualitative properties of traveling waves are examined, and the types of traveling waves are identified. By using Melnikov functions, the sufficient conditions for chaos are derived. Numerical simulations of Lyapunov exponents, bifurcation diagrams, trajectories, and time travel curves are presented to illustrate the process of the associated dynamical system from order to chaotic motions in some bifurcation sets. The numerical results are consistent with those of the theoretical analysis. The thresholds of physical parameters and sufficient conditions for chaos may provide critical information for nondestructive testing of structures. [ABSTRACT FROM AUTHOR]

The main idea of this paper is to study the chaotic behavior of Zakharov–Kuznetsov equation with perturbation. By taking the traveling wave transformation, we transform the perturbed Zakharov–Kuznetsov equation with dual-power law and triple-power law nonlinearity into planar dynamic systems, and then analyze how the external perturbed terms affect the chaotic behavior. We emphasize here that there is no chaotic phenomenon for the non-perturbed ZK equation, thus it is only caused by the external perturbed terms. [ABSTRACT FROM AUTHOR]

This study explores the modified Oskolkov equation, which depicts the behavior of the incompressible viscoelastic Kelvin–Voigt fluid. The primary focus of this research lies in several key areas. Firstly, the Lie symmetries of the considered equation are identified. These symmetries are utilized to transform the discussed model into an ordinary differential equation. Analytical solutions are subsequently derived using the new auxiliary equation technique. Next, a comprehensive analysis of the equation’s dynamic nature is undertaken from multiple aspects. Bifurcation is carried out at fixed points within the system, and chaotic behavior is unveiled by introducing an external force to the dynamic system. Various tools, including 3D and 2D phase plots, time series, Poincaré maps, and multistability analysis, are employed to identify the chaotic nature of the system. Furthermore, the sensitivity of the model is explored across diverse initial conditions. In general, comprehending the dynamic characteristics of systems holds immense significance in forecasting outcomes and innovating new technologies.

This paper proposes a novel magnetically excited variable cross section bending beam piezoelectric energy harvester (ME-VCB-PEH) to overcome the limitations of current piezoelectric energy harvesters, such as narrow bandwidth and single-directional collection. Firstly, a nonlinear distributed parameter model of the ME-VCB-PEH is established based on Hamilton's principle, and the optimal distance between magnets is derived, providing a theoretical basis for subsequent research. Secondly, through simulation and experimentation, the optimal structural parameters of the ME-VCB-PEH are determined. Finally, an energy harvesting test platform is constructed to test its output performance and application effectiveness. The results show that compared to rectangular and trapezoidal beams, the VCB structure exhibits lower frequency, wider bandwidth, and higher output voltage. The designed ME-VCB-PEH effectively harvests vibration energy in three axial directions below an external frequency of 80 Hz, effectively meeting the auxiliary power supply needs of temperature sensors and wireless transmission modules for the roadheader under normal operating conditions. This technology provides an important technical reference for self-powered wireless sensing technology and plays a significant role in IoT and sensing applications. • A magnetically excited variable cross section bending beam piezoelectric energy harvester is first proposed. • A nonlinear distribution parameter model for energy harvesting was developed based on Hamilton's principle. • Satisfying the vibration spectrum of the roadheader and capable of providing auxiliary power supply. [ABSTRACT FROM AUTHOR]

This study explores the intrinsic characteristics of the (2+1)-dimensional Schwarz-Korteweg-de Vries equation used to describe shallow water waves. The multiple solitons are successfully constructed using a technique involving multiple exponential functions. Graphical representations of the results are provided in 3D, 2D, and density plots to assess the compatibility of the solutions. Also, the dynamic nature of studied equation is examined based on bifurcation and chaos theory for nonlinear systems. Bifurcation signifies how our dynamical system is affected by physical parameters in planar dynamical system. After that, we apply the external force on planar dynamical system to show the chaotic like behavior of the studied model. Such behavior is confirmed by utilizing different chaos detecting tools. The obtained results serve to explain the effectiveness and applicability of the utilized methodologies in comprehend the exact solution and qualitative behavior of nonlinear physical models. [ABSTRACT FROM AUTHOR]

In this paper, the abundant nonlinear dynamical behaviors of a fractional-order time-delayed Duffing system under harmonic excitation are studied. By constructing Melnikov function, the necessary conditions of chaotic motion in horseshoe shape are detected, and the chaos threshold curve is obtained by comparing the results obtained through the Melnikov theory and numerical iterative algorithm. The results show that the trend of change is the same, which confirms the accuracy of the chaos threshold curve. It could be found that when the excitation frequency ω is larger than a certain value, the Melnikov theory is not valid for these values. Furthermore, by numerical simulation, some numerical results are obtained, including phase portraits, the largest Lyapunov exponents, and the bifurcation diagrams, Poincare maps, time histories, and frequency spectrograms at some typical points. These numerical simulation results show that the system exhibits some new complex dynamical behaviors, including entry into the state of chaotic motion from single period to period-doubling bifurcation and chaotic motion and periodic motion alternating under the necessary condition of chaotic occurrence. In addition, the effects of time delay, fractional-order coefficient, fractional order, linear viscous damping coefficient, and linear stiffness coefficient on the chaotic threshold curve are discussed, respectively. Those results reveal that there exist abundant nonlinear dynamic behaviors in this fractional-order system, and by adjusting these parameters reasonably, the system could be transformed from chaotic motion to non-chaotic motion.

Dynamics of ion-acoustic wave (IAW) are observed in a magnetized collisional four-component dusty plasma comprising cold ions, inertialess two-temperature electrons, and immobile negatively charged dust grains. Employing the reductive perturbation technique (RPT), the damped forced modified Zakharov–Kuznetsov (DFMZK) equation is derived and assuming that the conservation principle holds in the system, an approximate analytical solution of the DFMZK equation is obtained; however, the solution is valid for very small values of perturbed damping and forcing term. In that circumstance, using the bifurcation theory of the dynamical system, a qualitative study on ZK medium in the presence of damping and forcing terms has been performed. The significant effects from small damping and forcing terms are presented from a numerical standpoint. Further, utilizing the bifurcation diagram the periodic and quasi-periodic orbits of IAW in the DFMZK framework are analyzed. It is found that both the initial condition and the damping term play a significant role in forming complexity in the dynamic system. Finally, the effective role of the strength ( f 0 ) of the externally applied force perturbation is studied in detail and the existence of chaotic motion of the DFMZK equation is confirmed through time series analysis and Poincare analysis. [ABSTRACT FROM AUTHOR]

Slight variations induced by thermal effects may bring unexpected discrepancies in both the system’s linear and non-linear responses. The present study investigates the temperature effects on the non-linear coupled motions of suspended cables subject to one-to-one internal resonances between the in-plane and out-of-plane modes. The classical non-linear flexible system is excited by a uniform distributed harmonic excitation with the primary resonance. Introducing a two-mode expansion and applying the multiple scale method, the polar and Cartesian forms of modulation equations are obtained. Several parametric investigations have highlighted the qualitative and quantitative discrepancies induced by temperature through the curves of force/frequency-response amplitude, time history diagrams, phase portraits, frequency spectrum, and Poincaré sections. Based on the bifurcation and stability analyses, temperature effects on the multiple steady-state solutions, as well as static and dynamic bifurcations, it is observed that the periodic motions may be bifurcated into the chaotic motions in thermal environments. The saddle-node, pitch-fork, and Hopf bifurcations are sensitive to temperature changes. Finally, our perturbation solutions are confirmed by directly integrating the governing differential equations, which yield excellent agreement with our results and validate our approach.

Energetic charged particles trapped in planetary radiation belts are hazardous to spacecraft. Planned missions to iron‐rich asteroids with possible strong remanent magnetic fields require an assessment of trapped particles energies. Using laboratory measurements of iron meteorites, we estimate the largest possible asteroid magnetic moment. Although weak compared to moments of planetary dynamos, the small body size may yield strong surface fields. We use hybrid simulations to confirm the formation of a magnetosphere with an extended quasi‐dipolar region. However, the short length scale of the field implies that energetic particle motion would be nonadiabatic, making existing radiation belt theories not applicable. Our idealized particle simulations demonstrate that chaotic motions lead to particle loss at lower energies than those predicted by adiabatic theory, which may explain the energies of transiently trapped particles observed at Mercury, Ganymede, and Earth. However, even the most magnetized asteroids are unlikely to stably trap hazardous particles. Plain Language Summary: Radiation belts are regions in space filled with energetic charged particles trapped by a planetary magnetic field. To date, radiation belts were found around Earth, Jupiter, Saturn, Uranus, and Neptune. Understanding radiation belts is important because they can be hazardous to spacecraft. In addition, their emission, if detected remotely, can reveal a planet's magnetic field. The future exploration of asteroids and exoplanets may lead to the discovery of more magnetized bodies, including the upcoming NASA mission to asteroid (16) Psyche, the largest known metal‐rich planetary object. Existing radiation belt theory was developed for known radiation belts surrounding the large, highly magnetized planets in the solar system. However, the nature of particle trapping should differ for small bodies like asteroids. We examine these differences to provide predictive tools for the existence of belts that can be applied to any kind of magnetosphere. We find a class of bodies that can sustain belt‐like structures but where particle trajectories are chaotic, preventing prolonged stable trapping and associated energization. We show that Mercury and Ganymede's transient trapping is consistent with our predictions. Finally, we show that although asteroids may stably trap low‐energy protons and electrons, even the most strongly magnetized metallic asteroids cannot form belts that pose a risk to typical spacecraft hardware. Key Points: Meteorite paleomagnetism and hybrid modeling imply that asteroids may form magnetospheres and trap charged particlesPlanning of missions to asteroids requires assessing trapped particle energies, but existing models are not valid for such magnetospheresA new nonadiabatic trapping criterion shows that asteroids should not have hazardous particle radiation belts [ABSTRACT FROM AUTHOR]

Theoretical analysis using mathematical models is often used to understand a mechanism of collective motion in a self-propelled system. In the experimental system using camphor disks, several kinds of characteristic motions have been observed due to the interaction of two camphor disks. In this paper, we understand the emergence mechanism of the motions caused by the interaction of two self-propelled bodies by analyzing the global bifurcation structure using the numerical bifurcation method for a mathematical model. Finally, it is also shown that the irregular motion, which is one of the characteristic motions, is chaotic motion and that it arises from periodic bifurcation phenomena and quasi-periodic motions due to torus bifurcation. [ABSTRACT FROM AUTHOR]

This study examines the dynamics in a brushless DC motor (BLDCM) and methods used to control potentially chaotic behavior or behavior similar to chaotic processes in these systems. Bifurcation diagrams revealed complex nonlinear behaviors over a range of parameter values. In the resulting bifurcation diagram, period-doubling bifurcation, period-three bifurcation, and chaotic behavior can clearly be seen. We used Lyapunov exponents and Lyapunov dimensions to show the occurrence of chaos in a BLDCM. We then used the state feedback method to control chaos behaviors in the same BLDCM. Numerical simulations show the feasibility of the suggested means. Analysis of robustness against parametric perturbation in a BLDCM was performed from the perspective of Lyapunov stability theory and by using numerical simulations. We believe that studying the nonlinear dynamics and controlling chaos in BLDCMs will help to advance the development of high-performance electric vehicles. [ABSTRACT FROM AUTHOR]

This paper presents a nonlinear stability analysis of an unbalanced rotor-bearing system using the numerical continuation method and the numerical integration method. In this study, the effect of unbalance on journal motion is highlighted and a relationship is established between the bifurcation diagram of a balanced rotor and that of an unbalanced rotor. The results show that the stable operating speed range, the shaft motion type, the whirl speed and the chaotic motion occurrence depend on the unbalance level, the bearing geometry, the oil viscosity, and the speed range of unstable limit cycles existence.

This article presents an energy-based method for examining the dynamics of an autoparametric double pendulum with quadratic coupling. The method is based on decomposing the total energy (Hamiltonian) into partial terms and studying these terms in the time domain. The article then demonstrates using this method to analyse internal resonances in various regimes, including regular periodic, quasiperiodic, and chaotic motion. This study also explores higher-order resonances linked to the geometric tuning parameter and initial energy, which enrich the complexity of the system. The work focuses on identifying conditions that lead to the development of internal resonances and analysing energy exchange mechanisms at various initial energy levels. This analysis uses Fast Lyapunov Indicators and Poincaré maps for these higher energy levels to identify chaotic regimes. Within these regimes, the assessment of coupling energy contributes to a deeper understanding of the studied system. Moreover, the research demonstrates how energy-based methods can be used to investigate auto-parametric systems. • Energy-based approach for autoparametric pendulum with quadratic coupling. • In-depth exploration of internal resonances across motion types. • Chaos detection using Fast Lyapunov Indicators. • Higher-order resonances linked to system parameters revealed. • Method applicable for various autoparametric systems. [ABSTRACT FROM AUTHOR]

Many problems in practical engineering can be simplified as the cantilever beam model, which is generally studied by theoretical analysis, experiment, and numerical simulation. This paper discusses the local activity of the nonlinear nonplanar motion of a cantilever beam at the equilibrium point. Firstly, the equilibrium point of the model and the Jacobian matrix have been calculated. The stability of the characteristic root corresponding to the characteristic polynomial has been analyzed. Secondly, the corresponding complexity function of the model at the equilibrium point has been given. Then, the local activity region of the model at the equilibrium point can be obtained by using the theory of the local activity. Based on the actual engineering research background, the damping coefficient is generally taken as 0 < c < 1. The cantilever beam model is the local activity at the equilibrium point only if the parameters of the model satisfy a certain condition. In the numerical simulation, it is found that when the proper parameters are selected in the local activity region, the cantilever beam can exhibit different types of chaotic motion. The local activity theory provides a theoretical basis for the parameter selection of the chaotic motion in the cantilever beam. [ABSTRACT FROM AUTHOR]

Artificial satellites are widely used in different areas such as communication, position systems, and agriculture. The number of satellites orbiting Earth is becoming huge, and many are set to be launched soon. This huge number of satellites in addition to space debris are sources of concern. Indeed, some incidents have occurred either between satellites or because of space debris. These incidents are a threat for the hit satellite and can be a source of irreversible damages. A hit satellite may diverge to a chaotic motion with all the entailed consequences. The inertia moment of a satellite is a main factor to determine if the hit satellite is heading toward a chaotic motion or not. The inertia moment is determined over the mass density function. In this paper, a circularly orbiting artificial satellite was modeled as a thin rotating rod. The objective was to determine a suitable mass density function for this satellite allowing the prevention as much as possible of the chaotic motion after being hit. This unknown density mass function satisfies a system of equations reflecting some physical constraints. Conventional procedures are not applicable to solve this system of equations. The presented resolution method is based on several mathematical transformations, allowing converting this system into a highly nonlinear one with several unknowns. Several mathematical techniques were applied, and an analytical solution was obtained. Finally, from the mechanical engineering point of view, the obtained mass density function corresponds to a Functionally Graded Material (FGM). [ABSTRACT FROM AUTHOR]

The dynamical characteristics of a gyrostat system with intermittent forcing are investigated, the main work and contributions are given as follows: (1) The gyrostat system with an intermittent forcing is studied, and its dynamical characteristics are investigated by the corresponding Lyapunov exponent spectrums and bifurcation diagrams with respect to the amplitude of intermittent forcing. The modified gyrostat system exists chaotic motion when the amplitude of intermittent forcing belongs to a certain interval, and it can be at a state of stable point or periodic motion by the design of amplitude. (2) The gyrostat system with multiple intermittent forcings is also investigated through the combination of Lyapunov exponent spectrums and bifurcation diagrams, and it behaves periodic motion or chaotic motion when the amplitude or forcing width is different. (3) By the selection of parameters in intermittent forcings, the modified gyrostat system is at a state of stable point, periodic motion or chaotic motion. Numerical simulations verify the feasibility and effectiveness of the modified gyrostat system. [ABSTRACT FROM AUTHOR]

This paper addresses the problem of designing an algorithm capable of generating chaotic trajectories for mobile robots. In particular, the chaotic behavior is induced in the linear and angular velocities of a differential drive mobile robot by infusing them with the states of the Hénon chaotic map. A possible application of patrolling a work area by using trajectories with chaotic motion in differential drive mobile robots is presented. In this work, obtained numerical and experimental results are reported and analyzed. In addition, two quantitative numerical tests are applied in order to measure the chaoticity of the generated trajectories. With the intention of expanding this work further, we explore the case where multiple mobile robots move simultaneously in the same work area with different trajectories, and the final results are exhaustively analyzed. In addition, experimental results are presented. [ABSTRACT FROM AUTHOR]

Gear wear failure is one of the important failures of the gear system in the shearer cutting section. To reveal the influence mechanism of shearer cutting gear wear on the system dynamic characteristics, considering coupling factors such as time-varying meshing stiffness, dynamic gear clearance, internal error excitation, end load constraint and bearing radial clearance under wear failure, an improved dynamic model of shearer drive gear system is introduced to present an in-depth investigation of uniform wear of gear teeth effect. The dynamic meshing stiffness of gears under different degrees of wear is analysed. Furthermore, the bifurcation diagram is utilized to observe the motion state of the system experiencing different excitation frequencies, support damping as well as terminal loads. It is demonstrated that the gear surface wear could bring a change in gear dynamic transmission error, vibration impact state and amplitude, which is mainly manifested in increasing the unstable area and the vibration amplitude of the gear system, providing a method for monitoring and diagnosing of gear surface faults. [ABSTRACT FROM AUTHOR]

In an attempt to improve the current low efficiency and high consumption situation of vibration mills, this paper analyses the chaotic motion characteristics of the system and the movement of vibration mill. The complex stiffness-dispersion coupling of the system is also studied, so as to investigate the effect of the system's chaotic motion characteristics on the efficiency improvement and energy consumption reduction. Based on the ADAMS software, this paper establishes a simplified vibration mill mechanical model, analyzes the singularity and stability of the system, and determines the critical speed at which the vibration motor becomes chaotic according to the bifurcation diagram. Then the chaotic state of the grinding machine with sinusoidal variation in its motor speed is studied based on the Poincaré principle, singular attractor and maximum Lyapunov exponent. Lastly, a 200 h vibration test on diamond powder with an average particle size of 10 μ m was carried out. Test results under the two operating conditions of variable and constant speeds are compared and analyzed. Our results show that with variable speed the vibration mill achieved higher grinding efficiency but smaller particle grain size. The research elaborated in this paper provides a valuable reference for the engineering application of the chaotic characteristics of vibration mill. [ABSTRACT FROM AUTHOR]