Chaotic Motion Research Articles

Clustering and chaotic motion of heavy inertial particles in an isolated non-axisymmetric vortex

We investigate the dynamics of heavy inertial particles in a flow field due to an isolated, non-axisymmetric vortex. For our study, we consider a canonical elliptical vortex – the Kirchhoff vortex and its strained variant, the Kida vortex. Contrary to the anticipated centrifugal dispersion of inertial particles, which is typical in open vortical flows, we observe the clustering of particles around co-rotating attractors near the Kirchhoff vortex due to its non-axisymmetric nature. We analyse the inertia-modified stability characteristics of the fixed points, highlighting how some of the fixed points migrate in physical space, collide and then annihilate with increasing particle inertia. The introduction of external straining, the Kida vortex being an example, introduces chaotic tracer transport. Using a Melnikov analysis, we show that particle inertia and external straining can compete, where chaotic transport can be suppressed beyond a critical value of particle inertia.

Chaotic control of a simply supported beam in a multidimensional system

In this work, a control strategy based on fuzzy sliding mode control (FSMC) is applied to effectively manage the large-amplitude chaotic vibrations exhibited by a simply supported beam. The analysis considers the nonlinear beam structure, which is subjected to an external load. Hamilton’s principle is employed, and the equations of motion are derived for the studied structure. The 3rd Galerkin discretization is employed to derive the ordinary differential governing equation of the structure. A control strategy is presented to decrease the response of the obtained multidimensional system. The vibration of a one-dimensional system is compared with that of the derived multidimensional system. As shown throughout the study, a multidimensional nonlinear system of the structure must be considered for accurate dynamic estimation. By using the recurrent neural network (RNN) model, we can accurately predict chaotic motion and effectively apply control strategies to suppress chaotic motion. The efficacy and suitability of the employed control strategy have been demonstrated by controlling chaos in the beam’s multidimensional system.

Nonlinear dynamic of flexible rotor supported by turbulent bearings with quadratic damping and temperature dependent viscosity

A nonlinear dynamic model of a rotor supported by the turbulent journal bearing with quadratic damping under nonlinear suspension is presented in this study, and the lubricating oil is assumed to be temperature-dependent. The bifurcation diagrams, dynamic trajectories, power spectra, Poincaré map, Lyapunov exponent, and fractal dimension are used to verify the dynamic characteristics of the rotor-bearing system, and abundant subharmonic, and even chaotic motions are found. Temperature is one of the key parameters affecting the dynamic responses of the system, and the dynamic responses will perform non-periodic or even chaotic motions with higher working temperatures of lubricant fluid, as proved in this study. Nonlinear dynamic simulation analysis results will help engineers avoid unnecessary trial and error when designing and applying rotating machinery systems.

Orbits of a system of three point vortices and the associated chaotic mixing.

We study the general periodic motion of a set of three point vortices in the plane, as well as the potentially chaotic motion of one or more tracer particles. While the motion of three vortices is simple in that it can only be periodic, the actual orbits can be surprisingly complex and varied. This rich behavior arises from the existence of both co-linear and equilateral relative equilibria (steady motion in a rotating frame of reference). Here, we start from a general (unsteady) co-linear array with arbitrary vortex circulations. The subsequent motion may take the vortices close to a distinct co-linear relative equilibrium or to an equilateral one. Both equilibrium states are necessarily unstable, as we demonstrate by a linear stability analysis. We go on to study mixing by examining Poincaré sections and finite-time Lyapunov exponents. Both indicate widespread chaotic motion in general, implying that the motion of three vortices efficiently mixes the nearby surrounding fluid outside of small regions surrounding each vortex.

Two-parameter dynamics and multistability of a non-smooth railway wheelset system with dry friction damping.

A deep understanding of non-smooth dynamics of vehicle systems, particularly with dry friction damping offer valuable insights into the design and optimization of railway vehicle systems, ultimately enhancing the safety and reliability of railway operations. In this paper, the two-parameter dynamics of a non-smooth railway wheelset system incorporating dry friction damping are investigated. The effect of the crucial parameters on the complexity of the evolution process is comprehensively exposed by identifying different dynamic responses in the two-parameter plane. In addition, the multistability and the various routes transition to chaos for the system are also discussed. It is found that dry friction induces highly complex dynamics in the system, encompassing a range of behaviors such as periodic, quasi-periodic, and chaotic motions. These intricate dynamics are a direct result of the interplay between multiple parameters, such as speed and damping coefficients, which are critical in determining the system's stability and performance. The presence of multistability further complicates the system, resulting in unpredictable transitions between different motion states.

Reversible to irreversible transitions for ac driven skyrmions on periodic substrates

Abstract Using atomistic simulations, we investigate the dynamical behavior of magnetic skyrmions in dimer and trimer molecular crystal arrangements, as well as bipartite lattices at 3/2 and 5/2 fillings, under ac driving over a square array of anisotropy defects. For low ac amplitudes, at all fillings reversible motion appears in which the skyrmions return to their original positions at the end of each ac drive cycle and the diffusion is zero. We also identify two distinct irreversible regimes. The first is a translating regime in which the skyrmions form channels of flow in opposing directions and translate by one substrate lattice constant per ac drive cycle. The translating state appears in the dimer and trimer arrangements, and produces pronounced peaks in the diffusivity in the direction perpendicular to the external drive. For larger ac amplitudes, we find chaotic irreversible motion in which the skyrmions can randomly exchange places with each other over time, producing long-time diffusive behavior both parallel and perpendicular to the ac driving direction.

Radiative Back-Reaction on Charged Particle Motion in the Dipole Magnetosphere of Neutron Stars

The motion of charged particles under the Lorentz force in the magnetosphere of neutron stars, represented by a dipole field in the Schwarzschild spacetime, can be determined by an effective potential, whose local extrema govern circular orbits both in and off the equatorial plane, which coincides with the symmetry plane of the dipole field. In this work, we provide a detailed description of the properties of these “conservative” circular orbits and, using the approximation represented by the Landau-Lifshitz equation, examine the role of the radiative back-reaction force that influences the motion of charged particles following both the in and off equatorial circular orbits, as well as the chaotic orbits confined to belts centered around the circular orbits. To provide clear insight into these dynamics, we compare particle motion with and without the back-reaction force. We demonstrate that, in the case of an attractive Lorentz force, the back-reaction leads to the charged particles falling onto the neutron star's surface in all scenarios considered. For the repulsive Lorentz force, in combination with the back-reaction force, we observe a widening of stable equatorial circular orbits; the off-equatorial orbits shift toward the equatorial plane and subsequently widen if they are sufficiently close to the plane. Otherwise, the off-equatorial orbits evolve toward the neutron star surface. The critical latitude, which separates orbital widening from falling onto the surface, is determined numerically as a function of the electromagnetic interaction's intensity.

Enhanced artificial hummingbird algorithm with chaotic traversal flight

Tackling the shortcomings of slow convergence, imprecision, and entrapment in local optima inherent in traditional meta-heuristic algorithms, this study presents the enhanced artificial hummingbird algorithm with chaotic traversal flight (CEAHA), which employs chaotic ergodicity within the foundational framework of the conventional artificial hummingbird algorithm. This approach implements chaotic motion within local regions of the solution space, ensuring a thorough exploration of potential optima and preventing algorithmic stagnation at local maxima by guaranteeing a non-repetitive traversal of all search states. This study also analyzes the intrinsic mechanisms by which eight different chaotic mappings affect optimization performance, from the perspectives of invariant measures and traversal efficiency of ergodic chaotic motion. In comparative tests with 21 meta-heuristic algorithms on the CEC2014, CEC2019, and CEC2022 benchmark suites across various dimensions, CEAHA demonstrates superior optimization performance. Furthermore, the practicability and robustness of CEAHA have been confirmed in mechanical design optimization problems through 4 engineering instances: pressure vessel, gear trains, speed reducers, and piston levers.

A Novel Liquid–Solid Fluidized Bed of Large-Scale Phase-Changing Sphere for Thermal Energy Storage

The storage of thermal energy has been hindered by the low heat-transfer rate of the solid phase of the phase-changing materiel. With water being the heat-transfer fluid as well as the liquid phase in the liquid–solid two-phase system, a novel type of fluidized bed is designed in this study. Numerous hollow spheres are fabricated with phase-changing materiel encapsulated. Adding the solid–liquid phase-change material capsules to the flowing fluid, the capsules are dispersed suspended in the carrier. The large spheres, 25 mm in present experiment, possess the merits of guaranteeing energy-storage density and tolerating internal interface chaotic motion. Both the fluidization status and phase-changing process are recorded by photography combined with image-processing technology. It is found that the large spheres, with density less than water, can be fluidized by the downward flowing fluid. As the flow rate increases, the expansion ratio of the solid phase increases and the regimes of incipient fluidization and bubbling fluidization can be observed. In comparison to the fixed bed, the oscillation of pressure drop across a fluidized bed is more severe, but the averaged value is less than the fixed bed. The melting and solidifying can be accelerated by 22.6% and 50%, respectively, thus proving the superiority of the fluidized bed in improving the heat-transfer rate while charging/discharging the thermal energy. Three types of basic movement of the spheres are shown to contribute to the enhanced phase-changing rate, which are shifting, colliding and rotating.

The Phase Transition of Covariant Lyapunov Vector Precisely Locates a Stability Reversal of Quasi-Periodic Response

Abstract Assessing the stability of quasi-periodic (QP) response is crucial, as the bifurcation of QP response is usually accompanied by a stability reversal. The largest Lyapunov exponent (LLE), as an important indicator for chaotic motion, can also be used for the stability analysis of QP response. The precise location of a stability reversal, however, is tough to achieve as a poor convergence rate would be usually encountered when solving the LLE. Herein a straightforward and precise approach is suggested to identify the critical point when a stability reversal happens. Our approach is based on an explicit differential equation that provides the LLE straightforwardly via numerical integration, and the corresponding covariant Lyapunov vector is simultaneously obtained. The major finding consists in the phase transition of the covariant Lyapunov vector, which can happen much early before the LLE reaches a relatively convergent value. More importantly, the phase transition can serve as a strong indicator to locate a stability reversal of the QP response qualitatively. Numerical examples are provided to verify of the effectiveness and wide applicability the presented approach.

Evolution of the dual AGN in Mrk 266: a young AGN and a rotation-dominated disc in the SW nucleus

ABSTRACT Dual active galactic nuclei (AGNs) offer a unique opportunity to probe the relationship between supermassive black holes (SMBH) and their host galaxies as well as the role of major mergers in triggering AGN activity. The confirmed dual AGN Mrk 266 has been studied extensively with multiwavelength imaging. Now, high-spatial-resolution IFU spectroscopy of Mrk 266 provides an opportunity to probe the kinematics of both the merger event and AGN feedback. We present for the first time high-spatial-resolution kinematic maps for both nuclei of Mrk 266 obtained with the Keck OSIRIS IFU spectrograph, utilizing adaptive optics to achieve a resolution of $0.31$ and $0.20\,\mathrm{ arcsec}$ for the NE and SW nuclei, respectively. Using the $M_\text{BH} \!-\! \sigma _*$ relation for mergers, we infer an SMBH mass of approximately $7 \times 10^{7}\, \mathrm{M}_{\odot }$ for the south-western nucleus. Additionally, we report that the molecular gas kinematics of the south-western nucleus are dominated by rotation rather than large-scale chaotic motions. The south-west nucleus also contains both a circumnuclear ring of star formation from which an inflow of molecular gas is likely fuelling the AGN and a compact, AGN-dominated outflow of highly ionized gas with a time-scale of approximately 2 Myr, significantly shorter than the time-scale of the merger. The north-eastern nucleus, on the other hand, exhibits complex kinematics related to the merger, including molecular gas that appears to have decoupled from the rotation of the stars. Our results suggest that while the AGN activity in Mrk 266 was likely triggered during the merger, AGN feeding is currently the result of processes internal to each host galaxy, thus resulting in a strong asymmetry between the two nuclei.

Chaotic Motions of the van der Pol-Duffing Oscillator Subjected to Periodic External and Parametric Excitations with Delayed Feedbacks

Chaotic Motions of the van der Pol-Duffing Oscillator Subjected to Periodic External and Parametric Excitations with Delayed Feedbacks

Rydberg excitons in cuprous oxide: A two-particle system with classical chaos.

When an electron in a semiconductor gets excited to the conduction band, the missing electron can be viewed as a positively charged particle, the hole. Due to the Coulomb interaction, electrons and holes can form a hydrogen-like bound state called the exciton. For cuprous oxide, a Rydberg series up to high principle quantum numbers has been observed by Kazimierczuk et al. [Nature 514, 343 (2014)] with the extension of excitons up to the μm-range. In this region, the correspondence principle should hold and quantum mechanics turn into classical dynamics. Due to the complex valence band structure of Cu2O, classical dynamics deviates from a purely hydrogen-like behavior. The uppermost valence band in cuprous oxide splits into various bands resulting in yellow and green exciton series. Since the system exhibits no spherical symmetry, the angular momentum is not conserved. Thus, the classical dynamics becomes non-integrable, resulting in the possibility of chaotic motion. Here, we investigate the classical dynamics of the yellow and green exciton series in cuprous oxide for two-dimensional orbits in the symmetry planes as well as fully three-dimensional orbits. Our analysis reveals substantial differences between the dynamics of the yellow and green exciton series. While it is mostly regular for the yellow series, large regions in phase space with classical chaos do exist for the green exciton series.

The suppression effect of a vertical baffle on three-dimensional swirling and chaotic sloshing in a laterally excited square-based tank

In order to systematically investigate the suppression effect of a vertical baffle on three-dimensional (3D) swirling and chaotic sloshing in a square-based tank subjected to horizontal harmonic excitation, hundreds of experiments are conducted in a clean tank and baffled tanks with three different configurations. Specifically, the vertical baffle is mounted on the tank bottom parallel to the longitudinal direction, the transverse direction, or the diagonal direction. This experimental work finds that there are four sloshing wave regimes in a clean tank—planar, square-like, swirling, and chaotic—which can be described by the asymptotic multimodal theory. Furthermore, there are only two wave regimes in a longitudinal-baffle tank, the planar and swirling regimes, and the occurrence of swirling requires that the excitation amplitude is sufficiently large. It is confirmed that the longitudinal baffle has a significant suppression effect on the swirling and chaotic motions of the sloshing waves, even though it is parallel to the direction of tank movement. Furthermore, the suppression effect of the diagonal baffle is similar to but somewhat smaller than that of the longitudinal baffle. However, when the transverse baffle is mounted on the bottom of the tank, it is difficult to excite the rotation of the sloshing wave. Therefore, the suppression effect of a bottom-mounted baffle depends largely on the included angle between the vertical baffle and the tank movement direction.

Turbulent diffusion and air pollution: A comprehensive review of mechanisms, impacts, and modeling approaches

Turbulent diffusion is often a desirable phenomenon in many contexts, but it also presents significant challenges, particularly in the realm of air pollution. Air pollution, a universally recognized and frowned upon issue, degrades air quality and poses serious long-term health risks. Turbulent flows, characterized by chaotic and random fluid motions, give rise to turbulent diffusion, a process that acts as a transport mechanism for dispersing pollutants in the atmosphere. Despite its inherent complexities, the study of turbulent diffusion has advanced significantly, building on foundational concepts like Osborne Reynolds' decomposition in the 1850s and extending to Kolmogorov's length scale framework. These developments have enabled a deeper understanding of turbulent diffusion and its wide-ranging applications across various fields as new methods are continuously employed to broaden the scope of research and application.

A three-dimensional discrete fractional-order HIV-1 model related to cancer cells, dynamical analysis and chaos control

In this paper, we study a three-dimensional discrete-time model to describe the behavior of cancer cells in the presence of healthy cells and HIV-infected cells. Based on the Caputo-like difference operator, we construct the fractional-order biological system. This study's significance lies in developing a new approach to presenting a biological dynamical system. Since the qualitative analysis related to existence, uniqueness, and stability is almost the same as can be found in numerous existing papers, and comparing this study to other research, constructing a biological discrete system using the Caputo difference operator can be particularly important. Using powerful tools of nonlinear theory such as phase plots, bifurcation diagrams, Lyapunov exponent spectrum, and the 0-1 test, we establish that the proposed system can exhibit different biological states, including stable, periodic, and chaotic behaviors. Here, the route leading to chaos is period-doubling bifurcation. Furthermore, the level of chaos in the system is quantified using $C_{0}$ complexity and approximate entropy algorithms. The stabilization or suppression of chaotic motions in the fractional-order system is presented, where an efficient controller is designed based on the stability theory of the discrete-time fractional-order systems. Numerical simulations are provided to validate the theoretical results derived in this research paper.

Theoretical Analysis of Viscoelastic Friction System Characteristics of Robotic Arm Brake Based on Fractional Differential Theory

In engineering practice, the nonlinear vibration effect can easily lead to chaos in the system, which will not only reduce the performance of the system but also lead to premature fatigue of components, control failure, and increased safety risks. In view of the core position of the robotic arm in modern industry, this study relies on the robotic arm brake system to explore the theoretical basis of integrated viscoelastic materials as a vibration isolation layer. By analyzing the dynamic characteristics of the friction braking system with fractional differential terms, it aims to provide a new perspective for understanding and controlling the chaotic phenomena of a class of nonlinear friction systems. Firstly, we construct a model of a friction system and analyze its dynamic characteristics in detail. The self-excited vibration of the system under disturbance is studied. The relationship between amplitude and frequency is calculated by a nonlinear approximate analytical algorithm, and the accuracy of this relationship is verified by a numerical algorithm. Then, we compare the differences between non-fractional systems and fractional systems. It is found that with the increase in the fractional order term, the vibration amplitude of the system decreases significantly, which helps to reduce the nonlinear characteristics generated by the friction system and narrow the range of unstable solutions. Secondly, we also study the influence of parameter coefficients on the amplitude–frequency characteristics and analyze the local static bifurcation characteristics through singularity theory. Finally, we study the dynamic bifurcation behavior under different parameter perturbations and find that the change in system parameters will lead to the alternation of periodic motion and chaotic motion.

Weak nonlinear thermo bioconvection: Heat transfer via artificial neural network

The present study undertakes an analytical exploration of the onset of regular and chaotic thermo-bioconvection in a suspension of gravitactic microorganisms. Additionally, it employs a machine learning approach for numerical computation and prediction of heat transfer rates. The two-dimensional flow governing dynamics are modeled using the Hamilton-Crosser model for microorganisms. The amplitude of convection is determined by solving the cubic Ginzburg-Landau equation derived by applying the Lorenz technique. Stationary curves are plotted with thermal Rayleigh number and wave number, while Nusselt numbers are depicted over a range of time. The onset of chaotic motion is briefly discussed through Lyapunov plots and a bifurcation diagram. Further, to predict the heat transfer rate with multiple interconnected parameters, an artificial neural network is trained with the Levenberg-Marquardt algorithm to understand the underlying patterns of simulated data. The trained neural network is then employed to estimate the Nusselt number for various values of bioconvection Rayleigh number, bioconvection Lewis number, and bioconvection Péclet number. The values obtained from the artificial neural network models are compared with numerical data for validation and are found to be in good agreement. The findings of weak non-linear stability indicate that chaotic motion emerges in the system at the Hopf-Rayleigh number of 24.635, and fast-swimming microorganisms significantly increase the heat transfer rate. The coefficient of correlation is higher than 0.99, supporting the accuracy of developed ANN models to predict the Nusselt number with high accuracy. This observation supports the potential utilization of microbes in heat transfer applications.

Optical frequency combs and chaos in a hybrid atom–cavity optomagnonical system via the synergy of double-probe fields

We theoretically investigate the modulation of optical frequency combs and chaos by the synergy of double-probe fields in the hybrid atom–cavity optomagnonical system consisting of a yttrium iron garnet sphere, a two-level atom, and a tapered fiber. Our results show that in the case of a single-probe field, the atom-optical mode coupling strength and amplitude strength of the probe field can effectively modify the tooth spacing (from one to one-half tooth spacing) of the optical frequency combs, as well as the entry or withdrawal of chaos. In a chaotic region, the relative phases between the control and probe fields can also be adjusted to withdraw chaotic motion. Moreover, we demonstrate the generation of hybridized fraction-order frequency combs via the synergy of double-probe fields, which is caused by the sum and difference frequency effects of sidebands. Interestingly, it is found that through the synergy of the double-probe fields, the system can enter a chaotic state under lower applied field intensity, which is modulated by the amplitude strengths and relative phases of the dual probe fields. In addition to providing insight into the characteristics of optical frequency combs and chaos in the hybrid atom–cavity optomagnonical system, our research may offer a new perspective for the development of precision measurements and secret information processing.

Proportional-derivative control and motion stability analysis of a 16-pole legs rotor-active magnetic bearings system

This paper analyzes the motion stability of a 16-pole rotor-active magnetic bearings (rotor-AMB) system and investigates the complex vibrations under a proportional-derivative (PD) controller. First, electromagnetic theory and Newton’s second law are applied to derive the two-degree-of-freedom differential governing equations for the 16-pole rotor-AMB system, incorporating the PD control terms. The resulting differential equations include parametrically excited, quadratic nonlinear, and cubic nonlinear terms. Subsequently, the multiple time scales perturbation analysis method is performed on the obtained governing equations, yielding four-dimensional averaged equations in both Cartesian and polar coordinates. Finally, numerical simulations including the amplitude–frequency response characteristics, motion trajectories, energy–amplitude relationships, as well as bifurcation and chaotic motion of the system are studied. The results indicate that the PD controller affects the softening and hardening spring characteristics of the system and has significant control effects on the system’s amplitude, energy, and stability. Additionally, increasing the differential gain coefficient [Formula: see text] can change the system’s motion from chaotic to periodic.