Quasi-periodic Motions Research Articles

Sun–Jupiter–Saturn System May Exist: A Verified Computation of Quasiperiodic Solutions for the Planar Three-Body Problem

AbstractIn this paper, we present evidence of the stability of a model of our Solar System when taking into account the two biggest planets, a planar (Newtonian) Sun–Jupiter–Saturn system with realistic data: masses of the Sun and the planets, their semiaxes, eccentricities and (apsidal) precessions of the planets close to the real ones. (We emphasize that our system is not in the perturbative regime but for fixed parameters.) The evidence is based on convincing numerics that a KAM theorem can be applied to the Hamiltonian equations of the model to produce quasiperiodic motion (on an invariant torus) with the appropriate frequencies. To do so, we first use KAM numerical schemes to compute translated tori to continue from the Kepler approximation (two uncoupled two-body problems) up to the actual Hamiltonian of the system, for which the translated torus is an invariant torus. Second, we use KAM numerical schemes for invariant tori to refine the solution giving the desired torus. Lastly, the convergence of the KAM scheme for the invariant torus is (numerically) checked by applying several times a KAM–iterative lemma, from which we obtain that the final torus (numerically) satisfies the existence conditions given by a KAM theorem.

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.

Morphology and dynamics of the liquid jet in high-speed gas-assisted atomization retrieved through synchrotron-based high-speed X-ray imaging

The breakup of a liquid jet by a surrounding high-speed gas jet with different liquid Reynolds numbers and gas Weber numbers was studied using high speed phase-contrast X-ray imaging technique. Focusing on the liquid core, the portion of liquid that is connected to the nozzle, four distinct morphologies were observed and can be associated with changes in the large-scale configurations of the two-phase flow are reported in a phase diagram. Gas-to-liquid kinetic energy balance arguments capture several transitions between the liquid core regimes. The temporal evolution of the center of mass of the liquid core is extracted to quantify its motion, whose statistics can be utilized as a signature to distinguish different regimes. At low to moderate gas Weber numbers, the dynamics are strongly influenced by flapping, while long-time dynamics develop at high Weber numbers, that give way to quasi-periodic motions when swirl is impeded to the gas jet.

Dynamic Characteristics Analysis of the DI-SO Cylindrical Spur Gear System Based on Meshing Conditions

The dual-input single-output (DI-SO) cylindrical spur gear system possesses advantages such as high load-carrying capacity, precise transmission, and low energy loss. It is increasingly becoming a core component of power transmission systems in maritime vessels, aerospace, marine engineering, and construction machinery. In practical operation, the stability of the DI-SO cylindrical spur gear system is influenced by complex excitations. These excitations lead to nonlinear vibration, meshing instability, and noise, which affect the performance and reliability of the entire equipment. Hence, the dynamic performance of the DI-SO cylindrical spur gear system is thoroughly investigated in this research. The impact of excitations and nonlinear factors on the dynamic characteristics was investigated comprehensively. A comparative analysis of the gear system was conducted by establishing a bending–torsional coupling vibration analysis model under synchronous and asynchronous meshing conditions. Nonlinear factors such as periodic time-varying meshing stiffness, meshing damping, friction coefficient, friction arms, load sharing ratio, comprehensive transmission error, and backlash were considered in the proposed model. Then, the effect laws of excitations and nonlinear factors such as meshing frequency, driving load fluctuation, backlash, and comprehensive transmission error were analyzed. The results indicate that the dynamic characteristics exhibited staged stable and unstable states under different meshing frequencies and meshing conditions. At the medium-frequency meshing stage (0.96 × 104~1.78 × 104 Hz), alternating phenomena of multi-periodic, quasi-periodic, and chaotic motion states were observed. Moreover, the root mean square value (RMS) of the dynamic transmission error (DTE) in the asynchronized gear system was generally higher than that in the synchronized gear system. It was found that selecting the appropriate meshing condition could effectively reduce the amplitude of the DTE. Additionally, the dynamic performance could be significantly improved by adjusting control parameters such as driving load fluctuation (0~179 N), backlash (0.8 × 10−4~0.9 × 10−4 m), and comprehensive transmission error (7.9 × 10−4~9.4 × 10−4 m). The research results provide a theoretical guidance for the design and optimization of the DI-SO cylindrical spur gear system.

Active particle motion in Poiseuille flow through rectangular channels.

We investigate the dynamics of a pointlike active particle suspended in fluid flow through a straight channel. For this particle-fluid system, we derive a constant of motion for a general unidirectional fluid flow and apply it to an approximation of Poiseuille flow through channels with rectangular cross- sections. We obtain a 4D nonlinear conservative dynamical system with one constant of motion and a dimensionless parameter describing the ratio of maximum flow speed to intrinsic active particle speed. Applied to square channels, we observe a diverse set of active particle trajectories with variations in system parameters and initial conditions which we classify into different types of swinging, trapping, tumbling, and wandering motion. Regular (periodic and quasiperiodic) motion as well as chaotic active particle motion are observed for these trajectories and quantified using largest Lyapunov exponents. We explore the transition to chaotic motion using Poincaré maps and show "sticky" chaotic tumbling trajectories that have long transients near a periodic state. We briefly illustrate how these results extend to rectangular cross-sectionswith a width-to-height ratio larger than one. Outcomes of this paper may have implications for dynamics of natural and artificial microswimmers in experimental microfluidic channels that typically have rectangular cross sections.

An investigation on hydrodynamic behavior of horizontal gas–liquid intermittent flow by S-PLIF and parallel-wire conductance sensors

Horizontal gas–liquid two-phase flows widely exist in a variety of industrial fields such as the environment, chemical industry and energy. Intermittent flow, characterized by the quasi-periodic alternation motion of liquid slugs and large-scale Taylor bubbles, is the most prevalent flow pattern in horizontal gas–liquid two-phase flows. Comprehensive research on the hydrodynamic behavior of gas–liquid intermittent flow is essential for revealing the mass and heat transfer characteristics between phases, thereby establishing physical models of flow parameters. In this study, a structured planar laser-induced fluorescence (S-PLIF) system and paired parallel-wire conductance sensors (PWCSs) are designed to detect the gas–liquid interface structures and the hydrodynamic parameters. The S-PLIF system consists of a laser, a visualization box, a Ronchi ruling-plate and a pair of high-speed cameras. An optical linear prism forms a laser sheet in the transverse direction and the middle of the test section, and the cameras are positioned at 70° and 90° respectively to the plane of the laser sheet, referred to as S-PLIF70 and S-PLIF90. Under the laser excitation, bright and dark stripes generated by the Ronchi ruling-plate are deflected at the gas–liquid interface. The influence of total reflection can be effectively avoided by identifying the deflection position of the stripes, allowing for accurate detection of the gas–liquid interfacial structures. The effects of the interface fluctuations and detached bubbles on the S-PLIF70&90 results are investigated. Fluctuation and evolution characteristics of the gas–liquid interface are analyzed by probability density function (PDF) and power spectral density (PSD). Meanwhile, hydrodynamic parameters, i.e., Taylor bubble nose and tail velocity, Taylor bubble and slug length, and slug frequency, are detected by the PWCSs, and compared with previous correlations. The evolution characteristics of the interfacial structures and hydrodynamic parameters are uncovered as the flow condition changes.

Bifurcation boundaries analysis of the thin-walled internal resonance energy harvester

The dynamic instability of the thin-walled 1: 2 internal resonant piezoelectric vibration energy harvester is investigated. Previously work (Nie et al., 2019) derived the nonlinear electromechanical-coupled governing equations and validated them experimentally. The modulation equations are performed by the method of multiple scales. The system stability is obtained from the eigenvalues of the Jacobi matrix. Then, three types of bifurcation boundaries of the system are determined analytically and simulated numerically via Routh–Hurwitz criterion. Results show that internal resonance complicates the dynamic properties of the bifurcation region. For a large external excitation, new saddle node bifurcation regions appear in the stability diagram. Moreover, the new saddle node and Hopf bifurcation regions may intertwine. Furthermore, as the acceleration excitation increases, the system can be observed to transition to stable–unstable–stable states. The system’s high and low branch responses are fed back by the system’s large and small initial conditions. The system’s periodic and quasi-periodic motions are characterized by time response, FFT, phase trajectory, and Poincaré map. The study of the stability boundary of the piezoelectric energy harvester contributes to the harvesting of energy from ambient vibrations over a broader frequency range.

A complete dynamical analysis of discrete electric lattice coupled with modified Zakharov–Kuznetsov equation

The behavior of nonlinear waves within a modified Zakharov–Kuznetsov equation and their interactions with discrete electric lattice structures are examined in this study. The ϕ6−model expansion method is utilized to acquire substantial knowledge into the complex dynamics of the system under consideration, particularly with regard to the discrete electric lattice and analytical electrical solitons. By incorporating higher-order effects and improving accuracy in representing specific physical conditions, the study achieves a more realistic portrayal of nonlinear wave dynamics. The investigation also sheds light on the relationship between non-linearity, discreteness, and equation dynamics by exploring the conditions that lead to the formation of solitons and other nonlinear structures. In addition, a unique set of electrical solitons is defined to explore dynamic behaviors such as chaotic, quasi-periodic, and periodic motions under various parameterized conditions, including an external damping force. Phase plane analysis is visualized by using dynamic structure 3D and 2D phase plots, is used for bifurcation and sensitivity inspections. Finally, time series graphs are offered as mathematical depictions of solitary waves, and Lyapunov exponents with real and complex eigenvalues are used to study the stability and chaotic behaviors of the system.

Nonlinear thermal flutter of variable angle tow composite laminates in supersonic airflow

Compared with traditional composite materials, variable stiffness composite materials with curvilinear fiber paths have higher design flexibility and the potential to improve structural efficiency. In view of this, the nonlinear flutter characteristics of Variable Angle Tow (VAT) composite laminates with thermal effect in supersonic airflow are investigated in this paper. The proposed panel flutter model is developed based on the von Kármán large deformation plate theory. The thermal stress is approximated according to the quasi-steady thermal stress theory, whilst the aerodynamic pressure is calculated based on the first-order piston theory. The aerothermoelastic equations of motion are first established using Hamilton’s variational principle, and the Rayleigh-Ritz method is then applied to solve the governing equations in the space domain. The time-domain nonlinear equations are solved through the fourth-order Runge-Kutta method. The accuracy and convergence of the proposed method are verified through the comparisons with the results available in the literature. The effects of several parameters such as fiber orientation angle, thermal load, and aerodynamic pressure on the nonlinear flutter responses of VAT laminates are comprehensively discussed. Several dynamic behaviors of VAT laminates, including stable flat, buckled, Limit Cycle Oscillation (LCO), multi-periodic motion, quasi-periodic motion, and chaotic motion, are revealed by using the time history, phase portrait, Poincaré map, and bifurcation diagram. The results presented herein may be beneficial for the design of VAT laminates in supersonic airflow.

Investigating pseudo parabolic dynamics through phase portraits, sensitivity, chaos and soliton behavior

This research examines pseudoparabolic nonlinear Oskolkov-Benjamin-Bona-Mahony-Burgers (OBBMB) equation, widely applicable in fields like optical fiber, soil consolidation, thermodynamics, nonlinear networks, wave propagation, and fluid flow in rock discontinuities. Wave transformation and the generalized Kudryashov method is utilized to derive ordinary differential equations (ODE) and obtain analytical solutions, including bright, anti-kink, dark, and kink solitons. The system of ODE, has been then examined by means of bifurcation analysis at the equilibrium points taking parameter variation into account. Furthermore, in order to get insight into the influence of some external force perturbation theory has been employed. For this purpose, a variety of chaos detecting techniques, for instance poincaré diagram, time series profile, 3D phase portraits, multistability investigation, lyapounov exponents and bifurcation diagram are implemented to identify the quasi periodic and chaotic motions of the perturbed dynamical model. These techniques enabled to analyze how perturbed dynamical system behaves chaotically and departs from regular patterns. Moreover, it is observed that the underlying model is quite sensitivity, as it changing dramatically even with slight changes to the initial condition. The findings are intriguing, novel and theoretically useful in mathematical and physical models. These provide a valuable mechanism to scientists and researchers to investigate how these perturbations influence the system’s behavior and the extent to which it deviates from the unperturbed case.

Existence and stability of a quasi-periodic two-dimensional motion of a self-propelled particle

The mechanism of self-propelled particle motion has attracted much interest in mathematical and physical understanding of the locomotion of living organisms. In a top-down approach, simple time-evolution equations are suitable for qualitatively analyzing the transition between the different types of solutions and the influence of the intrinsic symmetry of systems despite failing to quantitatively reproduce the phenomena. We aim to rigorously show the existence of the rotational, oscillatory, and quasi-periodic solutions and determine their stabilities regarding a canonical equation proposed by Koyano et al. (J Chem Phys 143(1):014117, 2015) for a self-propelled particle confined by a parabolic potential. In the proof, the original equation is reduced to a lower dimensional dynamical system by applying Fenichel’s theorem on the persistence of normally hyperbolic invariant manifolds and the averaging method. Furthermore, the averaged system is identified with essentially a one-dimensional equation because the original equation is O(2)-symmetric.

Chaotic vibration of a curved CNT conveying magnetic fluid in the thermo-magnetic field considering the surface effects

The nonlinear chaotic vibration of curved single-walled carbon nanotube (CSWCNT) conveying magnetic fluid is studied based on the nonlocal Euler-Bernoulli beam model. The governing equation of CSWCNT is presented considering the axial thermo-magnetic load and the surface effect. By employing the Galerkin decomposition approximation method along with the admissible beam shape function satisfying the cantilevered beam boundary conditions, the nonlinear partial differential equation of the system is reduced to a nonlinear ordinary differential equation and numeric integration procedures are used to solve it. By considering the non-dimensional damping, cubic and quadratic terms, and amplitude of external force as controlling parameters, the bifurcation diagram and the largest Lyapunov exponent are employed to distinguish the chaotic, periodic, and quasi-periodic critical parameters of the curved CNT dynamics and validate the predicted results. Also, the phase plane and Poincare map for these critical parameters are provided. The results show that decreasing values of damping coefficient and quadratic term leads to quasi-periodic or chaotic motion, while system has periodic behaviour for smaller values of the external force and nonlinear cubic term. Also, magnetic field intensity and length of the CSWCNT help preventing the chaotic motion. Furthermore, adverse effects of higher values of temperature change, flow velocity and its correction factor are seen based on the obtained results.

Nonlinear dynamic analysis of parked large wind turbine blade considering parametric excitation

In this study, the nonlinear dynamics of parked large wind turbine blades are investigated, with a focus on the consideration of parametric excitations induced by the combined effect of wind and support motion. A continuum mathematical model for the blade is adopted. The time-varying damping terms that may induce parametric excitation are derived in this study. Nonlinear dynamic responses of NREL 5-MW wind turbine blade are numerically investigated using the multiple-scale method. Results show that the primary resonance of the first flapwise mode can induce the super-harmonic resonance of the first edgewise mode, with considering parametric excitation. Notably, when the ratio between the first flapwise to first edgewise modal eigenfrequencies is tuned to 1:2, the super-harmonic resonance leads to internal resonance between the two modes. In this case, the effects of incident wind speed, excitation direction and amplitude on the blade response are discussed. Quasi-periodic motions can also be induced because of the Neimark-Snack bifurcation in the case of large excitation amplitude.

Dynamics and non-integrability of the double spring pendulum

This paper investigates the dynamics and integrability of the double spring pendulum, which has great importance in studying nonlinear dynamics, chaos, and bifurcations. Being a Hamiltonian system with three degrees of freedom, its analysis presents a significant challenge. To gain insight into the system’s dynamics, we employ various numerical methods, including Lyapunov exponents spectra, phase-parametric diagrams, and Poincaré cross-sections. The novelty of our work lies in the integration of these three numerical methods into one powerful tool. We provide a comprehensive understanding of the system’s dynamics by identifying parameter values or initial conditions that lead to hyper-chaotic, chaotic, quasi-periodic, and periodic motion, which is a novel contribution in the context of Hamiltonian systems. In the absence of gravitational potential, the system exhibits S1 symmetry, and the presence of an additional first integral was identified using Lyapunov exponents diagrams. We demonstrate the effective utilization of Lyapunov exponents as a potential indicator of first integrals and integrable dynamics. The numerical analysis is complemented by an analytical proof regarding the non-integrability of the system. This proof relies on the analysis of properties of the differential Galois group of variational equations along specific solutions of the system. To facilitate this analysis, we utilized a newly developed extension of the Kovacic algorithm specifically designed for fourth-order differential equations. Overall, our study sheds light on the intricate dynamics and integrability of the double spring pendulum, offering new insights and methodologies for further research in this field.

Data-driven reconstruction of chaotic dynamical equations: The Hénon–Heiles type system

In this study, the classical two-dimensional potential VN=12mω2r2+1NrNsin(Nθ), N∈Z+, is considered. At N=1,2, the system is superintegrable and integrable, respectively, whereas for N>2 it exhibits a richer chaotic dynamics. For instance, at N=3 it coincides with the Hénon–Heiles system. The periodic, quasi-periodic and chaotic motions are systematically characterized employing time series, Poincaré sections, symmetry lines and the largest Lyapunov exponent as a function of the energy E and the parameter N. Concrete results for the lowest cases N=3,4 are presented in complete detail. This model is used as a benchmark system to estimate the accuracy of the Sparse Identification of Nonlinear Dynamical Systems (SINDy) method, a data-driven algorithm which reconstructs the underlying governing dynamical equations. We pay special attention at the transition from regular motion to chaos and how this influences the precision of the algorithm. In particular, it is shown that SINDy is a robust and stable tool possessing the ability to generate non-trivial approximate analytical expressions for periodic trajectories as well.

Negative differential resistance and quasiperiodic vortex-antivortex motion in a superconducting constriction

Negative differential resistance and quasiperiodic vortex-antivortex motion in a superconducting constriction

Analysis of quasi-periodic and chaotic motion of a dielectric elastomer shell under alternating voltage

Analysis of quasi-periodic and chaotic motion of a dielectric elastomer shell under alternating voltage

Dynamic analysis of multi-DOF acoustic resonant system involving highly viscous fluids based on fluid-structure interaction

This paper investigates the dynamic characteristics of a multi-degree-of-freedom (multi-DOF) acoustic resonant system involving highly viscous fluids. First, a fluid-structure interaction (FSI) model is established to describe the interaction between the multi-DOF acoustic resonant system and the two-phase fluids, and then the effects of various excitation parameters on the dynamic response of the coupled system are studied. Under the influence of vertical acoustic vibration, the fluid in the container undergoes violent and irregular motions, which leads to a decrease and fluctuation in the equivalent mass of the fluid. The reduction in fluid equivalent mass causes the excitation frequency to deviate from the natural frequency of the multi-DOF acoustic resonant system, thus significantly reducing the dynamic response of the coupled system. Furthermore, the fluctuation of the fluid equivalent mass induces a quasi-periodic motion pattern of the acoustic resonant system. Although increasing the excitation amplitude can effectively increase the dynamic response of the coupled system, it can also increase the fluctuation level of the dynamic response to a certain extent. By appropriately increasing the excitation frequency, the coupled system can operate at a new resonant frequency, thereby reducing the influence of fluid motion on the dynamic response of the system.

Adaptive iterative learning control of soft robot for beating heart tracking

PurposeSoft robots are known for their excellent safe interaction ability and promising in surgical applications for their lower risks of damaging the surrounding organs when operating than their rigid counterparts. To explore the potential of soft robots in cardiac surgery, this paper aims to propose an adaptive iterative learning controller for tracking the irregular motion of the beating heart.Design/methodology/approachIn continuous beating heart surgery, providing a relatively stable operating environment for the operator is crucial. It is highly necessary to use position-tracking technology to keep the target and the surgical manipulator as static as possible. To address the position tracking and control challenges associated with dynamic targets, with a focus on tracking the motion of the heart, control design work has been carried out. Considering the lag error introduced by the material properties of the soft surgical robotic arm and system delays, a controller design incorporating iterative learning control with parameter estimation was used for position control. The stability of the controller was analyzed and proven through the construction of a Lyapunov function, taking into account the unique characteristics of the soft robotic system.FindingsThe tracking performance of both the proportional-derivative (PD) position controller and the adaptive iterative learning controller are conducted on the simulated heart platform. The results of these two methods are compared and analyzed. The designed adaptive iterative learning control algorithm for position control at the end effector of the soft robotic system has demonstrated improved control precision and stability compared with traditional PD controllers. It exhibits effective compensation for periodic lag caused by system delays and material characteristics.Originality/valueTracking the beating heart, which undergoes quasi-periodic and complex motion with varying accelerations, poses a significant challenge even for rigid mechanical arms that can be precisely controlled and makes tracking targets located at the surface of the heart with the soft robot fraught with considerable difficulties. This paper originally proposes an adaptive interactive learning control algorithm to cope with the dynamic object tracking problem. The algorithm has theoretically proved its convergence and experimentally validated its performance at the cable-driven soft robot test bed.

A novel approach for estimating lung tumor motion based on dynamic features in 4D-CT

Due to the high expenses involved, 4D-CT data for certain patients may only include five respiratory phases (0%, 20%, 40%, 60%, and 80%). This limitation can affect the subsequent planning of radiotherapy due to the absence of lung tumor information for the remaining five respiratory phases (10%, 30%, 50%, 70%, and 90%). This study aims to develop an interpolation method that can automatically derive tumor boundary contours for the five omitted phases using the available 5-phase 4D-CT data. The dynamic mode decomposition (DMD) method is a data-driven and model-free technique that can extract dynamic information from high-dimensional data. It enables the reconstruction of long-term dynamic patterns using only a limited number of time snapshots. The quasi-periodic motion of a deformable lung tumor caused by respiratory motion makes it suitable for treatment using DMD. The direct application of the DMD method to analyze the respiratory motion of the tumor is impractical because the tumor is three-dimensional and spans multiple CT slices. To predict the respiratory movement of lung tumors, a method called uniform angular interval (UAI) sampling was developed to generate snapshot vectors of equal length, which are suitable for DMD analysis. The effectiveness of this approach was confirmed by applying the UAI-DMD method to the 4D-CT data of ten patients with lung cancer. The results indicate that the UAI-DMD method effectively approximates the lung tumor’s deformable boundary surface and nonlinear motion trajectories. The estimated tumor centroid is within 2 mm of the manually delineated centroid, a smaller margin of error compared to the traditional BSpline interpolation method, which has a margin of 3 mm. This methodology has the potential to be extended to reconstruct the 20-phase respiratory movement of a lung tumor based on dynamic features from 10-phase 4D-CT data, thereby enabling more accurate estimation of the planned target volume (PTV).