Fractional-Order Discrete-Time SIR Epidemic Model with Vaccination: Chaos and Complexity

This research presents a new fractional-order discrete-time susceptible-infected-recovered (SIR) epidemic model with vaccination. The dynamical behavior of the suggested model is examined analytically and numerically. Through using phase attractors, bifurcation diagrams, maximum Lyapunov exponent and the 0−1 test, it is verified that the newly introduced fractional discrete SIR epidemic model vaccination with both commensurate and incommensurate fractional orders has chaotic behavior. The discrete fractional model gives more complex dynamics for incommensurate fractional orders compared to commensurate fractional orders. The reasonable range of commensurate fractional orders is between γ = 0.8712 and γ = 1, while the reasonable range of incommensurate fractional orders is between γ2 = 0.77 and γ2 = 1. Furthermore, the complexity analysis is performed using approximate entropy (ApEn) and C0 complexity to confirm the existence of chaos. Finally, simulations were carried out on MATLAB to verify the efficacy of the given findings.

Nonlinear Dynamics Analysis of Multifactor Low Speed Heavy Load Gear System with Temperature Effect Considered

Low-speed and heavy-duty gears will generate a lot of heat during meshing transmission, which will cause thermal deformation of the gears and affect the transmission performance of the gear system. It is of great significance to explore the influence of temperature effects on the nonlinear dynamics of the gear system. Taking the spur gear system as the research object, considering the nonlinear factors such as time-varying meshing stiffness, tooth backlash and comprehensive error, and introducing the influence of temperature change, the nonlinear dynamic model of the gear system is established, using 4~5th order Runge -Kutta algorithm performs simulation calculation on the model, combined with bifurcation diagram, maximum Lyapunov exponent diagram, phase diagram and Poincare section diagram, etc., to analyze the influence of temperature changes and time-varying stiffness coefficients on the motion characteristics of the gear system. The results show that the influence of temperature change on the gear system is related to the value of the time-varying stiffness coefficient. The larger the value, the more obvious the influence of temperature change; the system will show different dynamic response with the change of the time-varying stiffness coefficient, including four states of single-period motion, multiple-period motion, bifurcation and chaotic motion. The relevant conclusions can provide references for the design of gear systems under special working conditions.

Dynamic Response and Chaotic Behavior of a Controllable Flexible Robot

Flexible robots with controllable mechanisms have advantages over common tandem robots in vibration magnitude, residual vibration time, working speed, and efficiency. However, abnormal vibration can sometimes occur during their use, affecting their normal operation. In order to better understand the causes of this abnormal vibration, our work takes a controllable flexible robot as a research object, and uses a combination of Lagrangian and finite element methods to establish its nonlinear elastic dynamics. The effectiveness of the model is verified by comparing the frequency of the numerical calculation and the test. The time-domain diagram, phase diagram, Poincaré map, and maximum Lyapunov exponent of the elastic motion of the robot wrist are studied, and the chaotic phenomena in the system are identified through the phase diagram, Poincaré map, and the maximum Lyapunov exponent. The relationship between the parameters of the robot motion and the maximum Lyapunov exponent is discussed, including trajectory angular speed and radius. The results show that chaotic behavior exists in the controllable flexible robot, and that trajectory angular speed and radius all have an influence on the chaotic motion, which provides a theoretical basis for further research on the control and optimal design of the mechanism.

Chaotic vibrations of carbon nanotubes subjected to a traversing force considering nonlocal elasticity theory

The existence of chaos in the lateral vibration of the carbon nanotube (CNT) can contribute to source of instability and inaccuracy within the nano mechanical systems. So, chaotic vibrations of a simply supported CNT which is subjected to a traversing harmonic force are studied in this paper. The model of the system is formulated by using nonlocal Euler–Bernoulli beam theory. The equation of motion is solved using the Rung–Kutta method. The effects of the nonlocal parameter, velocity and amplitude of the traversing harmonic force on the nonlinear dynamic response of the system are analyzed by the bifurcation diagrams, phase plane portrait, power spectra analysis, Poincaré map and the maximum Lyapunov exponent. The results indicate that the nonlocal parameter, velocity and amplitude of the traversing harmonic force have considerable effects on the bifurcation behavior and can be used as effective control parameters for avoiding chaos.

Extreme Multistability of a Fractional-Order Discrete-Time Neural Network

At present, the extreme multistability of fractional order neural networks are gaining much interest from researchers. In this paper, by utilizing the fractional ℑ-Caputo operator, a simple fractional order discrete-time neural network with three neurons is introduced. The dynamic of this model are experimentally investigated via the maximum Lyapunov exponent, phase portraits, and bifurcation diagrams. Numerical simulation demonstrates that the new network has various types of coexisting attractors. Moreover, it is of note that the interesting phenomena of extreme multistability is discovered, i.e., the coexistence of symmetric multiple attractors.

Examining optimum prediction time of rainfall dynamics based on chaotic perspective at different temporal scales: a case study in Bojonegoro, Indonesia

As the driving force of the hydrological system, rain has severe impact when dealing with petroleum mining activities, especially in protecting assets and safety. Rainfall has high variability, both spatial and temporal (chaotic data). Due to this reason, ones can only create long-range prediction using the stochastic method. Here we use the Lyapunov exponent to analyze the nonlinear pattern of rainfall dynamics. This method is useful for identifying chaotic deportment in rainfall data. This study uses rainfall data for six years obtained from one of the largest petroleum mining sites in Bojonegoro, Indonesia. Rainfall dynamics have been analyzed on three different time scales, namely daily data, 5-day, and 10-day. The time delay (τ) was obtained by using the Average Mutual Information (AMI) method for the three-rainfall series (3, 2, 3, respectively). The observed rainfall data in Bojonegoro show signs of chaos as the finite correlation dimensions (m) attain values about 4 for all time scales. The maximum Lyapunov exponent λmax for each of three-rainfall series in Bojonegoro is 0.111, 0.057, 0.062, respectively. These values were analyzed to find the optimum prediction time of rainfall occurrence to perform better forecasting. The result shows that the optimum range of prediction time for daily, 5-day, and 10-day have 9, 18, and 16 times longer than their temporal scale.

A novel memristor-based chaotic system with line equilibria and its complex dynamics

Memristor, as a nonlinear element, provides many advantages thanks to its superior properties to design different chaotic circuits. Thus, a novel four-dimensional double-scroll chaotic system with line equilibria as well as two unstable equilibria based on the flux-memristor model is proposed in this paper. The effects of initial values and parameters on the dynamic characteristics of the system are studied in detail by means of phase diagrams, Lyapunov exponent spectrums, bifurcation diagrams, two-parameter bifurcation diagrams and basins of attraction. Besides, a series of complex phenomena in the system, such as sustained chaos, bistability, transient chaos and coexisting attractors are observed, proving that the chaotic system has rich dynamic characteristics. Also, spectral entropy (SE) complexity algorithm and [Formula: see text] complexity algorithm based on structure complexity are adopted to analyze the complexity of the system. Additionally, PSPICE circuit simulation and Micro-Controller Unit (MCU) hardware experiment are carried out to verify the correctness of theoretical analysis and numerical simulation. Finally, the pulse chaos synchronization is achieved from the perspective of maximum Lyapunov exponent, and numerical simulations demonstrate the occurrence of the proposed system and practicability of the pulse synchronization control.

Experimental Evidence of Chaos Generated by a Minimal Universal Oscillator Model

Detection of chaos in experimental data is a crucial issue in nonlinear science. Historically, one of the first evidences of a chaotic behavior in experimental recordings came from laser physics. In a recent work, a Minimal Universal Model of chaos was developed by revisiting the model of laser with feedback, and a first electronic implementation was discussed. Here, we propose an upgraded electronic implementation of the Minimal Universal Model, which allows for a precise and reproducible analysis of the model’s parameters space. As a marker of a possible chaotic behavior the variability of the spiking activity that characterizes one of the system’s coordinates was used. Relying on a numerical characterization of the relationship between spiking activity and maximum Lyapunov exponent at different parameter combinations, several potentially chaotic settings were selected. The analysis via divergence exponent method of experimental time series acquired by using those settings confirmed a robust chaotic behavior and provided values of the maximum Lyapunov exponent that are in very good agreement with the theoretical predictions. The results of this work further uphold the reliability of the Minimal Universal Model. In addition, the upgraded electronic implementation provides an easily controllable setup that allows for further developments aiming at coupling multiple chaotic systems and investigating synchronization processes.

Effects of arm swing amplitude and lower limb asymmetry on motor variability patterns during treadmill gait

Motor variability is a fundamental feature of gait. Altered arm swing and lower limb asymmetry (LLA) may be contributing factors having been shown to affect the magnitude and dynamics of variability in spatiotemporal and trunk motion. However, the effects on lower limb joints remain unclear. Full-body kinematics of 15 healthy young adults were recorded during treadmill walking using the Computer-Assisted Rehabilitation Environment system. Participants completed six trials, combining three arm swing (AS) amplitude (normal, active, held) and two LLA (symmetrical, asymmetrical) conditions. The mean standard deviation (meanSD), maximum Lyapunov exponent (λmax), detrended fluctuation analysis scaling exponent of range of motion (DFAα), and sample entropy (SaEn) were computed for tridimensional trunk, pelvis, and lower limb joint angles, and compared using repeated-measures ANOVAs. Relative to normal AS, active AS increased meanSD of all joint angles, λmax of frontal plane hip and ankle angles, and SaEn of sagittal plane ankle angles. Active AS, however, did not affect λmax or SaEn of trunk or pelvis angles. LLA increased meanSD of sagittal plane joint angles, λmax of Euclidean norm trunk angle and of lower limb joint angles, and SaEn of ankle dorsiflexion/ plantarflexion, but decreased SaEn of tridimensional trunk angles and hip rotation in the slower moving leg. Alterations in lower limb variability with active AS and LLA suggest that young adults actively exploit their lower limb redundancies to maintain gait. This appears to preserve trunk stability and regularity during active AS but not during LLA.

Chaotic Characteristic Analysis of Vibration Response of Pumping Station Pipeline Using Improved Variational Mode Decomposition Method

The measured vibrational responses of the pumping station pipeline in the irrigation site were chosen to confirm the chaotic characteristics of the pumping station pipeline vibration and to determine the vibrational excitation that makes it chaotic. First, the chaotic properties of the pipeline vibration responses were investigated using a saturation correlation dimension and the maximum Lyapunov exponent. The vibration excitation with chaotic features was obtained using an improved variational mode decomposition (IVMD) method to examine the multi-time-scale chaotic characteristics of the pipeline vibration responses. The results show that the vibrational responses of each measuring point of the pipeline under different operating conditions have clear chaotic characteristics, where the chaotic characteristics of the axial points and bifurcated pipe points are relatively strong. The vibration of the operating conditions and measurement points affected by the unit’s operation and flow state change is further complicated. The intrinsic mode function (IMF) produces a low-dimensional chaotic attractor after the IVMD disrupts the vibration response. Still, the vibration excitation of the remaining components on behalf of the units does not have chaotic properties, implying that water pulsation excitation makes the pumping station pipeline vibrations chaotic. The vibration excitation caused by the unit’s operation covers the chaotic characteristics of the pipeline vibration and increases its uncertainty. The outcomes of this study provide a theoretical basis for further exploration of the vibration characteristics of pumping station pipelines, and a new method of chaos analysis is proposed.