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Direct yaw moment control (DYC) and differential drive-assist steering (DDAS) for distributed-drive vehicles are both realized by allocating the in-wheel motor torque. To address the interference caused by overlapping control objectives, this paper proposes a multilayer control strategy that integrates DYC and DDAS, consisting of an upper controller, a coordinated decision layer, and a torque distribution layer. The upper controller, designed based on the vehicle's dynamic characteristics, incorporates an adaptive fuzzy control DYC system and a dual PID control DDAS system. The coordinated decision layer is developed utilizing a phase-plane dynamic weighting method, delineating region boundaries by applying the double-line and limit cycle methods. The torque distribution strategy is formulated considering motor peak torque and road adhesion conditions. Multi-condition joint simulation experiments indicate that the proposed multilayer control strategy, integrating the advantages of DYC and DDAS, reduces peak steering wheel torque by approximately 10%, peak yaw rate by around 25%, peak sideslip angle by roughly 29%, and peak sideslip angle rate by about 19%, significantly improving driving stability and maneuvering flexibility. [ABSTRACT FROM AUTHOR]

For the sake of enhancing the handling and stability of distributed drive electric vehicles (DDEVs) under four-wheel steering (4WS) conditions, this study proposes a novel hierarchical control strategy based on a phase plane analysis. This approach involves a meticulous comparison of the stable region in the phase plane to thoroughly analyze the intricate influence of the front wheel angle, rear wheel angle, road adhesion coefficient, and longitudinal speed on the complex dynamic performances of DDEVs and to accurately determine the critical stable-state parameter. Subsequently, a hierarchical control strategy is presented as an integrated solution to achieve the coordinated control of maneuverability and stability. On the upper control level, a model predictive control (MPC) motion controller is developed, wherein the real-time adjustment of the control weight matrix is ingeniously achieved by incorporating the crucial vehicle stable-state parameter. The lower control level is responsible for the optimal torque allocation among the four wheel motors to minimize the tire load rate, thereby ensuring a sufficient tire grip margin. The optimal torque distribution for the four wheel motors is achieved using a sophisticated two-level allocation algorithm, wherein the friction ellipse is employed as a judgement condition. Finally, this developed control strategy is thoroughly validated through co-simulation utilizing the CarSim 2019 and Simulink 2020b commercial software, demonstrating the validity of the developed control strategy. The comparative results indicate that the presented controller ensures a better tracking capability to the desired vehicle state while exhibiting improved handling stability under both the double lane shifting condition and the serpentine working condition. [ABSTRACT FROM AUTHOR]

To improve the handling stability of distributed drive electric vehicles, this paper introduces an electronic differential control strategy based on the stability criterion of the phase plane method. The strategy first plots the distributed electric vehicle's center of mass side angle and center of mass angular speed on the phase plane, and then it analyzes the vehicle's stability under various working conditions to determine the parameters that ensure the stability performance. Subsequently, an adaptive fuzzy control strategy is employed to achieve fast and accurate distribution of the torque to each wheel, thereby enhancing the vehicle's stability. A joint simulation platform is constructed using MATLAB/Simulink and CarSim. A comparison with the traditional electronic differential strategy demonstrates that the proposed distribution strategy based on phase plane stability exhibited excellent stability. [ABSTRACT FROM AUTHOR]

The tethered asteroid probe system refers to a method of connecting a probe to an asteroid using a tether. This system allows the probe to hover near the asteroid's surface without consuming fuel, making it valuable for conducting high-precision exploration, sampling, and other exploration tasks. However, various physical characteristics, such as the tether length, flexibility, gravity, internal tension, and the initial velocity state of the probe, affect the probe's hovering position and stability. In this article, the dynamic model and stability of tether probes near irregular asteroids are investigated. By considering the mass and elasticity of the tether, a multi-flexible body dynamic equation is established to describe the behavior of the probe and tether under the influence of the gravitational field of the asteroid. Dynamic equations can describe the dynamical response of the probe when switching between the tensed and relaxed states. Moreover, the changes in the position and stability of the stable range caused by the tether are analyzed. The findings suggest that connecting the probe to the asteroid through tethers significantly expands the stable operation range of the probe. The length of the tether and the amplitude of the tension significantly impact the dynamic characteristics of the probe and the stable range. Additionally, compared with the assumption that the tether is assumed to be massless and rigid, the initial range in which the tethered system can be stable is significantly reduced. Hence, the influence of the tether mass, elasticity, and damping should not be overlooked. The research presented in this article holds great significance for asteroid proximity exploration and sampling tasks that utilize tether systems. By understanding the dynamics and stability of tethered probes near irregular asteroids, human capabilities in exploring and studying these celestial bodies can be enhanced. [ABSTRACT FROM AUTHOR]

For surfaces of revolution we prove the existence of infinitely many sign-changing rotationally symmetric solutions to a wide class of $ p $-Laplace-Beltrami equations with polynomial like nonlinearities. We combine the methods in [10] where the semilinear case $ (p = 2) $ with positive weight was studied with those in [5] and [6] where radial solutions to a $ p $-Laplacian Dirichlet problem in a ball was considered. Our equations lead to ordinary differential equations with two singularities and a sign changing weight function which leads to an initial value problem that may blow up in the region of definition of the equation. [ABSTRACT FROM AUTHOR]

With the rapid development of today's society, the traffic environment has become more and more complex. As an essential part of intelligent vehicles, trajectory tracking has attracted significant attention for its stability and safety. It is prone to poor accuracy and instability in extreme working conditions like high speed. In this paper, a trajectory tracking control strategy to ensure lateral stability at a high speed and low attachment limit conditions is proposed for distributed drive vehicles. The model predictive controller (MPC) was used to control the front wheel angle, and the particle swarm optimization (PSO) algorithm was designed to optimize the MPC control parameters adaptively. The sliding mode controller controls the rear wheel angle, and the vehicle instability degree is judged by analyzing the β − β ˙ phase plane. The controllers of different instability degrees are designed herein. Finally, a torque divider is designed to consider the actuation anti-slip. The designed controller is verified by Carsim and MATLAB-Simulink co-simulation. The results show that the trajectory tracking controller designed in this paper effectively improves the tracking accuracy under the premise of ensuring stability. [ABSTRACT FROM AUTHOR]

The dynamics of four component dusty plasma with nonthermal electrons and positrons are examined. The bifurcation analysis of small amplitude dust ion acoustic (DIA) waves in this multicomponent plasma is investigated. Employing the reductive perturbation technique (RPT), the two nonlinear equations viz. KdV and mKdV are derived and using the travelling wave transformation, respective planar dynamical systems are deduced. Based on the bifurcation theory of this dynamical system, all possible phase portraits, including nonlinear homoclinic orbit, nonlinear periodic orbit and supernonlinear periodic orbit are presented. The pseudopotential profiles for the nonlinear equations are plotted for different parametric values to illustrate and confirm the phase plane analysis. It is found that the system parameter values affect the bifurcation of the DIA waves. It is also shown that the system supports both nonlinear and supernonlinear DIA periodic waves. [ABSTRACT FROM AUTHOR]

To ensure precise phase estimation within the q-axis of the phase-locked loop (PLL), integrating a filter into the q-axis loop is essential to mitigate grid-voltage harmonics. Nevertheless, the intrinsic delay characteristics of this filter impede PLL synchronization during significant grid disturbances. This study begins by developing mathematical models for three types of filters—moving-average filter (MAF) for eliminating odd harmonic components, dq-frame cascaded delayed signal cancellation (dqCDSC) filter, and notch filter (NF). Following the reduction in filter orders, a third-order nonlinear large-signal model of the PLL, incorporating an additional q-axis internal filter, is formulated. Using phase plane analysis, this study investigates the transient synchronism of the grid-following converter (GFL) and explores the influence of delay time constants from the three PLL filters on its behavior while delineating the boundaries of their basins of attraction. Theoretical findings indicate that, relative to the traditional SRF-PLL, incorporating an internal filter into the PLL compromises the transient synchronous stability of GFL. Specifically, greater filter delay time constants exacerbate the GFL's vulnerability to transient instability amid substantial grid disturbances. Hence, careful consideration is essential when using MAF-PLL and NF-PLL in situations demanding high synchronization stability. The theoretical analyses are validated using Matlab/Simulink to verify their accuracy. [ABSTRACT FROM AUTHOR]

Intelligentization is the development trend of the future automobile industry. Intelligentization requires that the dynamic control of the vehicle can complete the trajectory tracking according to the trajectory output of the decision planning the driving state of the vehicle and ensure the driving safety and stability of the vehicle. However, trajectory limit planning and harsh road conditions caused by emergencies will increase the difficulty of trajectory tracking and stability control of unmanned vehicles. In view of the above problems, this paper studies the trajectory tracking and stability control of distributed drive unmanned vehicles. This paper applies a hierarchical control framework. Firstly, in the upper controller, an adaptive prediction time linear quadratic regulator (APT LQR) path following algorithm is proposed to acquire the desired front-wheel-steering angle considering the dynamic stability performance of the tires. The lateral stability of the DDAUV is determined based on the phase plane, and the sliding surface, in the improved sliding mode control (SMC), is further dynamically adjusted to obtain the desired additional yaw moment for coordinating the path following and lateral stability. Then, in the lower controller, considering the slip and the working load of four tires, a comprehensive cost function is established to reasonably distribute the driving torque of four in-wheel motors (IWMs) for producing the desired additional yaw moment. Finally, the proposed control algorithm is verified by the hardware-in-the-loop (HIL) experiment platform. The results show the path following and lateral stability can be coordinated effectively under different driving conditions. [ABSTRACT FROM AUTHOR]

In order to improve transport efficiency and enhance the safety of the crane system, it is required that the trolley can reach the desired position fast enough, while the load swing needs to be within the bounded limits. To achieve these goals, a novel double-step acceleration strategy with equal acceleration amplitude and acceleration time, and unequal acceleration amplitude and acceleration time is proposed for input shaping anti-sway control based on phase plane trajectory planning. Specifically, the desired double-step acceleration signal is designed using an input shaper, and then the phase plane trajectory of the load swing angle is planned using the phase plane method. The geometric constraints of the phase plane trajectory and the physical constraints of the crane allow the amplitude and switching time of the double-step acceleration to be solved. The final simulation results show that the proposed control method achieves an effective anti-sway effect on the container crane. [ABSTRACT FROM AUTHOR]

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]

Lateral control is an essential safety control technology for autonomous vehicles, but the effectiveness of lateral control technology relies heavily on the precision of vehicle motion state judgements. In order to achieve accurate judgements of the vehicle motion state and to improve the control effectiveness of vehicle maneuverability and the stability controller, this paper starts with an analysis of phase plane stability. A simulation analysis is conducted to investigate the effect of the vehicle steering angle of the front wheels, the longitudinal velocity, and the tire–road adhesion coefficient on the boundary of the stability area. The stable area of the phase plane was partitioned using the proposed novel quadrilateral method, and we established a stability area regression model using machine learning methods. We analyzed the inherent connection between the lateral tire forces and the principles of vehicle maneuverability and stability control, indirectly combining the characteristics of tire forces with vehicle maneuverability and stability control. An allocation algorithm for maneuverability and stability control was designed. A co-simulation indicates that the vehicle stability controller not only accurately assesses the motion state of the vehicle but also demonstrates a considerably better performance in maneuverability and stability control compared to a controller using the traditional partitioning method of stable regions. The suggested allocation method enhances vehicle maneuverability and stability by enabling a seamless transition between the two and improving the effectiveness of stability control. [ABSTRACT FROM AUTHOR]

The task of managing unstable systems is a critically important management problem, as an unstable object can pose significant danger to humans and the environment when it fails. In this paper, a neural network was trained to determine the optimal control for an unstable system, based on a comparative analysis of two control methods: the implicit Euler method and the linearization method. This neural network identifies the optimal control based on the position of a point on the phase plane. [ABSTRACT FROM AUTHOR]

For sample return missions from deep space, a thin aeroshell capsule was proposed aimed at mitigating the aerodynamic heating during its reentry into the earth's atmosphere. The functional requirements of the capsule mandate that it aerodynamically decelerate sufficiently and land in the appropriate attitude without a parachute. To evaluate its motion attitude during transonic conditions in a wind tunnel, we employed the 1-degree of freedom (1-DOF) oscillation method. Furthermore, to predict its motion during an actual flight based on the wind tunnel test results, we examined the impact of the moment of inertia (MOI) on the motion. In all the MOI cases investigated, the limit cycle oscillation (LCO) was observed with a constant amplitude of angle. The variations in the MOI resulted in differences in the amplitude of the angle and time to reach the LCO. This study focused on the growth phase until the LCO and evaluated the characteristics on the phase plane. Consequently, a gradual approach to acceleration attributed to the static moment was observed in the growth history. A new modeling approach, centered on the nullcline during the growth phase, was adopted. Here, an approximation using a third-order function of angular velocity was applied to reproduce the nullcline obtained from the experiment. Upon non-dimensionalization of this model equation using the motion period, it was confirmed that the non-dimensional parameters were affected by the motion period, while this model focused solely on the growth phase. Furthermore, upon parameter tuning to replicate the amplitude of the angle and time to reach the LCO, and using these parameters, we predicted the impact of the MOI. The predicted results qualitatively reproduced larger MOI leading to larger amplitudes of angle and longer times to reach the LCO, which was consistent with the experimental results.

For the sake of enhancing the handling and stability of distributed drive electric vehicles (DDEVs) under four-wheel steering (4WS) conditions, this study proposes a novel hierarchical control strategy based on a phase plane analysis. This approach involves a meticulous comparison of the stable region in the phase plane to thoroughly analyze the intricate influence of the front wheel angle, rear wheel angle, road adhesion coefficient, and longitudinal speed on the complex dynamic performances of DDEVs and to accurately determine the critical stable-state parameter. Subsequently, a hierarchical control strategy is presented as an integrated solution to achieve the coordinated control of maneuverability and stability. On the upper control level, a model predictive control (MPC) motion controller is developed, wherein the real-time adjustment of the control weight matrix is ingeniously achieved by incorporating the crucial vehicle stable-state parameter. The lower control level is responsible for the optimal torque allocation among the four wheel motors to minimize the tire load rate, thereby ensuring a sufficient tire grip margin. The optimal torque distribution for the four wheel motors is achieved using a sophisticated two-level allocation algorithm, wherein the friction ellipse is employed as a judgement condition. Finally, this developed control strategy is thoroughly validated through co-simulation utilizing the CarSim 2019 and Simulink 2020b commercial software, demonstrating the validity of the developed control strategy. The comparative results indicate that the presented controller ensures a better tracking capability to the desired vehicle state while exhibiting improved handling stability under both the double lane shifting condition and the serpentine working condition.

In application-oriented mathematics, particularly in the context of nonlinear system analysis, phase plane analysis through SageMath offers a visual display of the qualitative behaviour of solutions to differential equations without inundating students with cumbersome calculations of the plane-phase. A variety of examples is usually given to illustrate phase-plane behaviour. We approach these problems by considering a problem containing a single real parameter that exemplifies the various situations clearly and simply. We developed two computer programs in SageMath: one program calculates aspects of the Jacobian matrix and displays the phase plane portraits, the other determines the centre manifold. Computations and images generated with computer codes are useful in understanding dynamic models in biology, physics and engineering that involve planar non-linear autonomous differential equations. This paper covers analytical and computational skills that are helpful for students and teachers. [ABSTRACT FROM AUTHOR]

In this study, a model of two nonlinear spring masses subjected to a smooth friction-velocity curve is examined. The second order governing system of motion is obtained in view of the system friction forces. This system is transformed to another suitable one of first order to investigate the points of equilibrium using Hurwitz's theory. The excitation and critical stick-slip (SS) speeds are determined in accordance with the characteristic equation for the Jacobian matrix of the system. The stability and behavior of the system motion, along with the behavior of the SS movement, are examined. The effect of various parameters, such as excitation speed, damping and fraction coefficients, linear and nonlinear stiffness of springs, and masses that affect the motion and stability of the system is analyzed and studied. This research has numerous practical applications in a variety of industries, including airports, modern car brakes, well drilling, explaining and understanding seismic events, mechanical transportation, friction in bridges, and civil systems, which all exhibit impacted friction in dynamics and stick-slip phenomena. [ABSTRACT FROM AUTHOR]

To improve the handling stability of distributed drive electric vehicles, this paper introduces an electronic differential control strategy based on the stability criterion of the phase plane method. The strategy first plots the distributed electric vehicle’s center of mass side angle and center of mass angular speed on the phase plane, and then it analyzes the vehicle’s stability under various working conditions to determine the parameters that ensure the stability performance. Subsequently, an adaptive fuzzy control strategy is employed to achieve fast and accurate distribution of the torque to each wheel, thereby enhancing the vehicle’s stability. A joint simulation platform is constructed using MATLAB/Simulink and CarSim. A comparison with the traditional electronic differential strategy demonstrates that the proposed distribution strategy based on phase plane stability exhibited excellent stability.

With the rapid development of today’s society, the traffic environment has become more and more complex. As an essential part of intelligent vehicles, trajectory tracking has attracted significant attention for its stability and safety. It is prone to poor accuracy and instability in extreme working conditions like high speed. In this paper, a trajectory tracking control strategy to ensure lateral stability at a high speed and low attachment limit conditions is proposed for distributed drive vehicles. The model predictive controller (MPC) was used to control the front wheel angle, and the particle swarm optimization (PSO) algorithm was designed to optimize the MPC control parameters adaptively. The sliding mode controller controls the rear wheel angle, and the vehicle instability degree is judged by analyzing the β − β˙ phase plane. The controllers of different instability degrees are designed herein. Finally, a torque divider is designed to consider the actuation anti-slip. The designed controller is verified by Carsim and MATLAB-Simulink co-simulation. The results show that the trajectory tracking controller designed in this paper effectively improves the tracking accuracy under the premise of ensuring stability.

Intelligentization is the development trend of the future automobile industry. Intelligentization requires that the dynamic control of the vehicle can complete the trajectory tracking according to the trajectory output of the decision planning the driving state of the vehicle and ensure the driving safety and stability of the vehicle. However, trajectory limit planning and harsh road conditions caused by emergencies will increase the difficulty of trajectory tracking and stability control of unmanned vehicles. In view of the above problems, this paper studies the trajectory tracking and stability control of distributed drive unmanned vehicles. This paper applies a hierarchical control framework. Firstly, in the upper controller, an adaptive prediction time linear quadratic regulator (APT LQR) path following algorithm is proposed to acquire the desired front-wheel-steering angle considering the dynamic stability performance of the tires. The lateral stability of the DDAUV is determined based on the phase plane, and the sliding surface, in the improved sliding mode control (SMC), is further dynamically adjusted to obtain the desired additional yaw moment for coordinating the path following and lateral stability. Then, in the lower controller, considering the slip and the working load of four tires, a comprehensive cost function is established to reasonably distribute the driving torque of four in-wheel motors (IWMs) for producing the desired additional yaw moment. Finally, the proposed control algorithm is verified by the hardware-in-the-loop (HIL) experiment platform. The results show the path following and lateral stability can be coordinated effectively under different driving conditions.

This paper is dedicated to identifying stable equilibrium positions of the tether systems attached to the L1 or L2 libration points of the Mars–Phobos system. The orbiting spacecraft deploying the tether is at the L1 or L2 libration point and is held at one of these unstable points by the low thrust of its engines. In this paper, the analysis is performed assuming that the tether length is constant. The equation of motion for the system in the polar reference frame is obtained. The stable equilibrium positions are found and the dependence of the tether angular oscillation period on the tether length is determined. An analytical solution in the vicinity of the stable equilibrium positions for small angles of deflection of the tether from the local vertical is obtained in Jacobi elliptic functions. The comparison of the numerical and analytical solutions for small angles of deflection is performed. The results show that the dependencies of the oscillation period on the length of the tether are fundamentally different for L1 and L2 points. Analytical expressions for the tether tension are derived, and the influence of system parameters on this force is investigated for static and dynamic cases. [ABSTRACT FROM AUTHOR]

Ultradiscretization enables us to construct a piecewise linear equation which approximates a given subtraction-free difference equation. Recently proposed "ultradiscretization with parity variables" (pUD) can treat an equation with subtraction. However, its solution may have an infinite number of branches under some specific conditions. In this paper, solutions of pUD for the hard-spring equation is investigated. The pUD equation is reinterpreted as the mapping which maps a set on the phase plane to another one, and the behaviour of the solutions are analyzed through approximative transition diagrams.As a result, the infinite branches are translated into a few finite branches and allowed easier understanding. [ABSTRACT FROM AUTHOR]

We investigated the thermal buckling temperature and nonlinear free vibration of functionally graded fiber-reinforced composite laminated (FG-FRCL) beams. The governing nonlinear partial differential equations were derived from the Euler–Bernoulli beam theory, accounting for the von Kármán geometrical nonlinearity. Such equations were then reduced to a single equation by neglecting the axial inertia. Thus, the Galerkin method was applied to discretize the governing nonlinear partial differential equation in the form of a nonlinear ordinary differential equation, which was then solved analytically according to the He's variational method. Three different boundary conditions were selected, namely simply, clamped and clamped-free supports. We also investigated the effect of power-index, lay-ups, and uniform temperature rise on the nonlinear natural frequency, phase trajectory and thermal buckling of FG-FRCL beams. The results showed that FG-FRCL beams featured the highest fundamental frequency, whereas composite laminated beams were characterized by the lowest fundamental frequency. Such nonlinear frequencies increase for an increased power index and a decreased temperature. Finally, it was found that FG-FRCL beams with [0/0/0] lay-ups featured the highest nonlinear natural frequency and the highest thermal buckling temperature, followed by [0/90/0] and [90/0/90] lay-ups, while a [90/90/90] lay-up featured the lowest nonlinear natural frequency and critical buckling temperature. [ABSTRACT FROM AUTHOR]

To address the time-optimal trajectory planning (TOTP) problem with joint jerk constraints in a Cartesian coordinate system, we propose a time-optimal path-parameterization (TOPP) algorithm based on nonlinear optimization. The key insight of our approach is the presentation of a comprehensive and effective iterative optimization framework for solving the optimal control problem (OCP) formulation of the TOTP problem in the (s , s ˙) -phase plane. In particular, we identify two major difficulties: establishing TOPP in Cartesian space satisfying third-order constraints in joint space, and finding an efficient computational solution to TOPP, which includes nonlinear constraints. Experimental results demonstrate that the proposed method is an effective solution for time-optimal trajectory planning with joint jerk limits, and can be applied to a wide range of robotic systems. [ABSTRACT FROM AUTHOR]

Synaptic strengths between neurons are plastic and modified by spontaneous activity and information from the outside. There is increasing interest in the impact of correlated neuron activity and learning rules on global network structure. Here the networks of exponential integrate-and-fire neurons with spike timing-dependent plasticity (STDP) learning rules are considered, by providing the theoretical approximation of spiking cross-covariance between connected neurons and the theory for the evolution of synaptic weights. Background input mean and variance highly affect the spiking covariance, even for the fixed baseline firing rate and connection. Through analyzing the effects of covariance and STDP on vector fields for pairwise correlated neurons under fixed baseline firing rate, we show that the connections from a neuron with lower input mean to that with higher one will strengthen for balanced Hebbian STDP. However, this situation is reversed for Anti-Hebbian cases. Moreover, for potentiation dominated STDP, the synaptic weights for the networks of neurons with lower input mean are more likely to be enhanced. In addition, these properties found from coupled neurons also hold for large recurrent networks in both theories and simulations. This study provides a self-consistent theoretical method for understanding how correlated spiking activity and STDP shape the network structure and an approach for predicting structures of large networks through the analysis of simple neural circuits. [ABSTRACT FROM AUTHOR]

The Chen-Lee -Liu model has many applications in assorted fields, particularly in the study of nonlinear dynamics, chaos theory, circuit design, signal processing, secure communications, encryption and decryption of chaotic signals, as well as cryptography. The modified extended auxiliary equation mapping method has been applied to the Chen-Lee-Liu model in this article and explores new wave profiles, such as singular periodic solutions, periodic solutions, and kink-type soliton solutions. The complex wave conversion is considered to make a simple differential equation. Three- and two-dimensional images are plotted using Mathematica and MATLAB, and their dispersion and nonlinearity effects are discussed. We also discuss the bifurcation analysis of the studied model. The stability of the equilibrium points is studied, and the phase portrait of the system is presented graphically. The obtained wave profiles might play an important role in telecommunication systems, fiber optics, and nonlinear optics.

The rapid development of computer biometrics over the past 2-3 decades is largely due to the development and widespread introduction into clinical practice of new methods of studying human body health, including pulse methods. It is possible to judge changes in hemodynamic characteristics, heart rate, and blood flow rate in the studied part of the body based on the parameters of the pulse wave signal. At the same time, the physical processes of the formation of the pulse wave shape have not been fully studied, although the number of biophysical models of blood circulation is quite significant. The development of such a model will allow us to effectively apply modern developments in digital signal processing to the pulse wave and increase its diagnostic value. A qualitative model of pulse signal can be entrusted to the development of the base unit of the biotechnical system as a type of edge device. The work is devoted to the improvement of methods of rapid diagnosis of the cardiovascular system based on the analysis of model pulse-grams. An adequate mathematical model of the pulse wave, which corresponds to real pulse signals in different states of the human body and contains mathematical relationships between the main parameters of pulse-grams, has been refined. The algorithm of express diagnostics with the established criteria of the analysis of pulsograms is offered.

There are presented the results of analysis of some irregularities in the series of displacement coordinates on the Earth’s surface for the geographic North Pole in order to relate them temporally to some global jerks in the geomagnetic field. The calculations were made using the International Earth Rotation and Reference Systems Service (IERS) data on the average daily motions of the North Pole from 1962 to 2021 and the information on global jerks in the geomagnetic field measured by the magnetic observatories throughout the world. The identification and analysis of the irregularities in the pole displacement along the Earth’s surface were performed by the Fourier Method and wavelet methods for time series analysis, methods of fixed threshold determination and minimax analysis of the non-Gaussian noise, phase and pseudo-phase space methods, and stroboscopic methods of construction of Poincare maps.The analysis of local Fourier spectra and wavelet-spectra reveals irregularities in the displacement of the North Pole in the time intervals from 1967.04.09 to 1967.11.30, from 1974.03.29 to 1974.09.12, and from 2005.11.03 to 2006.03.07. It is supposed that energy reconstructions which caused short-term pole deviations from the trajectory and starting to follow its previous trajectory at the return points are related to the specific features of oscillatory interactions in the process of the translational-rotational motion of the Earth’s in the Solar System. The time of marking specific points on graphs of the displacement of the pole along the surface of the Earth in 1967 and 1974 is ahead of global geomagnetic jerk occurrence times.

The FitzHugh-Nagumo (FHN) spiking model has rich dynamics behaviors and can imitate the firing process of a neuron. The memristor is a nonvolatile and resistance tunable device, which gradually becomes a potential candidate for performing dynamic behaviors and implementing neuromorphic computation in the nervous system. In this paper, the memristive FitzHugh-Nagumo (MFHN) spiking model is presented. Firstly, we introduce a memristor to the FHN spiking model to build the MFHN spiking model and analyze the phase plane trajectory of the MFHN model. Secondly, we couple two MFHN models with a memristor and experimentally simulate the dynamic behaviors of two coupled neurons. The synchronous and asynchronous phenomena of two coupled neurons are discussed. Thirdly, the feasibility of the MFHN spiking model is demonstrated by the realization of binary logical operations and the implementation of binary adders. The comparison between the MFHN binary adder and the FHN binary adder is conducted. The simulation results illustrate that the proposed model efficiently performs rich, dynamic behaviors and higher firing frequency. The coupled MFHN models show the effectiveness of a memristor acting as an electric coupling synapse. The threshold logic computation can be completed efficiently by the MFHN spiking model. The MFHN binary adders reproduce the computing functions and behaviors of the biological neuron. [ABSTRACT FROM AUTHOR]

Basic Oxygen Furnace (BOF) steelmaking is an important way for steel production. Correctly recognizing different blowing periods and abnormal refining states is significant to ensure normal production process, while accurately predicting the end-point time helps to increase the first-time qualification rate of molten steel. Since the decarburization products CO and CO2 are the main compositions of off-gas, information of off-gas is explored for BOF steelmaking control. However, the problem is that most of the existing research directly gave the proportions of CO and CO2 as model input but barely considered the variation information of off-gas to describe the production state. At the same time, the off-gas information can be expected to recognize the last blowing period and predict the end-point time earlier than the existing methods that are based on sub-lance or furnace flame image, but little literature makes an attempt. Therefore, this work proposes a new method based on functional data analysis (FDA) and phase plane (PP), defined as FDA-PP, to describe and predict the BOF steelmaking process from the metallurgical dynamics viewpoint. This method extracts the total proportion of CO and CO2 and its first-order derivative as dynamics features of steelmaking process via FDA, which indicate the reaction velocity and acceleration of decarburization reaction, and describes the evolution of dynamics features via PP. Then, the FDA-PP method extracts the features of phase trajectories for production state recognition and end-point time prediction. Experiments on a real production dataset demonstrate that the FDA-PP method has higher production state recognition accuracy than the classical phase space, SVM, and BP methods, which is 87.78% for blowing periods of normal batches, 90.94% for splashing anomaly, and 81.29% for drying anomaly, respectively. At the same time, the FDA-PP method decreases the mean relative prediction error (MRE) of the end-point time prediction for abnormal batches by about 10% compared with the SVM and BP methods. [ABSTRACT FROM AUTHOR]

These lecture notes address the controllability under state constraints of reaction-diffusion equations arising in socio-biological contexts. We restrict our study to scalar equations with monostable and bistable nonlinearities.The uncontrolled models describing, for instance, population dynamics, concentrations of chemicals, temperatures, etc., intrinsically preserve pointwise bounds of the states that represent a proportion, volume-fraction, or density. This is guaranteed, in the absence of control, by the maximum or comparison principle.We focus on the classical controllability problem, in which one aims to drive the system to a final target, for instance, a steady-state. In this context the state is required to preserve, in the presence of controls, the pointwise bounds of the uncontrolled dynamics.The presence of constraints introduces significant added complexity for the control process. They may force the needed control-time to be large enough or even make some natural targets to be unreachable, due to the presence of barriers that the controlled trajectories might not be able to overcome.We develop and present a general strategy to analyze these problems. We show how the combination of the various intrinsic qualitative properties of the systems' dynamics and, in particular, the use of traveling waves and steady-states' paths, can be employed to build controls driving the system to the desired target.We also show how, depending on the value of the Allee parameter and on the size of the domain in which the process evolves, some natural targets might become unreachable. This is consistent with empirical observations in the context of endangered minoritized languages and species at risk of extinction.Further recent extensions are presented, and open problems are settled. All the discussions are complemented with numerical simulations to illustrate the main methods and results. [ABSTRACT FROM AUTHOR]

The article considers dynamical systems on the plane, defined by continuous piecewise smooth vector fields. Such systems are used as mathematical models of real processes with switching. An important task is to find the conditions for the generation of periodic trajectories when the parameters change. The paper describes the bifurcation of the birth of a periodic trajectory from the loop of the separatrix of a sewn saddle-node — an analogue of the classical bifurcation of the separatrix loop of a saddle-node of a smooth dynamical system. Consider a one-parameter family $\{ X_\varepsilon \} $ of continuous piecewise-smooth vector fields on the plane. Let $z^0 $ be a point on the switching line. Let's choose the local coordinates $x,y$ in which $z^0 $ has zero coordinates, and the switching line is given by the equation $y = 0$. Let the vector field $X_0 $ in a semi-neighborhood $y \ge 0$ ($y \le 0$) coincide with a smooth vector field $X_0^ + $ ($X_0^ - $), for which the point $z^0 $ is a stable rough node (rough saddle), and the proper subspaces of the matrix of the linear part of the field in $z^0 $ do not coincide with the straight line $y = 0$. The singular point $z^0 $ is called a sewn saddle-node. There is a single trajectory $L_0 $ that is $\alpha $-limit to $z^0 $ — the outgoing separatrix of the point $z^0 $. It is assumed that $L_0 $ is also $\omega $-limit to $z^0$, and enters $z^0 $ in the leading direction of the node of the field $X_0^ + $. For generic family, when the parameter $\varepsilon $ changes, the sewn saddle-node either splits into a rough node and a rough saddle, or disappears. In the paper it is proved that in the latter case the only periodic trajectory of the field $X_\varepsilon $ is generated from the contour $L_0 \cup \{ z^0 \} $ — a stable limit cycle.

A new methodology for constructing stability maps (phase-plane analysis) is presented and validated for application to complex multibody vehicle models implemented in Multibody Dynamics simulation software (Adams®). Traditional methodologies are developed to be applied to explicit mathematical models. Given the complexity of some special multibody systems, particularly in vehicle dynamics, simplifications are needed to apply this stability analysis technique. The main limitation when using simplified models is the need to neglect components which could have a significant influence on the dynamic behavior of the system and therefore on its stability. In the proposed methodology it is not necessary to have access to explicit mathematical models of multibody systems. Thus, the stability map of a vehicle model can be constructed by considering highly nonlinear dynamic elements, such as tires and silent-blocks components, modeled using the nonlinear finite element technique. [ABSTRACT FROM AUTHOR]

This paper is dedicated to identifying stable equilibrium positions of the tether systems attached to the L1 or L2 libration points of the Mars–Phobos system. The orbiting spacecraft deploying the tether is at the L1 or L2 libration point and is held at one of these unstable points by the low thrust of its engines. In this paper, the analysis is performed assuming that the tether length is constant. The equation of motion for the system in the polar reference frame is obtained. The stable equilibrium positions are found and the dependence of the tether angular oscillation period on the tether length is determined. An analytical solution in the vicinity of the stable equilibrium positions for small angles of deflection of the tether from the local vertical is obtained in Jacobi elliptic functions. The comparison of the numerical and analytical solutions for small angles of deflection is performed. The results show that the dependencies of the oscillation period on the length of the tether are fundamentally different for L1 and L2 points. Analytical expressions for the tether tension are derived, and the influence of system parameters on this force is investigated for static and dynamic cases.

In this paper, we investigated the resonant curves in the motion of the geo-synchronous satellite under the gravitational effect of the Sun, the Moon and the Earth. We have taken into consideration the effect of Earth's equatorial ellipticity and resistive force on the resonant curve. By defining the perturbations in (radial distance of the satellite from the center of the Earth) and (spin rate of Earth), we reduced equations of motion of satellite to a second order linear non-homogeneous differential equation in (radial deviation). With the help of different graphs, we have analyzed the effect of (rate of change of Earth's equatorial ellipticity parameter) and (angular velocity of the bary-center system around the Sun) on the oscillatory amplitudes. We observed that the oscillatory amplitudes decrease when the value of (Earth's equatorial ellipticity parameter) and (orbital angle of the bary-center system around the Sun) increase. Finally, we have studied the phase portrait, phase space and bifurcation theory using the method of Poincaré section. [ABSTRACT FROM AUTHOR]

We investigated the thermal buckling temperature and nonlinear free vibration of functionally graded fiber-reinforced composite laminated (FG-FRCL) beams. The governing nonlinear partial differential equations were derived from the Euler–Bernoulli beam theory, accounting for the von Kármán geometrical nonlinearity. Such equations were then reduced to a single equation by neglecting the axial inertia. Thus, the Galerkin method was applied to discretize the governing nonlinear partial differential equation in the form of a nonlinear ordinary differential equation, which was then solved analytically according to the He’s variational method. Three different boundary conditions were selected, namely simply, clamped and clamped-free supports. We also investigated the effect of power-index, lay-ups, and uniform temperature rise on the nonlinear natural frequency, phase trajectory and thermal buckling of FG-FRCL beams. The results showed that FG-FRCL beams featured the highest fundamental frequency, whereas composite laminated beams were characterized by the lowest fundamental frequency. Such nonlinear frequencies increase for an increased power index and a decreased temperature. Finally, it was found that FG-FRCL beams with [0/0/0] lay-ups featured the highest nonlinear natural frequency and the highest thermal buckling temperature, followed by [0/90/0] and [90/0/90] lay-ups, while a [90/90/90] lay-up featured the lowest nonlinear natural frequency and critical buckling temperature.

To address the time-optimal trajectory planning (TOTP) problem with joint jerk constraints in a Cartesian coordinate system, we propose a time-optimal path-parameterization (TOPP) algorithm based on nonlinear optimization. The key insight of our approach is the presentation of a comprehensive and effective iterative optimization framework for solving the optimal control problem (OCP) formulation of the TOTP problem in the (s,s˙)-phase plane. In particular, we identify two major difficulties: establishing TOPP in Cartesian space satisfying third-order constraints in joint space, and finding an efficient computational solution to TOPP, which includes nonlinear constraints. Experimental results demonstrate that the proposed method is an effective solution for time-optimal trajectory planning with joint jerk limits, and can be applied to a wide range of robotic systems.

This article summarizes the authors' work on mathematical modeling of the non-stationary stress-strain state (SSS) of shells under impact actions on their front surfaces and ends. Mathematical modeling is based on the use of an asymptotic method for constructing approximate theories for the components in different areas of the phase plane of the solution, where its fundamentally different properties take place and different values of the variability and dynamism indicators are observed. The areas where the two-dimensional bending component of the Kirchhoff-Love theory, the quasi-plane problem of the theory of elasticity, the hyperbolic boundary layer in the vicinity of the shear wave, and the elliptical boundary layer in the vicinity of the conditional front of Rayleigh surface waves are identified. The derivation of asymptotically optimal equations for an elliptic boundary layer, which takes place both in the case of tangential surface impact actions and in the case of normal type end impacts, is given. In this case, in the case of end action, the problem for a semi-infinite shell is reduced by extracting a particular solution to an equivalent problem for an infinite shell. As an example, solutions are constructed for an elliptical boundary layer in cylindrical shells for both loading cases. The integral Laplace transforms in time and Fourier transforms in the longitudinal coordinate are used. The inversion of integral transformations made it possible to present solutions using the elementary function arctg of complex arguments. An analysis of these solutions in a small neighborhood of the quasifront (conditional front of Rayleigh surface waves) made it possible to determine their properties as they move away from the quasifront along the longitudinal coordinate. [ABSTRACT FROM AUTHOR]

Nonlinear synaptic integration in dendrites is a fundamental aspect of neural computation. One such key mechanism is the Ca2+ spike at the apical tuft of pyramidal neurons. Characterized by a plateau potential sustained for tens of milliseconds, the Ca2+ spike amplifies excitatory input, facilitates somatic action potentials (APs), and promotes synaptic plasticity. Despite its essential role, the mechanisms regulating it are largely unknown. Using a compartmental model of a layer 5 pyramidal cell (L5PC), we explored the plateau and termination phases of the Ca2+ spike under input current perturbations, long-step current-injections, and variations in the dendritic high-voltage-activated Ca2+ conductance (that occur during cholinergic modulation). We found that, surprisingly, timed excitatory input can shorten the Ca2+ spike duration while inhibitory input can either elongate or terminate it. A significant elongation also occurs when the high-voltage-activated Ca2+ channels (CaHVA) conductance is increased. To mechanistically understand these phenomena, we analyzed the currents involved in the spike. The plateau and termination phases are almost exclusively controlled by the CaHVA inward current and the Im outward K+ current. We reduced the full model to a single-compartment model that faithfully preserved the responses of the Ca2+ spike to interventions and consisted of two dynamic variables: the membrane potential and the K+-channel activation level. A phase-plane analysis of the reduced model provides testable predictions for modulating the Ca2+ spike and reveals various dynamical regimes that explain the robust nature of the spike. Regulating the duration of the Ca2+ spike significantly impacts the cell synaptic-plasticity window and, as we show, its input-output relationship. [ABSTRACT FROM AUTHOR]

Vehicle driving stability is the basis of chassis integrated control and automatic driving. In this paper, we construct an 18 degrees of freedom (DOF) unified dynamics model of vehicle chassis. The unified dynamics model includes three subsystems: steering subsystem, brake subsystem and suspension subsystem, in which the coupling relationships among the three subsystems above are considered. Combined with phase plane analysis method, choosing the sideslip angle of vehicle center of gravity (c.g.) and yaw rate as the evaluation indexes, the influences of front wheel steering angle, road adhesion coefficient, braking mode of front and rear wheels and initial speed on vehicle driving stability are analyzed. In particular, phase plane and bifurcation analysis methods are used to study the influence of uneven road on vehicle driving stability, which has not been reported yet. The main contribution of this paper is to make a comprehensive analysis of the vehicle driving stability by the 18-DOF model above. Through the comprehensive simulation analysis of the above factors by Matlab/Simulink, we can fully grasp the driving stability law of the vehicle under different working conditions. With the above analysis method, the stability of various control systems designed for vehicle chassis can be evaluated more objectively. [ABSTRACT FROM AUTHOR]

Basic Oxygen Furnace (BOF) steelmaking is an important way for steel production. Correctly recognizing different blowing periods and abnormal refining states is significant to ensure normal production process, while accurately predicting the end-point time helps to increase the first-time qualification rate of molten steel. Since the decarburization products CO and CO2 are the main compositions of off-gas, information of off-gas is explored for BOF steelmaking control. However, the problem is that most of the existing research directly gave the proportions of CO and CO2 as model input but barely considered the variation information of off-gas to describe the production state. At the same time, the off-gas information can be expected to recognize the last blowing period and predict the end-point time earlier than the existing methods that are based on sub-lance or furnace flame image, but little literature makes an attempt. Therefore, this work proposes a new method based on functional data analysis (FDA) and phase plane (PP), defined as FDA-PP, to describe and predict the BOF steelmaking process from the metallurgical dynamics viewpoint. This method extracts the total proportion of CO and CO2 and its first-order derivative as dynamics features of steelmaking process via FDA, which indicate the reaction velocity and acceleration of decarburization reaction, and describes the evolution of dynamics features via PP. Then, the FDA-PP method extracts the features of phase trajectories for production state recognition and end-point time prediction. Experiments on a real production dataset demonstrate that the FDA-PP method has higher production state recognition accuracy than the classical phase space, SVM, and BP methods, which is 87.78% for blowing periods of normal batches, 90.94% for splashing anomaly, and 81.29% for drying anomaly, respectively. At the same time, the FDA-PP method decreases the mean relative prediction error (MRE) of the end-point time prediction for abnormal batches by about 10% compared with the SVM and BP methods.

We prove the existence of infinitely many sign changing radial solutions for a p-Laplacian Dirichlet problem in a ball. Our problem involves a weight function that is positive at the center of the unit ball and negative in its boundary. Standard initial value problems-phase plane analysis arguments do not apply here because solutions to the corresponding initial value problem may blow up near the boundary due to the fact that our weight function is negative at the boundary. We overcome this difficulty by connecting the solutions to a singular initial value problem with those of a regular initial value problem that vanishes at the boundary. [ABSTRACT FROM AUTHOR]

We consider the relationship between symmetries of two-dimensional autonomous dynamical systems in two common formulations; as a set of differential equations for the derivative of each state with respect to time, and a single differential equation in the phase plane representing the dynamics restricted to the state space of the system. Both representations can be analysed with respect to their symmetries, and we establish the correspondence between the set of infinitesimal generators of the respective formulations. We show that every generator of a symmetry of the autonomous system induces a well-defined vector field generating a symmetry in the phase plane and, conversely, that every symmetry generator in the phase plane can be lifted to a generator of a symmetry of the original system, which is unique up to constant translations in time. We exemplify the lift of symmetries in two cases; a mass conserved linear model and a non-linear oscillator. • Symmetries of two-dimensional ODEs and their phase plane realisations are related. • The relation between generating vector fields in the two domains is established. • Symmetries of the original ODE induce symmetries in the phase plane. • Symmetries in the phase plane can be lifted to symmetries of the original ODE. [ABSTRACT FROM AUTHOR]

This study develops a high-speed protection method for a modular multilevel converter (MMC)-based DC grids. The method utilizes the single-end derivative of common-mode (CM) and differential-mode (DM) currents to construct a protection phase-plane that divides different DC fault types into different regions. When the protection starts, the first sampling point that enters the fault regions of the protection phase-plane has the largest value. When five sampled data satisfy the corresponding fault conditions, the protection signal will be sent. This scheme ensures the rapidity of protection. CM and DM circuits of MMC, DC transmission line, and fault transition resistance under different fault types are derived following the principle of modulus decomposition. Finally, the synthesized equivalent circuits are obtained. On this basis, the expressions of CM and DM currents are derived to analyze the characteristics of fault currents on the protection phase-plane and provide the rules for distinguishing different fault types. A four-terminal MMC-based DC grid simulation model is established in PSCAD/EMTDC. The rapidness and reliability of the proposed protection method are then verified using different simulation cases. The proposed method is also validated on a scaled-down hardware setup in the laboratory. [ABSTRACT FROM AUTHOR]

Previous authors have proposed two basic hypotheses about the factors that form the basis of locomotor rhythms in walking insects: sensory feedback only or sensory feedback together with rhythmic activity of small neural circuits called central pattern generators (CPGs). Here we focus on the latter. Following this concept, to generate functional outputs, locomotor control must feature both rhythm generation by CPGs at the level of individual joints and coordination of their rhythmic activities, so that all muscles are activated in an appropriate pattern. This work provides an in-depth analysis of an aspect of this coordination process based on an existing network model of stick insect locomotion. Specifically, we consider how the control system for a single joint in the stick insect leg may produce rhythmic output when subjected to ascending sensory signals from other joints in the leg. In this work, the core rhythm generating CPG component of the joint under study is represented by a classical half-center oscillator constrained by a basic set of experimental observations. While the dynamical features of this CPG, including phase transitions by escape and release, are well understood, we provide novel insights about how these transition mechanisms yield entrainment to the incoming sensory signal, how entrainment can be lost under variation of signal strength and period or other perturbations, how entrainment can be restored by modulation of tonic top-down drive levels, and how these factors impact the duty cycle of the motor output. [ABSTRACT FROM AUTHOR]

A method for controlling a shale-shaker electrical drive system that enables trajectory changes of the shaker frame to be minimized is described. The control method is based on using a self-synchronization effect. The trajectories of the shaker frame are determined indirectly by the phase shift angles of the unbalanced rotation. The phase shift angle of unbalanced vibration exciter rotation was controlled by changing the effective phase stresses. The proposed control method could significantly expand the technological capabilities of drilling shale shakers. [ABSTRACT FROM AUTHOR]

Under consideration is the mathematical model of a clock frequency generator in which some high-frequency oscillations of a movable electrode are excited by a sequence of concentrated electrostatic pulses; wherein the times of pulse action are coordinated with the oscillations of the movable electrode by analogy with the well-known theory of a trigger clock. The results of studying the mathematical model provide a fairly complete understanding of the properties of the oscillations that are excited in the generator. [ABSTRACT FROM AUTHOR]