Markus–Yamabe Conjecture for Asymptotic Stability of Planar Piecewise Linear Systems

We present in this paper a detailed study on the Markus–Yamabe conjecture in planar piecewise linear systems. We consider discontinuous piecewise linear systems with two zones separated by a straight line, in which every subsystem is asymptotically stable. We prove the existence of limit cycles under explicit parameter conditions and give more different counterexamples to the Markus-Yamabe conjecture in addition to the counterexamples given by Llibre and Menezes. In particular, we consider continuous planar piecewise linear systems. For such a system with n + 1 zones separated by n parallel straight lines in phase space, we prove that if each of subsystems is asymptotically stable, then this system has a globally asymptotically stable equilibrium point, therefore the Markus–Yamabe conjecture still holds. Some examples are given to illustrate the main results.Mathematics Subject Classification (2020) 34C05 · 34C07 · 37G15

Stability analysis of two predators and one prey population model with harvesting in fisheries management

The discussion is focussed in the interaction between two predators and one prey population model in fishery management. Mathematically model is built by involving harvesting with constant efforts in the two predators and one prey populations. The positive equilibrium point of the model is analyzed via linearization and Routh-Hurwitz stability criteria. From the analysis, there exists a certain condition that makes the positive equilibrium point is asymptotically stable. The stable equilibrium point is then related to the maximum profit problem. With suitable value of harvesting efforts, the maximum profit is reached and the predator and prey populations remain stable. Finally, a numerical simulation is carried out to find out how much the maximum profit is obtained and to visualize how the trajectories of predator and prey tend to the stable equilibrium point.

Research on the Pollutant Emission Reduction Strategy and Simulation of Paper-Making Enterprises under the Reward and Punishment Mechanism

Environmental pollution has become an important obstacle on the path of ecological civilization construction, and it is urgent to control environmental pollution. By establishing an evolutionary game model, this thesis focuses on analyzing how paper-making enterprises choose their own emission reduction strategies under the reward and punishment mechanism. It further analyzes how social welfare changes under the reward and punishment mechanism, and finally through simulation research, this thesis analyzes the evolutionary paths of paper-making enterprises’ pollution emission strategies under the reward and punishment mechanism. The results of the reward and punishment mechanism are as follows: under the static reward and punishment mechanism, the game system will repeatedly oscillate around a point. There is no stable equilibrium point at this time. However, under the dynamic reward and punishment mechanism, the game system will tend to a stable equilibrium point. The results of social welfare analysis show that high-intensity rewards will reduce the amount of pollution discharged by paper-making enterprises, thereby maximizing social welfare. On the contrary, when paper-making enterprises discharge a large amount of pollution, they will be subject to high-intensity penalties. When facing high-intensity punishments, paper-making enterprises will tend to not to discharge. So social welfare is also maximized. The simulation research results show that reasonable punishment strategies are more effective than reward ones. Based on this, the author proposes countermeasures, such as establishing a reasonable reward and punishment mechanism, reasonably determining the reward and punishment intensity for polluting enterprises. The emission reduction strategies of paper-making enterprises will be affected by the government’s reward and punishment mechanism. A deep study of its internal mechanism is not only of great significance for pollution control but also of great significance for the development of a green economy.

Global Stabilization of a Single-Species Ecosystem with Markovian Jumping under Neumann Boundary Value via Laplacian Semigroup

By applying impulsive control, this work investigated the global stabilization of a single-species ecosystem with Markovian jumping, a time delay and a Neumann boundary condition. Variational methods, a fixed-point theorem, and Laplacian semigroup theory were employed to derive the unique existence of the global stable equilibrium point, which is a positive number. Numerical examples illuminate the feasibility of the proposed methods.

The Dynamic Game of Knowledge Hiding Behavior from Organizational Members: To Hide or Not to Hide?

In the era of the knowledge economy, it is urgent for organizations to solve the problem of knowledge hiding of internal members to accelerate the speed and efficiency of knowledge dissemination and innovation and adapt to rapid changes in the market. At present, research on knowledge hiding has received extensive attention from Western countries, but there are few relevant studies in China. Based on the hypothesis of bounded rationality, this paper constructs an evolutionary game model of second-level knowledge hiding of organizational members and analyzes the main factors affecting the stable equilibrium point using MATLAB numerical simulation. The results show that knowledge leakage risk is positively correlated with knowledge hiding. The ability of knowledge absorption and transformation is positively correlated with the behavior of knowledge hiding. There is a negative correlation between collaborative innovation ability and knowledge hiding. There is a negative correlation between knowledge stock and knowledge hiding. Only when the incentive reaches a certain level can organizational members be encouraged to give up knowledge hiding. This paper provides a more comprehensive and dynamic picture of the evolutionary game of knowledge hiding among members in the organization and provides a new idea of knowledge management for organizational managers.

Research on the Potential of Front Wheel Steering Control for Vehicle Dynamics Control

The development of active steering control technology not only provides key actuators for intelligent vehicle motion control, but also expands vehicle stability and safety. This paper studies the potential control ability of the front-wheel steering control to the vehicle plane dynamics, and the controllable area boundary is designed on the phase plane of side slip angle and yaw rate. Previous studies have defined a dynamics stable area on the vehicle states phase plane, in which the vehicle state can autonomously return to a stable equilibrium point. The area outside the stable area are divided into the controllable area and the uncontrollable area in this paper. In the controllable area, the front-wheel steering control has the ability to pull the vehicle states back towards the stable area. Considering actuator constraints and model errors, based on the principle of safety design, a band-shaped critical area is designed to separate the controllable area from the uncontrollable area, and the linear mathematical model of the controllable area boundary is designed. In order to verify the rationality of the controllable area definition, nonlinear model predictive controller is designed to control the vehicle outside the dynamics stable area. The controller uses the high-fidelity nonlinear vehicle model and the magic formula tire model as the state equation constraints, and the practical steering actuator constraints are used as the control input constraints, and the nonlinear numerical optimization solver is used to solve the optimal steering input sequence. The phase plane analysis of the controlled vehicle verifies the rationality of the controllable area defined in this paper.

Robotic Assembly for Irregular Shaped Peg-in-Hole with Partial Constraints

The tight tolerance peg-in-hole process brings great challenges for robotic assembly. Force control-based methods have been proposed to generate complex compliant behavior to deal with the shape and clearance variances. However, existing solutions are based on the assumption that the peg and hole parts are fixed during the assembly process and can absorb the contact force completely. For this purpose, customized fixtures have to be designed and utilized, which greatly affect the system’s deployment cost, time, and flexibility. Considering the fact that in an assembly, the parts are naturally related to each other, this paper studies the irregular-shaped peg-in-hole assembly problem with partial constraints. Firstly, geometric and force model are developed for the natural constraints between the parts; by analyzing the behavior of the partial constraint, a control policy is proposed to compensate the position errors and drive the parts to a stable equilibrium point; For the irregular-shaped parts, a multiple-stage searching method is developed to efficiently search for the real hole location; Finally, a switching force/position hybrid controller is designed to coordinate the alignment, searching and insertion processes. The method is implemented in a real platform. The experiment results verify the effectiveness of the proposed methods.

Equilibrium Resolution Mechanism for Multidimensional Conflicts in Farmland Expropriation Based on a Multistage Van Damme’s Model

Multidimensional conflicts in farmland expropriation originate from the game of multidimensional interests between the local government and farmers. The strategy choices and equilibrium results of the two sides have evolved with changes to the situation and policy adjustments. Focusing on different types of farmland expropriation conflicts, this paper constructs a multistage Van Damme’s model of multidimensional conflicts in farmland expropriation, analyzes the stable equilibrium point of the behavior evolution of the local government and farmers under litigation settlement and nonlitigation settlement, and conducts simulation analysis on the behavior evolution and conflict resolution of both sides at different stages through MATLAB numerical simulation. The results show that (1) the interests’ game between the local government and farmers has changed periodically due to the evolution of the farmland expropriation system; (2) under litigation settlement, there is only the “government rent-seeking” conflict: in order to resolve the conflict, the cost of litigation for farmers can be reduced, while other policy interventions, such as controlling the rent-seeking ceiling of the local government and increasing the rent-seeking costs of the local government, can be implemented; (3) under nonlitigation settlement, there are three types of conflicts: to resolve the “government rent-seeking” conflict, we should control the rent-seeking ceiling of the local government and increase the rent-seeking costs of the local government or its positive social externality benefits under reasonable expropriation; to resolve the “nail household dilemma” conflict, we should increase the rent-seeking costs of farmers or their positive social externality benefits under reasonable compensation; to resolve the “extreme controversy” conflict, on the one hand, we should control the rent-seeking ceiling of farmers, and on the other hand, while controlling the rent-seeking ceiling of the local government, we should increase the farmers’ positive social externality benefits under reasonable compensation or negative social externality losses of both sides under rent seeking.

Practical finite time vibration suppression of mechatronic systems using proportional–integral–derivative variable structure controls with dead-band nonlinearity

Vibration is an intrinsic phenomenon in many mechanical and mechatronic applied devices and undesirable vibration can either degrade the performance of the system or lead to unpredictable outputs. The main purpose of this article is to introduce a novel second-order proportional–integral–derivative sliding mode control methodology to suppress the undesirable vibrations of a class of applied dynamical systems with applications to mechatronic and mechanical devices. After designing a nonlinear proportional–integral–derivative terminal sliding manifold, rigorous mathematics are provided to guarantee that the origin is a practical finite time stable equilibrium point. Consequently, two efficient control laws are proposed to ensure the occurrence of the sliding motion with and/or without system unknown parameters. Motivated by situations encountered in practice, unknown lumped uncertainties are also added to the system and their impacts are tackled using adaptive control techniques. Furthermore, a hard nonlinear dead-band function is utilized in the control input and its effects such as lags and delays appeared on the control signals as well as on the system outputs are dealt with by the proposed proportional–integral–derivative variable structure controller. The proposed second-order variable structure controller not only utilizes the simple effective design approach of the proportional–integral–derivative controllers to ensure a reasonable transient performance, but also displays fast convergence properties demonstrated in non-singular terminal sliding modes. Finally, through simulation studies, it is confirmed that the proposed control strategy is effective in vibration attenuation of microelectromechanical resonators.

Dynamics of a Bistable Current-Controlled Locally-Active Memristor

This paper presents a novel current-controlled locally-active memristor model to reveal the switching and oscillating characteristics of locally-active devices. It is shown that the memristor has two asymptotically stable equilibrium points on its power-off plot and therefore exhibits nonvolatility. Switching from one stable equilibrium point to another is achieved by applying a suitable current pulse. The locally-active characteristic of the memristor is measured by the DC [Formula: see text]–[Formula: see text] plot. A small-signal equivalent circuit on a locally-active operating point with the bias current [Formula: see text] is constructed for describing the characteristic of the locally-active region of the memristor. A periodic oscillator circuit composed of the locally-active memristor, a compensation inductor and a resistor is proposed, whose dynamics is analyzed in detail by using the Hopf bifurcation and the zeros and poles of the impendence function of the circuit. It is found that the locally-active memristor based circuit with different current biases or different initial conditions can exhibit different dynamics such as periodic oscillation and stable equilibrium point. If an energy storage element (capacitor) is added to the periodic oscillation circuit, a chaotic oscillator is obtained, which can exhibit abundant dynamics. The oscillation mechanism of the memristor-based oscillator is analyzed via dynamic route map (DRM), showing that the memristor is an essential device for generating periodic and chaotic oscillations, and its local activity is the cause for complex oscillations.