Unstable Equilibrium Point Research Articles

Quantitative Studies on Asymptotic Growth Behaviors of Trajectories of Nonlinear Discrete Dynamical Systems

This technical note studies quantitatively asymptotic growth behaviors of trajectories (AGBT) of nonlinear autonomous discrete dynamical system that has unbounded domain, non-Lipschitz continuous nonlinear operator, and stable or unstable equilibrium point. We explain how trajectory motion speed is quantitatively determined in the system, and study exact computation and sharp estimation of the smallest exponential bound of trajectories. We characterize exponential stability and asymptotic stability of the system from a new point of view, and provide a simple condition to distinguish them from each other. These results extend existing results that were obtained in some special cases of the system, and are helpful for quantitative analysis and understanding of AGBT of the system.

Dynamical simulation and load control of a Jeffcott rotor system in Herbst maneuvering flight

In this paper, the dynamical simulation of a nonlinear aircraft rotor system in Herbst maneuvering flight is carried out, though which the load control is discussed for safety purposes. The equations of motion of the system are set up by using Lagrange’s equation, in which, the effects of maneuvering flight are formulated by the flight parameters of the aircraft. By employing the method of closest unstable equilibrium point, the stability of periodic solutions for the primary resonance of the system in steady flight is analyzed to demonstrate the existence of double steady solutions in some rotation speed region near the critical speed and the possibility of the switch of them due to the effect of maneuvering flight. By introducing the Herbst flight parameters from the results of a relative control simulation, the dynamical simulation of the system is carried out, from which the switch of the smaller solution of amplitude to the larger solution is observed. Furthermore, the load control is designed by adding a gain λ to the Herbst flight parameters to avoid the switch. Through numerical experiments, it is found that λ = 0.679 is the critical condition of the switch. The results in this paper will provide a better understanding for the effect of aircraft maneuvering flight on the dynamics of the rotor system.

Specific non-monotonous interactions increase persistence of ecological networks

The relationship between stability and biodiversity has long been debated in ecology due to opposing empirical observations and theoretical predictions. Species interaction strength is often assumed to be monotonically related to population density, but the effects on stability of ecological networks of non-monotonous interactions that change signs have not been investigated previously. We demonstrate that for four kinds of non-monotonous interactions, shifting signs to negative or neutral interactions at high population density increases persistence (a measure of stability) of ecological networks, while for the other two kinds of non-monotonous interactions shifting signs to positive interactions at high population density decreases persistence of networks. Our results reveal a novel mechanism of network stabilization caused by specific non-monotonous interaction types through either increasing stable equilibrium points or reducing unstable equilibrium points (or both). These specific non-monotonous interactions may be important in maintaining stable and complex ecological networks, as well as other networks such as genes, neurons, the internet and human societies.

On traveling waves in viscous-capillary Euler equations with thermal conductivity

This work establishes the stability of the equilibrium states corresponding to traveling waves in viscous-capillary Euler equations when a standard thermal conductivity coefficient is present. Due to the presence of the heat conduction, the associated system of ordinary differential equations is much more involved and complicated. Given a shock wave, we can obtain exactly the sign of the real part of each eigenvalue of the Jacobian matrix of the corresponding system of ODEs at the two equilibria associated with the left-hand and right-hand states of the shock. It turns out that one equilibrium point is asymptotically stable, why the other point is unstable and admits eigenvalues whose real parts have opposite signs. Suitable approximate connections between the unstable and stable equilibria are obtained by numerical tests for various ranges of thermal conductivity: low, medium, and high values. Moreover, numerical tests also suggest that a trajectory could leave the unstable equilibrium point and enters the attraction domain of the asymptotically stable equilibrium point. This work therefore may motivate for further study on the existence of the traveling waves of the viscous-capillary gas dynamics equations with the presence of heat conduction.

Multi-agent-system-based decentralized coordinated control for large power systems

In this paper, a decentralized coordinated control based on multi-agent system (MAS) is proposed to improve the transient stability of large power system. First, a two-hierarchical MAS is structured, which consists of upper level agent for implementing coordinated control and lower level generator unit agents for implementing decentralized control. Then, the transient stability of large power systems is analyzed based on controlling unstable equilibrium point (CUEP) method. According to transient stability assessment indexes, the upper level agent judges whether it need be activated to send coordinated control command. Afterwards, decentralized controller that is implemented by every generator unit agent, and decentralized coordinated controller is executed by the two-hierarchical MAS, are respectively designed by using H∞ robust control method integrated linear matrix inequality (LMI) technique. Finally, the effectiveness of the proposed control scheme is tested by means of simulation study.

Control of multi-scroll Chen system

This paper investigates the chaos control problem of a new multi-scroll chaotic system. Two nonlinear control methods are studied, namely high-order and predictive types of control. The proposed methodologies offer the possibility of stabilizing unstable periodic orbits and unstable equilibrium points from the state equations. For this purpose, we apply the high-order control method for stabilizing a desired unstable periodic orbit, while the predictive control method is applied for stabilization problem of unstable equilibrium points. In particular, these approaches are effective and easy to be implemented since we only need to apply small perturbations to the system dynamics. The multi-scroll Chen system is used as representative example to show the working principle of these methods. Numerical simulation results indicate the potential engineering applications of the proposed control methods for various multi-scroll chaos-based practical applications.

Linear control of the flywheel inverted pendulum

The flywheel inverted pendulum is an underactuated mechanical system with a nonlinear model but admitting a linear approximation around the unstable equilibrium point in the upper position. Although underactuated systems usually require nonlinear controllers, the easy tuning and understanding of linear controllers make them more attractive for designers and final users. In a recent paper, a simple PID controller was proposed by the authors, leading to an internally unstable controlled plant. To achieve global stability, two options are developed here: first by introducing an internal stabilizing controller and second by replacing the PID controller by an observer-based state feedback control. Simulation and experimental results show the effectiveness of the design.

Control global del péndulo con rueda de reacción mediante regulación de energía y linealización extendida de las variables de estado

En este documento se presenta el diseño y la simulación de un controlador global para el péndulo invertido con rueda de reacción empleando métodos de regulación de energía y linealización extendida de las variables de estado. La regulación de energía propuesta se fundamenta en la variación gradual de la energía del sistema para alcanzar la posición vertical invertida. La señal de entrada requerida se obtiene a partir de la aplicación del teorema de estabilidad de Lyapunov. El control por realimentación extendida de las variables de estado se emplea para obtener una función no lineal suave que extienda la región de operación del sistema en un rango mayor, en contraste con la realimentación estática de las variables de estado que se obtiene mediante métodos de linealización aproximada alrededor de un punto de operación. El controlador diseñado opera a partir de la conmutación de las señales de control dependiendo de la región de operación y para verificar su robustez y eficiencia se aplican perturbaciones en la señal de control y en las variables medidas. Finalmente las simulaciones y pruebas realizadas sobre el modelo del sistema físico, permiten observar la versatilidad y funcionalidad del controlador propuesto en toda la región de operación del péndulo.

Homoclinic orbits in a two-patch predator-prey model with Preisach hysteresis operator

Systems of operator-differential equations with hysteresis operators can have unstable equilibrium points with an open basin of attraction. Such equilibria can have homoclinic orbits attached to them, and these orbits are robust. In this paper a population dynamics model with hysteretic response of the prey to variations of the predator is introduced. In this model the prey moves between two patches, and the derivative of the Preisach operator is used to describe the hysteretic flow between the patches. A numerical example of a robust homoclinic loop is presented, and a mechanism creating this homoclinic trajectory is discussed.

Hamiltonian Function for Satellite Motion in an Elliptic Orbit under The Influence of Aerodynamic Torque

This paper deals with the non-linear oscillation of a satellite in an elliptic orbit around the Earth under the influence of aerodynamic and gravitational torque. It is assumed that the orbital plane coincides with the equatorial plane of the Earth. Using Euler's dynamical equation of motion, the equation of motion and Hamiltonian function for the problem is derived. Taking aerodynamic torque of the order of eccentricity, it is observed that Hamiltonian function is dependent on aerodynamic torque. The equilibrium points and Double Asymptotic solutions related to the problem are calculated. It is concluded that in the phase space (0, 2) and (π, 2) are the equilibrium points, out of which (0, 2) is a stable and (π, 2) is an unstable equilibrium point.

Control of the Fractional-Order Chen Chaotic System via Fractional-Order Scalar Controller and Its Circuit Implementation

A fractional-order scalar controller which involves only one state variable is proposed. By this fractional-order scalar controller, the unstable equilibrium points in the fractional-order Chen chaotic system can be asymptotically stable. The present control strategy is theoretically rigorous. Some circuits are designed to realize these control schemes. The outputs of circuit agree with the results of theoretical results.

Mathematical Modelling and Simulation of <i>β</i>-Cell Mass, Insulin and Glucose Dynamics: Effect of Genetic Predisposition to Diabetes

Worldwide, diabetes is affecting 370 million people, causing nearly five million deaths and absorbing more than 471 billion USD per year. Mathematical models have been developed to simulate, analyse and understand the dynamics of β-cells, insulin and glucose. In this paper, we consider the effect of genetic predisposition to diabetes on dynamics of β-cells, glucose and insulin. We assume that the β-cell dynamics is governed by the differential equation: . The model indicates different behaviours according to the presence or absence of genetic predisposition. In presence of predisposition (ε = 1), the model shows three equilibrium points: a stable physiological equilibrium point (G = 100, I = 20, β = 600), a stable trivial pathological equilibrium point (G = 600, I = 0, β = 0) and a saddle point (G = 250, I = 9.8, β = 129.36). In absence of predisposition (ε = 0), the model has only two equilibrium points: an unstable pathological equilibrium point (G = 600, I = 0, β = 0) and a stable physiological equilibrium point (G = 82.6, I = 23, β = 900). In order to see how physical activity, obesity and other factors affect insulin sensitivity, simulations are carried out with different values of insulin induced glucose uptake rate (c), β-cell maximum insulin secretory rate (d) and environmental capacity (K).

Improved Counter-Current and Co-Current Guidance of Underactuated Marine Vehicles with Semiglobal Stability Properties

A control technique for counter-current and co-current guidance of underactuated marine vehicles is revisited and improved. The control system is based on a pure integral guidance law and two feedback controllers, in a cascaded configuration. The sway component of the ocean current in the body frame is viewed as the error signal of the guidance law instead of the absolute sway velocity, as done in the original approach. This makes the vessel search for the two possible yaw angles giving zero current component in sway: the counter-current direction and the co-current direction. The closed loop system has multiple equilibria where the two mentioned directions represent a set of stable equilibrium points and a set of unstable equilibrium points, respectively. Compared to the approach based on the absolute sway velocity, it is possible to achieve stronger stability properties. In particular, while the previous guidance approach considered a simplified model and yielded local exponential stability only, the complete model is analyzed in this paper and uniform semiglobal asymptotic stability as well as uniform local exponential stability are proven. The theoretical results are supported by simulations.

Hybrid Dislocated Control and General Hybrid Projective Dislocated Synchronization for Memristor Chaotic Oscillator System

Some important dynamical properties of the memristor chaotic oscillator system have been studied in the paper. A novel hybrid dislocated control method and a general hybrid projective dislocated synchronization scheme have been realized for memristor chaotic oscillator system. The paper firstly presents hybrid dislocated control method for stabilizing chaos to the unstable equilibrium point. Based on the Lyapunov stability theorem, general hybrid projective dislocated synchronization has been studied for the drive memristor chaotic oscillator system and the same response memristor chaotic oscillator system. For the different dimensions, the memristor chaotic oscillator system and the other chaotic system have realized general hybrid projective dislocated synchronization. Numerical simulations are given to show the effectiveness of these methods.

The interaction of a screw dislocation with a circular inhomogeneity near the free surface

The interaction of a screw dislocation with a circular inhomogeneity near the free surface is discussed in this paper. By using the complex potential and conformal mapping technique, an explicit series solution is obtained. Then, the solution is cast into a new expression to separate the interaction effects between the dislocation, inhomogeneity, and free surface. The new expression is not only convenient to reveal the coupling interaction effects, but also helpful to improve the convergence of the solution. As an application of the new expression, a simple approximate formula is presented with high accuracy. Finally, the full-field interaction energy and image force are evaluated and studied graphically. It is found that when the screw dislocation, inhomogeneity, and free surface are close to each other, their interaction effects strongly and intricately couple in the near field. In the case of a soft inhomogeneity or a hole, there is an unstable equilibrium point of the screw dislocation between the inhomogeneity and free surface, whereas in the case of a hard or rigid inhomogeneity, there is an unstable equilibrium point on the opposite side of the inhomogeneity.

Negative, zero and positive stiffness in extended Eshelby plates

Structures under loading can show positive and negative stiffness (during buckling) with zero stiffness just being one point of transition. Material structures with extended zero stiffness regimes have also been reported. Using the example of an extended Eshelby plate with a positive and negative edge dislocation, we show the existence of finite regimes of positive, zero and negative stiffness in a single system. Given that stable states/configurations of negative stiffness cannot exist in an isolated system; we show the existence of metastable states of negative stiffness along with unstable states of positive stiffness. The system also exhibits one point of unstable equilibrium.

Designing Nonlinear Feedback Control for Controlling Tumor Growth

Tumor growth models as nonlinear systems have some equilibrium points. In clinical situations, the desirable equilibrium point is the state that the competition between the populations of tumor cells and immune cells tend to the state in which the sizes of these populations do not vary (dormancy mode) or populations of tumor are zero (free tumor mode). In this study, a feedback nonlinear control for tumor growth model is designed so as to stabilize the unstable desirable equilibrium points based on Lyapunov stability theory. The aim is to steer the state trajectories to the desirable equilibrium point in order to treat cancer disease. The efficiency of the proposed method is shown in the simulation part.

An Application of Critical Trajectory Method to BCU Problem for Transient Stability Studies

This paper proposes a new method for obtaining controlling unstable equilibrium point (CUEP) for transient stability analysis in electric power systems. A minimization problem is formulated to attain CUEP by applying the critical trajectory (CTrj) method to the boundary controlling unstable equilibrium point (BCU) method. CTrj method is a unique approach proposed by the authors to obtain critical condition for transient stability problem. The Proposed method simultaneously computes a trajectory on the stability boundary starting from the exit point reaching CUEP. The effectiveness of the method is demonstrated through simulations for various systems.

Orbits and manifolds near the equilibrium points around a rotating asteroid

We study the orbits and manifolds near the equilibrium points of a rotating asteroid. The linearised equations of motion relative to the equilibrium points in the gravitational field of a rotating asteroid, the characteristic equation and the stable conditions of the equilibrium points are derived and discussed. First, a new metric is presented to link the orbit and the geodesic of the smooth manifold. Then, using the eigenvalues of the characteristic equation, the equilibrium points are classified into 8 cases. A theorem is presented and proved to describe the structure of the submanifold as well as the stable and unstable behaviours of a massless test particle near the equilibrium points. The linearly stable, the non-resonant unstable, and the resonant equilibrium points are discussed. There are three families of periodic orbits and four families of quasi-periodic orbits near the linearly stable equilibrium point. For the non-resonant unstable equilibrium points, there are four relevant cases; for the periodic orbit and the quasi-periodic orbit, the structures of the submanifold and the subspace near the equilibrium points are studied for each case. For the resonant equilibrium points, the dimension of the resonant manifold is greater than 4, and we find at least one family of periodic orbits near the resonant equilibrium points. As an application of the theory developed here, we study relevant orbits for the asteroids 216 Kleopatra, 1620 Geographos, 4769 Castalia and 6489 Golevka.

Control Fractional-Order Continuous Chaotic System via a Simple Fractional-Order Controller

A universal fractional-order controller is proposed to asymptotically stable the unstable equilibrium points and the nonequilibrium points of continuous fractional-order chaos systems. The simple fractional-order controller is obtained based on the stability theorem of nonlinear fractional-order systems. The control scheme is simple and theoretically rigorous. The numerical simulations demonstrate the validity and feasibility of the proposed fractional-order controller.