Existence Of Multiple Equilibrium Points Research Articles

Event-triggered impulse control on reaction–diffusion Gilpin–Ayala competition model with multiple stationary solutions

In previous literature, the parameter restrictions for the reaction–diffusion Gilpin–Ayala competition ecosystem were limited to the control of the first eigenvalue of the Laplace operator. This paper considers extending the parameter to the control of the second eigenvalue. However, exceeding the first eigenvalue results in a loss of compactness in the energy functional of the determined ecosystem. Consequently, this paper discusses the orthogonal decomposition of a given Sobolev space and variational techniques, obtaining the existence of multiple equilibrium points. The presence of multiple equilibrium points is a common phenomenon in real-world ecosystems and implies that the system cannot be globally stable under normal circumstances. However, this study develops an effective event-triggered impulse mechanism (ETIM) that achieves global stability for the positive equilibrium solution in the ecosystem. Notably, the newly designed ETIM not only eliminates Zeno behaviour but also reduces the cost of impulse control. This is achieved by removing some unnecessary impulses in the fixed impulsive instants mechanism associated with the ecosystem in previous literature. Finally, a numerical simulation validates the effectiveness of the proposed method.

Short-Circuit Analysis of AC Distribution Systems Dominated by Voltage Source Converters Considering Converter Limitations

This paper deals with the short-circuit analysis of distribution systems dominated by Voltage Source Converters (VSCs). A novel methodology has been proposed in this paper for short-circuit calculation of power systems populated with VSCs where the control modes and potential current-saturated operation of power converters are considered. The studied distribution system has been modeled using an element-based steady-state formulation that includes the converter equations in both normal operation and fault scenarios. Then, equilibrium points of the studied distribution system are identified by solving the established systems of equations for different potential current-saturation states of the VSCs. The methodology identifies possible solutions involving different saturation states of different converters. The proposed methodology is applied to a specific case study of a distribution system penetrated with VSCs in both grid-forming and grid-following control. The short-circuit analysis results indicate that VSCs operation shows a significant impact on the short-circuit equilibrium points of the studied system. In addition, multiple equilibrium points may exist depending on whether the grid-forming converter is saturated or not. The existence of multiple equilibrium points is validated through dynamic simulations.

Online pricing for demand‐side management in a low‐voltage resistive micro‐grid via a Stackelberg game with incentive strategies

It has been demonstrated that online pricing mechanisms are a viable solution for demand side management in power systems. This study deals with the analysis and design of a droop-controlled low-voltage resistive AC micro-grid network system. Such a system is subjected to a dynamic demand obtained from an online pricing mechanism, which is proposed as a novelty in the study of micro-grids. This mechanism is derived from a variation of the Stackelberg game, which includes the use of incentive strategies. First, a configuration in which a supplier announces an incentive function and n-consumers’ reaction to the resulting personalised price is presented. Then, a detailed stability analysis of the micro-grid is presented as a result of the interaction with the aforementioned online pricing mechanism. The units of the micro-grid (generators as the supplier and loads as the consumers) operate under either conventional or bounded droop control. The novelty of the approach is that it bridges the gap between the physical and the market layers of the problem. The ways in which the existence of multiple equilibrium points is guaranteed for both the consumer's load and the supplier's announced incentive are shown. A detailed design process for the profit functions of the players is shown in conjunction with the parameter selection for their implementation. Finally, simulations that demonstrate the system stability and its convergence to different equilibria are implemented under scenarios with one and multiple consumers.

Mutiple [formula omitted]-type stability of fractional-order quaternion-valued neural networks

The multiple ψ−type stability of fractional-order quaternion-valued neural networks (FQVNNs) was investigated in this paper. Some new conditions ensuring the existence of multiple equilibrium points of the considered FQVNNs are provided. Meanwhile, the ψ−type stability for the proposed neural networks is studied by employing the fractional calculus theory and fractional derivative techniques into the system dynamics. Finally, an numerical simulation is given to show the effectiveness of the theoretical results.

Distance + Angle-Based Control of 2-D Rigid Formations.

A well-known problem with distance-based formation control is the existence of multiple equilibrium points not associated with the desired formation. This problem can be potentially mitigated by introducing an additional controlled variable. In this article, we generalize the distance + angle-based scheme for 2-D formations of single-integrator agents by using directed graphs and triangulation of the n-agent formation. We show that under certain conditions on the control gains and desired formation shape, our controller ensures the asymptotic stability of the correct formation for almost all initial agent positions.

Switching formation shape control with distance + area/angle feedback

A well known problem with distance-based formation control is the existence of multiple equilibrium points not associated with the desired formation. This problem can be mitigated by introducing an additional controlled variable. In this paper, we consider distance + area (or angle) schemes for 2D formations of single-integrator agents. By using directed graphs and triangulation of the n-agent formation, we introduce switching control laws that ensure the asymptotic stability of the desired formation for almost all initial agent positions. The state-dependent switching strategy is designed to force the formation to escape the subspaces of the incorrect equilibria. The switching strategy also allows us to remove restrictions on the shape of the desired formation that were present in previous results.

Existence of multiple equilibrium points in global attractor for damped wave equation

This paper is a continuation of Meng and Zhong in (Discrete Contin. Dyn. Syst., Ser. B 19:217–230, 2014). We go on studying the property of the global attractor for some damped wave equation with critical exponent. The difference between this paper and Meng and Zhong in (Discrete Contin. Dyn. Syst., Ser. B 19:217–230, 2014) is that the origin is not a local minimum point but rather a saddle point of the Lyapunov function F for the symmetric dynamical systems. Using the abstract result established in Zhang et al. in (Nonlinear Anal., Real World Appl. 36:44–55, 2017), we prove the existence of multiple equilibrium points in the global attractor for some wave equations under some suitable assumptions in the case that the origin is an unstable equilibrium point.

Multiple equilibrium points in global attractors for some p-Laplacian equations

In this paper, we continue to study the properties of the global attractor for some p-Laplacian equations with a Lyapunov function F in a Banach space when the origin is no longer a local minimum point but a saddle point of F. By using the abstract result established in our previous work, we prove the existence of multiple equilibrium points in the global attractor for some p-Laplacian equations under some suitable assumptions in the case that the origin is an unstable equilibrium point.

Game Theoretic Decentralized Feedback Controls in Markov Jump Processes

This paper studies a decentralized routing problem over a network, using the paradigm of mean-field games with large number of players. Building on a state-space extension technique, we turn the problem into an optimal control one for each single player. The main contribution is an explicit expression of the optimal decentralized control which guarantees the convergence both to local and to global equilibrium points. Furthermore, we study the stability of the system also in the presence of a delay which we model using an hysteresis operator. As a result of the hysteresis, we prove existence of multiple equilibrium points and analyze convergence conditions. The stability of the system is illustrated via numerical studies.

The existence of multiple equilibrium points in global attractors for some symmetric dynamical systems II

This paper is a continuation of Zhong et al. (2014), we go on discussing some properties of the global attractor for some symmetric dynamical system with a Lyapunov function F in a Banach space. The difference between this paper and Zhong et al. (2014) is that the origin is not a local minimum point but a saddle point of F. Under some suitable assumptions, we establish an abstract result about the existence of the multiple equilibrium points in the global attractor by estimating the lower bound of Z2-index of the global attractor. As an application of this abstract result, we consider the existence of multiple stationary solutions for some semilinear reaction–diffusion equation when the origin is an unstable equilibrium point.

The existence of multiple equilibrium points in a global attractor for some p-Laplacian equation

In this paper, we are mainly concerned with some properties of the global attractor for some p-Laplacian equation with a Lyapunov function in a Banach space. Under some suitable assumptions, we prove the existence of multiple equilibrium points in a global attractor for some p-Laplacian equation.

The existence of multiple equilibrium points in a global attractor for some symmetric dynamical systems

In this paper, we are mainly concerned with some properties of the global attractor for some symmetric dynamical systems with a Lyapunov function in a Banach space. Under some suitable assumptions, we give an estimate of lower bound of Z2 index of the global attractor and get the existence of the multiple equilibrium points in the global attractor for the symmetric dynamical systems. As an application of these results, we consider the existence of multiple stationary solutions for some semilinear reaction–diffusion equations.

Dynamical stability analysis of multiple equilibrium points in time-varying delayed recurrent neural networks with discontinuous activation functions

This paper is concerned with the dynamical stability analysis of multiple equilibrium points in recurrent neural networks with time-varying delays and discontinuous activation functions. Based on the decomposition of state space, some sufficient conditions for the existence of multiple equilibrium points are established, which ensure that n-dimensional recurrent neural networks with k-level discontinuous activation functions can have kn equilibrium points. Under these conditions, the equilibrium points are locally exponentially stable. Moreover, some conditions for the existence of sets of stable equilibrium points and unstable equilibrium points are derived for recurrent neural networks without delay and with discontinuous activation functions. Finally, three examples are given to illustrate the effectiveness of the results.

Multiple equilibrium points and their significance in the second breakdown of bipolar transistors

Suggests a simple one-dimensional model for the electrical and thermal behavior of a bipolar transistor. The proposed model can be embedded in an external circuit and analyzed by an appropriate circuit analysis program. Existence of multiple equilibrium points for certain biasing conditions is demonstrated and their implication upon the transient behavior of a device is discussed. The triggering mechanism for second breakdown in the case of forward base current, as well as reverse base current, is described in the light of these multiple equilibrium points. Prediction of that part of the safe operating area boundary limited by second breakdown is reported. The results compare favorably with the manufacturer's recommended limit.

Pressures toward Uniformity in Groups: A Dynamic Structural Equation Model

In this paper a dynamic structural equation model consisting of a set of difference equations has been constructed in order to examine the process by which groups generate pressures toward uniformity. The model is based on Festinger's hypotheses about communication resulting from such pressures. Parameters of the model have been estimated from a consideration of one of the equilibrium conditions of the model. Hypothetical time paths that describe the dynamic behaviour of the system have been generated by a series of computer simulation runs. These graphs describe the process by which group cohesiveness and discrepancy in opinions among group members adjust to each other over time. The model has lead to a number of major insights into this dynamic process. The results of the simulation indicate the existence of multiple equilibrium points for the system and that the relevance of the issue for the group's functioning is a critical factor in determining the rate at which a new equilibrium is reached once the initial equilibrium is disturbed. Groups can and do attempt to influence members in order to achieve uniformity of opinions, beliefs, values and behaviour (Asch, 1951; Deutsch and Gerard, 1955; Scott 1965). Cartwrght and Zander (1968) in reviewing the theoretical explanations of why groups attempt to achieve uniformity point out four functions that such pressure serves; namely goal attainment (Festinger, 1950), group maintenance (Stock, Whitman and Lieberman, 1958), the development of social reality through consensus (Festinger, 1954; Schachter, 1959), and achievement of consensus concerning group relations with its social surroundings (Burns, 1955; Cohen, 1955; Yinger, 1950). As a result discrepancies in opinions, beliefs, values and behaviour, give rise to pressures on members to communicate with one another. Festinger (1950) in his investigations of informal social communications has enumerated a set of hypotheses concerning such communication that results from group pressures to achieve uniformity. Several of his hypotheses are concerned with members of the group as a whole and form the basis of one model by Simon and Guetzkow (1955) involving a set of differential equations. In the following section of this paper Festinger's verbal theory will be recast as a causal model (Blalock, 1969). A dynamic structural equation model will be developed consisting of a set of difference equations. This structural model will form the basis of a computer simulation model which will be validated with data from studies by Festinger, Schachter and Back (1950), Festinger and Thibaut (1951) and Back (1951).