Equilibrium Manifold Research Articles

Catastrophes minimization on the equilibrium manifold

In a fixed total resources setting, we show that there exists a Riemannian metric g on the equilibrium manifold, which coincides with any (fixed) Riemannian metric with an economic meaning outside an arbitrarily small neighborhood of the set of critical equilibria, such that a minimal geodesic connecting two regular equilibria is arbitrarily close to a smooth path which minimizes catastrophes.

On the Stabilization of Nonholonomic Mechanical Systems via Immersion and Invariance

AbstractThis work presents some results on the application of the Immersion & Invariance (I&I) approach proposed for stabilization of nonlinear systems to the case of mechanical systems with kinematic constraints. We consider only smooth state–feedback control laws and thus, by Brockett's necessary condition, restrict our attention to deriving an I&I controller that achieves smooth stabilization of an equilibrium manifold. As is well–known, the design of I&I controllers requires the solution of some partial differential equations. A second contribution of our work is to obviate this obstacle, giving explicit control laws for a class of nonholonomic mechanical systems, which include the knife-edge and the rolling wheel. The approach is illustrated with the well-known knife-edge example.

Two-dimensional hyperbolic equilibrium manifold method

The computation of manifold plays an important role in understanding the dynamics of a nonlinear system. This paper presented a revised algorithm for computing two dimensional manifolds of vector fields. The trajectory was grown by solving appropriate initial value problem, then the manifold was grown step by step with equal arc-length along the trajectory and points on the trajectory were selected according to the local curvature. For the first time, both curvature-control technique and distance constraints were employed when interpolation between mesh points was needed. And in this way, the accuracy of interpolated points was guaranteed and it was the main advantage over the existing methods. The simulation results show that the proposed algorithm is effective.

Finding All Pure-Strategy Equilibria in Games with Continuous Strategies

Static and dynamic games are important tools for the analysis of strategic interactions among economic agents and have found many applications in economics. In many games equilibria can be described as solutions of polynomial equations. In this paper we describe state-of-the-art techniques for finding all solutions of polynomial systems of equations and illustrate these techniques by computing all equilibria of both static and dynamic games with continuous strategies. We compute the equilibrium manifold for a Bertrand pricing game in which the number of equilibria changes with the market size. Moreover, we apply these techniques to two stochastic dynamic games of industry competition and check for equilibrium uniqueness.

A Trajectory Planning Method for Output Tracking of Linear Flexible Systems Using Exact Equilibrium Manifolds

This paper describes a new trajectory planning method for the output tracking control of linear flexible systems; this method computes the exact solution of the equilibrium manifold. We establish a system of differential equations by combining kinematic and dynamic constraints and reformulate them in a singularly perturbed dynamics to obtain the equilibrium manifold in the form of an infinite series, which is our planned trajectory. We show that for the desired output defined by exponential functions, the equilibrium manifold becomes a converging geometric series that has a succinct form of summation. In addition, the inverse torque, which is the feedforward command, is easily produced by using the computed exact equilibrium manifold. We validate the effectiveness of the proposed method through simulations and experimental studies using a single-link flexible arm.

Curvature as a measure of thermodynamic interaction

We present a systematic and consistent construction of geometrothermodynamics by using Riemannian contact geometry for the phase manifold and harmonic maps for the equilibrium manifold. We present several metrics for the phase manifold that are invariant with respect to Legendre transformations and induce thermodynamic metrics on the equilibrium manifold. We review all the known examples in which the curvature of the thermodynamic metrics can be used as a measure of the thermodynamic interaction.

Modeling and Dynamic Exploration of a Tilt-Rotor VTOL Aircraft

In this paper we introduce and study a novel model of a Tilt-Rotor VTOL aircraft. The aircraft is structured as a blended wing body equipped with a tilting rotor (the propulsion unit) that gives the vehicle the capability of vertical take off and transition to a forward flight configuration. This model captures the main features of novel tilt-rotor aircraft architectures with more challenging control and maneuvering capabilities. We introduce a complex nine degrees of freedom model of the aircraft to explore the dynamics and the maneuvering capabilities of the vehicle. We perform an analysis of the equilibrium manifold of the aircraft (namely a parametrized family of trimming trajectories) and, as main contribution of the paper, we introduce a set of optimal control based strategies to explore the trajectory manifold of the vehicle in order to generate non-stationary and highly aggressive trajectories. In particular, we provide numerical computations showing how to generate a trajectory for transitions from near hover to forward flight.

Preface to the Special Issue on Computational Economics

Economics is the study of how scarce resources are allocated. Operations research studies how to accomplish goals in the least costly manner. These fields have much to offer each other in terms of challenging problems that need to be solved and the techniques to solve them. This was the case after World War II, partly because the individuals who went on to be the leading scholars in economics and operations research worked together during WWII. In fact, the two fields share many early luminaries, including Arrow, Dantzig, Holt, Kantorovich, Koopmans, Modigliani, Scarf, and von Neumann. Unfortunately, as new generations of scholars took charge, those ties weakened, resulting in little interaction between operations research scholars and economists in the past three decades. In recent years, some economists and applied mathematicians have worked to reestablish ties between these fields. This special issue of Operations Research is an important step in that effort. It is not meant to be just a collection of current research that is of interest to both economists and operations researchers. It is an example of what we hope is a future in which we will not need “special issues” to describe computational economics. The papers in this issue cover a broad range of material, including the introduction of advanced mathematical techniques to computational economics, algorithm development and analysis, examples where advanced computational methods are used to solve important economic problems, and examples of where computational ideas are used to motivate concepts of bounded rationality. “Tackling Multiplicity of Equilibria with Grobner Bases,” by Kubler and Schmedders, shows how algebraic geometry methods can be used to address the problem of multiplicity of equilibria. In the past 30 years, the field of algebraic geometry has been an example of where the mathematical theory has been spurred on by the exploding power of computers. The combination of theory, algorithms, and computer power now makes it possible to apply these methods to economic problems. Another important task of computational theory is to determine how hard problems are. “Sharing Supermodular Costs,” by Schulz and Uhan, looks at the complexity of computing the least core value in cooperative games with supermodular cost structures. Such games arise in a variety of operations research problems—scheduling, for example. The authors show that such problems are strongly NP hard, but for a special case of single-machine scheduling the Shapley value is in the least core. “Multigrid Techniques in Economics,” by Speight, is a good example of how advances in computational physics, in this case computational fluid dynamics, can be applied to economic problems. Multigrid methods are a relatively new approach to solving partial differential equations, and this paper explains how they can be applied to dynamic economic problems. Mechanism design has been an active field in game theory, but there has been relatively little work on developing computational methods that can be used to implement the theory in concrete ways. This issue contains two papers that improve our ability to apply mechanism design theory: “Multidimensional Mechanism Design: Finite-Dimensional Approximations and Efficient Computation” by Belloni, Lopomo, and Wang, and “Endogenous Selection and Moral Hazard in Compensation Contracts” by Armstrong, Larcker, and Su. Developing methods for computing Nash equilibria of games has been a staple of computational economics, and this issue contains three contributions. “On a Markov Game with One-Sided Information,” by Horner, Rosenberg, Solan, and Vieille, tackles the difficult computational problems presented by asymmetric information. In “A User’s Guide to Solving Dynamic Stochastic Games Using the Homotopy Method,” Borkovsky, Doraszelski, and Kryukov apply homotopy methods to explore the equilibrium manifold of stochastic games, illustrating a tool that could be used for many types of games. “Intertemporal Pricing and Consumer Stockpiling,” by Su, develops a solution method for the game between a seller and the consumers of durable goods. The key step is to decouple each buyer’s problem into a separate dynamic program.

Regular Infinite Economies

The main contribution of this paper is to place smooth infinite economies in the setting of the equilibrium manifold and the natural projection map à la Balasko. We show that smooth infinite economies have an equilibrium set that has the structure of a Banach manifold and that the natural projection map is smooth. We define regular and critical economies, and regular and critical prices, and we show that the set of regular economies coincides with the set of economies whose excess demand function has only regular prices. Generic determinacy of equilibria follows as a by-product.

Phase transitions in geometrothermodynamics

Using the formalism of geometrothermodynamics, we investigate the geometric properties of the equilibrium manifold for diverse thermodynamic systems. Starting from Legendre invariant metrics of the phase manifold, we derive thermodynamic metrics for the equilibrium manifold whose curvature becomes singular at those points where phase transitions of first and second order occur. We conclude that the thermodynamic curvature of the equilibrium manifold, as defined in geometrothermodynamics, can be used as a measure of thermodynamic interaction in diverse systems with two and three thermodynamic degrees of freedom.

A note on the structural stability of the equilibrium manifold

In a smooth pure exchange economy with fixed total resources we investigate whether the smooth selection property holds when endowments are redistributed across consumers through a continuous (non-local) redistribution policy. We show that if the policy is regular then there exists a unique continuous path of equilibrium prices which support it.

WITHDRAWN: Book Review: The Equilibrium Manifold by Yves Balasko

WITHDRAWN: Book Review: The Equilibrium Manifold by Yves Balasko

Computation of invariant manifolds with self-adaptive parameter and trajectories continuation method

Most work on manifold study focuses on two-dimensional manifolds and there have been proposed some good computing methods. However, the computation of two-dimensional manifold is still a hot research field. In this paper the two-dimensional manifold of hyperbolic equilibria for vector fields is computed by combining self-adaptive parameter with trajectories continuation, approximating the local manifold with an ellipse around the equilibria, extending the trajectory with equal distance, and adjusting the trajectory with self-adaptive parameter. This method is more accurate than the "trajectories and arc-length method", and better shows the trend of the manifolds than the "box covering method".

Mathematical modeling and simulation of seated stability

Various methods have been used to quantify the kinematic variability or stability of the human spine. However, each of these methods evaluates dynamic behavior within the stable region of state space. In contrast, our goal was to determine the extent of the stable region. A 2D mathematical model was developed for a human sitting on an unstable seat apparatus (i.e., the “wobble chair”). Forward dynamic simulations were used to compute trajectories based on the initial state. From these trajectories, a scalar field of trajectory divergence was calculated, specifically a finite time Lyapunov exponent (FTLE) field. Theoretically, ridges of local maxima within this field are expected to partition the state space into regions of qualitatively different behavior. We found that ridges formed at the boundary between regions of stability and failure (i.e., falling). The location of the basin of stability found using the FTLE field matched well with the basin of stability determined by an alternative method. In addition, an equilibrium manifold was found, which describes a set of equilibrium configurations that act as a low dimensional attractor in the controlled system. These simulations are a first step in developing a method to locate state space boundaries for torso stability. Identifying these boundaries may provide a framework for assessing factors that contribute to health risks associated with spinal injury and poor balance recovery (e.g., age, fatigue, load/weight, and distribution). Furthermore, an approach is presented that can be adapted to find state space boundaries in other biomechanical applications.

Asymptotic Stabilization of the Inverted Equilibrium Manifold of the 3-D Pendulum Using Non-Smooth Feedback

The 3-D pendulum consists of a rigid body, supported at a fixed pivot, with three rotational degrees of freedom; it is acted on by gravity and it is fully actuated by control forces. In , almost global stabilization of the inverted equilibrium manifold was studied using a smooth globally defined feedback. Here, we study the problem of almost global stabilization of the inverted equilibrium manifold using non-smooth feedback of angular velocity and a reduced attitude vector of the 3-D pendulum. The importance of the non-smooth feedback is that the almost global domain of attraction is a geometrically simple set that excludes the hanging attitude manifold. Unlike the closed-loop for a 3-D pendulum with a smooth controller, the closed-loop designed in this paper does not exhibit a performance constraint. These new results are based on Lyapunov analysis of the non-smooth closed-loop 3-D pendulum.

General equilibrium without utility functions: how far to go?

How far can we go in weakening the assumptions of the general equilibrium model? Existence of equilibrium, structural stability and finiteness of equilibria of regular economies, genericity of regular economies and an index formula for the equilibria of regular economies have been known not to require transitivity and completeness of consumers’ preferences. We show in this paper that if consumers’ non-ordered preferences satisfy a mild version of convexity already considered in the literature, then the following properties are also satisfied: (1) the smooth manifold structure and the diffeomorphism of the equilibrium manifold with a Euclidean space; (2) the diffeomorphism of the set of no-trade equilibria with a Euclidean space; (3) the openness and genericity of the set of regular equilibria as a subset of the equilibrium manifold; (4) for small trade vectors, the uniqueness, regularity and stability of equilibrium for two version of tatonnement; (5) the pathconnectedness of the sets of stable equilibria.

General Equilibrium Without Utility Functions: How Far to Go?

How far can we go in weakening the assumptions of the general equilibrium model? Existence of equilibrium, structural stability and finiteness of equilibria of regular economies, genericity of regular economies and an index formula for the equilibria of regular economies have been known not to require transitivity and completeness of consumers’ preferences. We show in this paper that if consumers’ non-ordered preferences satisfy a mild version of convexity already considered in the literature, then the following properties are also satisfied: 1) the smooth manifold structure and the diffeomorphism of the equilibrium manifold with a Euclidean space; 2) the diffeomorphism of the set of no-trade equilibria with a Euclidean space; 3) the openness and genericity of the set of regular equilibria as a subset of the equilibrium manifold; 4) for small trade vectors, the uniqueness, regularity and stability of equilibrium for two version of tatonnement; 5) the pathconnectedness of the sets of stable equilibria.

Evolution under the multilocus Levene model without epistasis

Evolution under the multilocus Levene model is investigated. The linkage map is arbitrary, but epistasis is absent. The geometric-mean fitness, w ̃ ( ρ ) , depends only on the vector of gene frequencies, ρ ; it is nondecreasing, and the single-generation change is zero only on the set, Λ , of gametic frequencies at gene-frequency equilibrium. The internal gene-frequency equilibria are the stationary points of w ̃ ( ρ ) . If the equilibrium points ρ ˆ of ρ ( t ) (where t denotes time in generations) are isolated, as is generic, then ρ ( t ) converges as t → ∞ to some ρ ˆ . Generically, ρ ( t ) converges to a local maximum of w ̃ ( ρ ) . Write the vector of gametic frequencies, p , as ( ρ , d ) T , where d represents the vector of linkage disequilibria. If ρ ˆ is a local maximum of w ̃ ( ρ ) , then the equilibrium point ( ρ ˆ , 0 ) T is asymptotically stable. If either there are only two loci or there is no dominance, then d ( t ) → 0 globally as t → ∞ . In the second case, w ̃ ( ρ ) has a unique maximum ρ ˆ and ( ρ ˆ , 0 ) T is globally asymptotically stable. If underdominance and overdominance are excluded, and if at each locus, the degree of dominance is deme independent for every pair of alleles, then the following results also hold. There exists exactly one stable gene-frequency equilibrium (point or manifold), and it is globally attracting. If an internal gene-frequency equilibrium exists, it is globally asymptotically stable. On Λ , (i) the number of demes, Γ , is a generic upper bound on the number of alleles present per locus; and (ii) if every locus is diallelic, generically at most Γ − 1 loci can segregate. Finally, if migration and selection are completely arbitrary except that the latter is uniform (i.e., deme independent), then every uniform selection equilibrium is a migration-selection equilibrium and generically has the same stability as under pure selection.

Evolution paths on the equilibrium manifold

In a pure exchange smooth economy with fixed total resources, we define the length between two regular equilibria belonging to the equilibrium manifold as the number of intersection points of the evolution path connecting them with the set of critical equilibria. We show that there exists a minimal path according to this definition of length.

Output Feedback Control of Single-Phase UPQC Based on a Novel Model

The single-phase unified power-quality conditioner can be treated as a typical switched linear system due to its typical switching dynamic characteristics. However, because the switched model is difficult for controller design, we first transform it to an equivalent discrete system model (EDSM) in light of the pulsewidth-modulation principle. The EDSM is further transformed to a linear equivalent discrete system model (LEDSM) by states reconstruction and linearization at some neighborhood of the equilibrium manifold. The LEDSM, where the pulsewidths are adopted as the controller variables, is a periodical discrete linear system with the constrained control. Since some states cannot be available, the output feedback periodical-switched controller is designed to meet suboptimal performance and stabilize the closed-loop system. Compared with the proportional-integral controller, the output feedback control strategy features better characteristics. The voltage reference for the series inverter is obtained by using an adaptive voltage detection method. Simultaneously based on the detected phase of the system voltage, the instantaneous reactive power theory is adopted to calculate the reference current for the shunt inverter. Additionally, the power flow and power consumption are described and analyzed. Simulation results demonstrate the effectiveness and reliability of the proposed control strategy.