Estimating the domain of attraction of a stable equilibrium state for certain delayed differential systems of order two

Meccanica ◽  
1976 ◽  
Vol 11 (1) ◽  
pp. 3-10
Author(s):  
Wiktor Baran ◽  
Tullio Trombetti
Keyword(s):  
Equilibrium State ◽  
Stable Equilibrium ◽  
Domain Of Attraction ◽  
Stable Equilibrium State ◽  
Differential Systems

scholarly journals Entropy Generation Rate in a Chemically Reacting System

1993 ◽  
Vol 115 (3) ◽  
pp. 208-212 ◽  
Author(s):  
E. P. Gyftopoulos ◽  
G. P. Beretta
Keyword(s):  
Equilibrium State ◽  
Entropy Generation ◽  
Chemical Potential ◽  
Stable Equilibrium ◽  
Entropy Generation Rate ◽  
Stable Equilibrium State ◽  
Microscopic Reversibility ◽  
Chemical Reaction Mechanisms ◽  
Chemically Reacting ◽  
Surrogate System

For a nonchemical-equilibrium state of an isolated system A that has r constituents with initial amounts na = {n1a, n2a, …, nra}, and that is subject to τ chemical reaction mechanisms, temperature, pressure, and chemical potentials cannot be defined. As time evolves, the values of the amounts of constitutents vary according to the stoichiometric relations ni(t) = nia + Σj=1τ νi(j) εj(t), where νi(j) is the stoichiometric coefficient of the ith constituent in the j-reaction mechanism and εj(t) the reaction coordinate of the jth reaction at time t. For such a state, we approximate the values of all the properties at time t with the corresponding properties of the stable equilibrium state of a surrogate system B consisting of the same constituents as A with amounts equal to ni(t) for i = 1, 2, …, r, but experiencing no chemical reactions. Under this approximation, the rate of entropy generation is given by the expression S˙irr = ε˙ · Y, where ε˙ is the row vector of the τ rates of change of the reaction coordinates, ε˙ = { ε˙1, …, ε˙τ }, Y the column vector of the τ ratios aj/Toff for j = 1, 2, …, τ, aj = −Σi=1r νi(j) μi,off, that is, the jth affinity of the stable equilibrium state of the surrogate system B, and μi,off, and Toff are the chemical potential of the ith constituent and the temperature of the stable equilibrium state of the surrogate system. Under the same approximation, by further assuming that ε˙ can be represented as a function of Y only that is, ε˙(Y), with ε˙(0) = 0 for chemical equilibrium, we show that ε˙ = L·Y + (higher order terms in Y), where L is a τ × τ matrix that must be non-negative definite and symmetric, that is, such that the matrix elements Lij satisfy the Onsager reciprocal relations, Lij = Lji. It is noteworthy that, for the first time, the Onsager relations are proven without reference to microscopic reversibility. In our view, if a process is irreversible, microscopic reversibility does not exist.

scholarly journals Finite-dimensional maps describing the dynamics of a logistic equation with delay

2019 ◽  
Vol 487 (6) ◽  
pp. 611-616
Author(s):  
S. D. Glyzin ◽  
S. A. Kashchenko
Keyword(s):  
Numerical Simulation ◽  
Equilibrium State ◽  
Stable Equilibrium ◽  
Dynamic Properties ◽  
Logistic Equation ◽  
Mathematical Ecology ◽  
Stable Equilibrium State ◽  
Finite Dimensional ◽  
Equation With Delay ◽  
Dynamics Of Populations

This article discusses a family of maps that are used in the numerical simulation of a logistic equation with delay. This equation and presented maps are widely used in problems of mathematical ecology as models of the dynamics of populations. The paper compares the dynamic properties of the trajectories of these mappings and the original equation with delay. It is shown that the behavior of the solutions of maps can be quite complicated, while the logistic equation with delay has only a stable equilibrium state or cycle.

Thermodynamic Definition and Quantum-Theoretic Pictorial Illustration of Entropy

1998 ◽  
Vol 120 (2) ◽  
pp. 154-160 ◽  
Author(s):  
E. P. Gyftopoulos
Keyword(s):  
Equilibrium State ◽  
Thermodynamic Equilibrium ◽  
Stable Equilibrium ◽  
Variable Temperature ◽  
Cyclic Change ◽  
Stable Equilibrium State ◽  
Change Of State ◽  
Laws Of Thermodynamics ◽  
External Forces ◽  
Work Done

Cannot analyzed an engine operating between two reservoirs. Through a peculiar mode of reasoning, he found the correct optimum shaft work done during a cyclic change of state of the engine. Clausius justified Carnot’s result by enunciating two laws of thermodynamics, and introducing the concept of entropy as a ratio of heat and temperature of a thermodynamic equilibrium state. In this paper, we accomplish five purposes: (i) We consider a Carnot engine. By appropriate algebraic manipulations we express Carnot’s optimum shaft work in terms of available energies or exergies of the end states of one reservoir with respect to the other, and Clausius’ entropy S in terms of the energies and available energies of the same and states. (ii) We consider the optimum shaft work done during a cyclic change of state of an engine operating between a reservoir, and a system with fixed amounts of constituents and fixed volume, but variable temperature. We express the optimum shaft work in terms of the available energies of the end states of the system, and Clausius’ entropy in terms of the energies and available energies of the same end states. Formally, the entropy expression is identical to that found for the Carnot engine, except that here the change of state of the system is not isothermal. (iii) We consider the optimum shaft work done during a cyclic change of state of a general engine operating between a reservoir R and system A which initially is in any state A1, stable or thermodynamic equilibrium or not stable equilibrium. In state A1, the values of the amounts of constituents are n1, and the value of the volume is V1 whereas, in the final state A0, n0 ≠ n1 and V0 ≠ V1 Using the laws of thermodynamics presented by Gyftopoulos and Beretta, we prove that such an optimum exists, call it generalized available energy with respect to R, and use it together with the energy to define a new property Σ1 We note that the expression for Σ is formally identical to and satisfies the same criteria as Clausius’ entropy S. The only difference is that Σ applies to all states, whereas Clausius’ S applies only to stable equilibrium states. So we call Σ entropy and denote it by S (iv) We use the unified quantum theory of mechanics and thermodynamics developed by Hatsopoulos and Gyftopoulos, and find a quantum theoretic expression for S in terms of the density operator ρ that yields all the probabilities associated with measurement results. (v) We note that the quantumtheoritic expression for S can be interpreted as a measure of the shape of an atom, molecule, or other system because ρ can be though of as such a shape, and provide pictorial illustrations of this interpretation. For given values of energy E, amounts of constituents n, and volume V, the value of the measure is zero for all shapes that correspond to projectors (wave functions), positive for density operators that are not projectors, and the largest for the ρ that corresponds to the unique stable equilibrium state determined by the given E, n, and V. Accordingly, spontaneous entropy generation occurs as a system adapts its shape to conform to the internal and external forces. Beginning with an arbitrary initial ρ this adaptation continues only until no further spontaneous change of shape can occur, that is, only until a stable equilibrium state is reached.

scholarly journals A Bi-Stable von-Mises Truss for Morphing Applications Actuated Using Shape Memory Alloys

2013 ◽  
Author(s):  
Silvestro Barbarino ◽  
Farhan S. Gandhi ◽  
Rodolphe Visdeloup
Keyword(s):  
Equilibrium State ◽  
Shape Memory ◽  
Stable Equilibrium ◽  
The Other ◽  
Stable Equilibrium State ◽  
Actuation Force ◽  
Numerical Predictions ◽  
System Equilibrium ◽  
Governing Differential Equation ◽  
Von Mises

The present study focuses on a bi-stable von-Mises truss (VMT), with integrated Shape Memory Alloy (SMA) wires which are resistively heated to provide the actuation force to transition the VMT from one stable equilibrium condition to the other, and back. This coupled VMT-SMA system can potentially be used in structural morphing applications. The paper considers in detail the design of the system, equilibrium between the VMT and the SMA wires, the initial pre-stress required in the two SMA wires, explains how the active (heated) SMA wire drives the VMT beyond the unstable equilibrium state, and the VMT in moving to the second stable equilibrium state pre-stresses the passive (unheated) SMA wire. The two SMA wires switch roles in moving the VMT back from the second to the first stable equilibrium state. A prototype is designed and fabricated and the transition of the VMT from one equilibrium state to the other, and back, is experimentally demonstrated. The governing differential equation representing the VMT behavior is coupled with equations representing the SMA behavior based on the Brinson thermo-mechanical model. The numerical predictions of system displacements versus temperature and time show good correlation with experimental results.

Stable Equilibrium State Analysis of the Differential Ring Oscillator with Even Number of Stages

2011 ◽  
Vol 33 (8) ◽  
pp. 1969-1974 ◽  
Author(s):  
Hui Zhang ◽  
Hai-gang Yang ◽  
Fa-biao Zhou ◽  
Fei Liu ◽  
Tong-qiang Gao
Keyword(s):  
Equilibrium State ◽  
Stable Equilibrium ◽  
Ring Oscillator ◽  
Stable Equilibrium State ◽  
Differential Ring ◽  
State Analysis

scholarly journals Haematopoietic stem cells: entropic landscapes of differentiation

2018 ◽  
Vol 8 (6) ◽  
pp. 20180040 ◽  
Author(s):  
K. Wiesner ◽  
J. Teles ◽  
M. Hartnor ◽  
C. Peterson
Keyword(s):  
Gene Expression ◽  
Equilibrium State ◽  
Shannon Entropy ◽  
Stable Equilibrium ◽  
Single Cells ◽  
Entropy Measure ◽  
Stable Equilibrium State ◽  
Time Resolved ◽  
Entry Points ◽  
Time Point

The metaphor of a potential epigenetic differentiation landscape broadly suggests that during differentiation a stem cell approaches a stable equilibrium state from a higher free energy towards a stable equilibrium state which represents the final cell type. It has been conjectured that there is an analogy to the concept of entropy in statistical mechanics. In this context, in the undifferentiated state, the entropy would be large since fewer constraints exist on the gene expression programmes of the cell. As differentiation progresses, gene expression programmes become more and more constrained and thus the entropy would be expected to decrease. In order to assess these predictions, we compute the Shannon entropy for time-resolved single-cell gene expression data in two different experimental set-ups of haematopoietic differentiation. We find that the behaviour of this entropy measure is in contrast to these predictions. In particular, we find that the Shannon entropy is not a decreasing function of developmental pseudo-time but instead it increases towards the time point of commitment before decreasing again. This behaviour is consistent with an increase in gene expression disorder observed in populations sampled at the time point of commitment. Single cells in these populations exhibit different combinations of regulator activity that suggest the presence of multiple configurations of a potential differentiation network as a result of multiple entry points into the committed state.

scholarly journals Research on evolution of collaboration mechanism of stakeholders in mining enterprises’ implementation of social responsibility

2020 ◽  
Author(s):  
Xingxin Nie ◽  
Zhang Jingjing ◽  
Yang Luhan
Keyword(s):  
Local Government ◽  
Social Responsibility ◽  
Equilibrium State ◽  
Local Community ◽  
Stable Equilibrium ◽  
Early Stage ◽  
Evolutionary Game ◽  
Stable Equilibrium State ◽  
Multiple Variables ◽  
The Relationship

Multiple stakeholders are involved when mining enterprises assume their social responsibility, so a tripartite game pattern among local government, mining enterprises and local community is formed after supervision is introduced. By combining evolutionary game model with system dynamics for simulation analysis, this paper studies the dynamic game process among the three parties and we find out that: (1) local government, mining enterprises and local community will eventually reach a stable equilibrium state (supervision, participation,  performance), and it will simplify the evolutionary path of the stable equilibrium state when local government chooses supervision at the early stage; (2) local government plays a leading role in supervision, and the strategy choices of mining enterprises and local community depend on local government's strategy and the relationship between the size of exogenous; (3) the formation of the eventual stabilization strategy is decided by the joint action of multiple variables, although every single variable will affect the strategy choice of all parties, so local government should deal with the relationship between various stakeholders properly.

Entropy: Part II — Thermodynamics and Perfect Order

1999 ◽  
Author(s):  
Elias P. Gyftopoulos
Keyword(s):  
Equilibrium State ◽  
Thermodynamic Equilibrium ◽  
Statistical Approach ◽  
Stable Equilibrium ◽  
Particle Systems ◽  
Second Law ◽  
Statistical Interpretation ◽  
Stable Equilibrium State ◽  
Perfect Order ◽  
Thermodynamic Equilibrium State

Abstract Part II of this two-part paper refutes the beliefs about the statistical interpretation of thermodynamics, and the association of entropy with disorder that are summarized in Part I. The refutation of the statistical approach is based on either a nonstatistical unified quantum theory of mechanics and thermodynamics, or an almost equivalent, novel, nonquantal exposition of thermodynamics. Entropy is shown to be: (i) valid for any system (both macroscopic and microscopic, including one-particle systems), and any state (both thermodynamic or stable equilibrium, and not stable equilibrium); (ii) a measure of the quantum-theoretic pliable shape of the molecules of a system; and (iii) a monotonic indicator of order. In contrast to statistics which associates a thermodynamic equilibrium macrostate with the largest number of compatible microstates, the second law avers that, for each set of values of energy, volume, and amounts of constituents of either a macroscopic or a microscopic system, there exists one and only one thermodynamic or stable equilibrium state. So, even if Boltzmann’s definition were used, a thermodynamic equilibrium state is one of perfect order.

On the Existence of Multiple Equilibrium States in Strained-Layer Superlattices

1987 ◽  
Vol 103 ◽  
Author(s):  
William C. Johnson
Keyword(s):  
Free Energy ◽  
Equilibrium State ◽  
Stable Equilibrium ◽  
Multiple Equilibrium ◽  
Equilibrium Thermodynamic ◽  
Relative Thickness ◽  
Two Phase ◽  
Strained Layer ◽  
Strained Layer Superlattice ◽  
Strained Layer Superlattices

ABSTRACTUsing recent results from the thermodynamics of stressed solids, two-phase coexistence in a simple binary strained-layer superlattice is examined. We show that for a given temperature and overall composition of the superlattice, there can exist more than one linearly stable, equilibrium thermodynamic state. That is, there may exist several combinations of relative thickness of the phases and corresponding phase compositions that minimize the free energy of the system. The equilibrium state observed experimentally can, therefore, be influenced by the processing path.

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