Academic literature on the topic 'Functional equilibrium equations'
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Journal articles on the topic "Functional equilibrium equations"
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Murakami, Satoru. "Stable equilibrium point of some diffusive functional differential equations." Nonlinear Analysis: Theory, Methods & Applications 25, no. 9-10 (November 1995): 1037–43. http://dx.doi.org/10.1016/0362-546x(95)00097-f.
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Tian, Xiaohong, and Rui Xu. "Global dynamics of a predator-prey system with Holling type II functional response." Nonlinear Analysis: Modelling and Control 16, no. 2 (April 25, 2011): 242–53. http://dx.doi.org/10.15388/na.16.2.14109.
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In this paper, a predator-prey system with Holling type II functional response and stage structure is investigated. By analyzing the corresponding characteristic equations, the local stability of each of feasible equilibria of the system is studied. The existence of the orbitally asymptotically stable periodic solution is established. By using suitable Lyapunov functions and the LaSalle invariance principle, it is proven that the predator-extinction equilibrium is globally asymptotically stable when the coexistence equilibrium is not feasible, and sufficient conditions are derived for the global stability of the coexistence equilibrium.
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Lotfi, El Mehdi, Mehdi Maziane, Khalid Hattaf, and Noura Yousfi. "Partial Differential Equations of an Epidemic Model with Spatial Diffusion." International Journal of Partial Differential Equations 2014 (February 10, 2014): 1–6. http://dx.doi.org/10.1155/2014/186437.
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The aim of this paper is to study the dynamics of a reaction-diffusion SIR epidemic model with specific nonlinear incidence rate. The global existence, positivity, and boundedness of solutions for a reaction-diffusion system with homogeneous Neumann boundary conditions are proved. The local stability of the disease-free equilibrium and endemic equilibrium is obtained via characteristic equations. By means of Lyapunov functional, the global stability of both equilibria is investigated. More precisely, our results show that the disease-free equilibrium is globally asymptotically stable if the basic reproduction number is less than or equal to unity, which leads to the eradication of disease from population. When the basic reproduction number is greater than unity, then disease-free equilibrium becomes unstable and the endemic equilibrium is globally asymptotically stable; in this case the disease persists in the population. Numerical simulations are presented to illustrate our theoretical results.
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Meng, Xin-You, and Jiao-Guo Wang. "Analysis of a delayed diffusive model with Beddington–DeAngelis functional response." International Journal of Biomathematics 12, no. 04 (May 2019): 1950047. http://dx.doi.org/10.1142/s1793524519500475.
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In this paper, a delayed diffusive phytoplankton-zooplankton model with Beddington–DeAngelis functional response and toxins is investigated. Existence of equilibria of the system are solved. The global asymptotic stability of the zooplankton-free equilibrium is obtained. The local stability of the coexistent equilibrium and existence of Hopf bifurcation are discussed. In addition, the properties of the Hopf bifurcation are studied based on the center manifold and normal form theory for partial differential equations. Finally, some numerical simulations are also carried out to confirm our theoretical analysis.
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BENKHALTI, R., and K. EZZINBI. "A HARTMAN-GROBMAN THEOREM FOR SOME PARTIAL FUNCTIONAL DIFFERENTIAL EQUATIONS." International Journal of Bifurcation and Chaos 10, no. 05 (May 2000): 1165–69. http://dx.doi.org/10.1142/s0218127400000839.
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We show that the flow of some partial functional differential equations has a global attractor. As a conseqsuence we prove that the flow near a hyperbolic equilibrium is equivalent to its variational equation.
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Arora, Vivek K., and George J. Boer. "Simulating Competition and Coexistence between Plant Functional Types in a Dynamic Vegetation Model." Earth Interactions 10, no. 10 (May 1, 2006): 1–30. http://dx.doi.org/10.1175/ei170.1.
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Abstract The global distribution of vegetation is broadly determined by climate, and where bioclimatic parameters are favorable for several plant functional types (PFTs), by the competition between them. Most current dynamic global vegetation models (DGVMs) do not, however, explicitly simulate inter-PFT competition and instead determine the existence and fractional coverage of PFTs based on quasi-equilibrium climate–vegetation relationships. When competition is explicitly simulated, versions of Lotka–Volterra (LV) equations developed in the context of interaction between animal species are almost always used. These equations may, however, exhibit unrealistic behavior in some cases and do not, for example, allow the coexistence of different PFTs in equilibrium situations. Coexistence may, however, be obtained by introducing features and mechanisms such as temporal environmental variation and disturbance, among others. A generalized version of the competition equations is proposed that includes the LV equations as a special case, which successfully models competition for a range of climate and vegetation regimes and for which coexistence is a permissible equilibrium solution in the absence of additional mechanisms. The approach is tested for boreal forest, tropical forest, savanna, and temperate forest locations within the framework of the Canadian Terrestrial Ecosystem Model (CTEM) and successfully simulates the observed successional behavior and the observed near-equilibrium distribution of coexisting PFTs.
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Faria, Teresa, and Luis T. Magalhães. "Realisation of ordinary differential equations by retarded functional differential equations in neighbourhoods of equilibrium points." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 125, no. 4 (1995): 759–76. http://dx.doi.org/10.1017/s030821050003033x.
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This paper addresses the realisation of ordinary differential equations (ODEs) by retarded functional differential equations (FDEs) in finite-dimensional invariant manifolds, locally around equilibrium points. A necessary and sufficient condition for realisability of C1 vector fields is established in terms of their linearisations at the equilibrium.It is also shown that any arbitrary finite jet of vector fields of ODEs can be realised without any further restrictions than those imposed by the realisability of its linear term, a fact of relevance for discussing the flows defined by FDEs around singularities, and their bifurcations. Besides, it is proved that such a realisation can always be achieved with FDEs whose nonlinearities are defined in terms of a finite number of delayed values of the solutions.
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Wang, Hanxiao, and Jiongmin Yong. "Time-inconsistent stochastic optimal control problems and backward stochastic volterra integral equations." ESAIM: Control, Optimisation and Calculus of Variations 27 (2021): 22. http://dx.doi.org/10.1051/cocv/2021027.
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An optimal control problem is considered for a stochastic differential equation with the cost functional determined by a backward stochastic Volterra integral equation (BSVIE, for short). This kind of cost functional can cover the general discounting (including exponential and non-exponential) situations with a recursive feature. It is known that such a problem is time-inconsistent in general. Therefore, instead of finding a global optimal control, we look for a time-consistent locally near optimal equilibrium strategy. With the idea of multi-person differential games, a family of approximate equilibrium strategies is constructed associated with partitions of the time intervals. By sending the mesh size of the time interval partition to zero, an equilibrium Hamilton–Jacobi–Bellman (HJB, for short) equation is derived, through which the equilibrium value function and an equilibrium strategy are obtained. Under certain conditions, a verification theorem is proved and the well-posedness of the equilibrium HJB is established. As a sort of Feynman–Kac formula for the equilibrium HJB equation, a new class of BSVIEs (containing the diagonal value Z(r, r) of Z(⋅ , ⋅)) is naturally introduced and the well-posedness of such kind of equations is briefly presented.
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MOAWAD, S. M. "Linear and nonlinear stability criteria for compressible MHD flows in a gravitational field." Journal of Plasma Physics 79, no. 5 (June 14, 2013): 873–83. http://dx.doi.org/10.1017/s0022377813000627.
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AbstractThe equilibrium and stability properties of ideal magnetohydrodynamics (MHD) of compressible flow in a gravitational field with a translational symmetry are investigated. Variational principles for the steady-state equations are formulated. The MHD equilibrium equations are obtained as critical points of a conserved Lyapunov functional. This functional consists of the sum of the total energy, the mass, the circulation along field lines (cross helicity), the momentum, and the magnetic helicity. In the unperturbed case, the equilibrium states satisfy a nonlinear second-order partial differential equation (PDE) associated with hydrodynamic Bernoulli law. The PDE can be an elliptic or a parabolic equation depending on increasing the poloidal flow speed. Linear and nonlinear Lyapunov stability conditions under translational symmetric perturbations are established for the equilibrium states.
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Hénot, Olivier. "On polynomial forms of nonlinear functional differential equations." Journal of Computational Dynamics 8, no. 3 (2021): 307. http://dx.doi.org/10.3934/jcd.2021013.
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<p style='text-indent:20px;'>In this paper we study nonlinear autonomous retarded functional differential equations; that is, functional equations where the time derivative may depend on the past values of the variables. When the nonlinearities in such equations are comprised of elementary functions, we give a constructive proof of the existence of an embedding of the original coordinates yielding a polynomial differential equation. This embedding is a topological conjugacy between the semi-flow of the original differential equation and the semi-flow of the auxiliary polynomial differential equation. Further dynamical features are investigated; notably, for an equilibrium or a periodic orbit and its embedded counterpart, the stable and unstable eigenvalues have the same algebraic and geometric multiplicity.</p>
Dissertations / Theses on the topic "Functional equilibrium equations"
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Hamaguchi, Yushi. "Extended backward stochastic Volterra integral equations and their applications to time-inconsistent stochastic recursive control problems." Doctoral thesis, Kyoto University, 2021. http://hdl.handle.net/2433/263434.
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Allanson, Oliver Douglas. "Theory of one-dimensional Vlasov-Maxwell equilibria : with applications to collisionless current sheets and flux tubes." Thesis, University of St Andrews, 2017. http://hdl.handle.net/10023/11916.
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Vlasov-Maxwell equilibria are characterised by the self-consistent descriptions of the steady-states of collisionless plasmas in particle phase-space, and balanced macroscopic forces. We study the theory of Vlasov-Maxwell equilibria in one spatial dimension, as well as its application to current sheet and flux tube models. The ‘inverse problem' is that of determining a Vlasov-Maxwell equilibrium distribution function self-consistent with a given magnetic field. We develop the theory of inversion using expansions in Hermite polynomial functions of the canonical momenta. Sufficient conditions for the convergence of a Hermite expansion are found, given a pressure tensor. For large classes of DFs, we prove that non-negativity of the distribution function is contingent on the magnetisation of the plasma, and make conjectures for all classes. The inverse problem is considered for nonlinear ‘force-free Harris sheets'. By applying the Hermite method, we construct new models that can describe sub-unity values of the plasma beta (βpl) for the first time. Whilst analytical convergence is proven for all βpl, numerical convergence is attained for βpl = 0.85, and then βpl = 0.05 after a ‘re-gauging' process. We consider the properties that a pressure tensor must satisfy to be consistent with ‘asymmetric Harris sheets', and construct new examples. It is possible to analytically solve the inverse problem in some cases, but others must be tackled numerically. We present new exact Vlasov-Maxwell equilibria for asymmetric current sheets, which can be written as a sum of shifted Maxwellian distributions. This is ideal for implementations in particle-in-cell simulations. We study the correspondence between the microscopic and macroscopic descriptions of equilibrium in cylindrical geometry, and then attempt to find Vlasov-Maxwell equilibria for the nonlinear force-free ‘Gold-Hoyle' model. However, it is necessary to include a background field, which can be arbitrarily weak if desired. The equilibrium can be electrically non-neutral, depending on the bulk flows.
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Edirisinghe, Pathirannehelage Neranjan S. "Charge Transfer in Deoxyribonucleic Acid (DNA): Static Disorder, Dynamic Fluctuations and Complex Kinetic." Digital Archive @ GSU, 2011. http://digitalarchive.gsu.edu/phy_astr_diss/45.
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The fact that loosely bonded DNA bases could tolerate large structural fluctuations, form a dissipative environment for a charge traveling through the DNA. Nonlinear stochastic nature of structural fluctuations facilitates rich charge dynamics in DNA. We study the complex charge dynamics by solving a nonlinear, stochastic, coupled system of differential equations. Charge transfer between donor and acceptor in DNA occurs via different mechanisms depending on the distance between donor and acceptor. It changes from tunneling regime to a polaron assisted hopping regime depending on the donor-acceptor separation. Also we found that charge transport strongly depends on the feasibility of polaron formation. Hence it has complex dependence on temperature and charge-vibrations coupling strength. Mismatched base pairs, such as different conformations of the G・A mispair, cause only minor structural changes in the host DNA molecule, thereby making mispair recognition an arduous task. Electron transport in DNA that depends strongly on the hopping transfer integrals between the nearest base pairs, which in turn are affected by the presence of a mispair, might be an attractive approach in this regard. I report here on our investigations, via the I –V characteristics, of the effect of a mispair on the electrical properties of homogeneous and generic DNA molecules. The I –V characteristics of DNA were studied numerically within the double-stranded tight-binding model. The parameters of the tight-binding model, such as the transfer integrals and on-site energies, are determined from first-principles calculations. The changes in electrical current through the DNA chain due to the presence of a mispair depend on the conformation of the G・A mispair and are appreciable for DNA consisting of up to 90 base pairs. For homogeneous DNA sequences the current through DNA is suppressed and the strongest suppression is realized for the G(anti)・A(syn) conformation of the G・A mispair. For inhomogeneous (generic) DNA molecules, the mispair result can be either suppression or an enhancement of the current, depending on the type of mispairs and actual DNA sequence.
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Arhinful, Daniel Andoh. "Lorenzův systém: cesta od stability k chaosu." Master's thesis, Vysoké učení technické v Brně. Fakulta strojního inženýrství, 2020. http://www.nusl.cz/ntk/nusl-417087.
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The theory of deterministic chaos has generated a lot of interest and continues to be one of the much-focused research areas in the field of dynamics today. This is due to its prevalence in essential parts of human lives such as electrical circuits, chemical reactions, the flow of blood through the human system, the weather, etc. This thesis presents a study of the Lorenz equations, a famous example of chaotic systems. In particular, it presents the analysis of the Lorenz equations from stability to chaos and various bifurcation scenarios with numerical and graphical interpretations. It studies concepts of non-linear dynamical systems such as equilibrium points, stability, linearization, bifurcation, Lyapunov function, etc. Finally, it discusses how the Lorenz equations serve as a model for the waterwheel (in detail), and the convection roll for fluid.
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Connell, R. J. "Unstable equilibrium : modelling waves and turbulence in water flow." Diss., Lincoln University, 2008. http://hdl.handle.net/10182/592.
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This thesis develops a one-dimensional version of a new data driven model of turbulence that uses the KL expansion to provide a spectral solution of the turbulent flow field based on analysis of Particle Image Velocimetry (PIV) turbulent data. The analysis derives a 2nd order random field over the whole flow domain that gives better turbulence properties in areas of non-uniform flow and where flow separates than the present models that are based on the Navier-Stokes Equations. These latter models need assumptions to decrease the number of calculations to enable them to run on present day computers or super-computers. These assumptions reduce the accuracy of these models. The improved flow field is gained at the expense of the model not being generic. Therefore the new data driven model can only be used for the flow situation of the data as the analysis shows that the kernel of the turbulent flow field of undular hydraulic jump could not be related to the surface waves, a key feature of the jump. The kernel developed has two parts, called the outer and inner parts. A comparison shows that the ratio of outer kernel to inner kernel primarily reflects the ratio of turbulent production to turbulent dissipation. The outer part, with a larger correlation length, reflects the larger structures of the flow that contain most of the turbulent energy production. The inner part reflects the smaller structures that contain most turbulent energy dissipation. The new data driven model can use a kernel with changing variance and/or regression coefficient over the domain, necessitating the use of both numerical and analytical methods. The model allows the use of a two-part regression coefficient kernel, the solution being the addition of the result from each part of the kernel. This research highlighted the need to assess the size of the structures calculated by the models based on the Navier-Stokes equations to validate these models. At present most studies use mean velocities and the turbulent fluctuations to validate a models performance. As the new data driven model gives better turbulence properties, it could be used in complicated flow situations, such as a rock groyne to give better assessment of the forces and pressures in the water flow resulting from turbulence fluctuations for the design of such structures. Further development to make the model usable includes; solving the numerical problem associated with the double kernel, reducing the number of modes required, obtaining a solution for the kernel of two-dimensional and three-dimensional flows, including the change in correlation length with time as presently the model gives instant realisations of the flow field and finally including third and fourth order statistics to improve the data driven model velocity field from having Gaussian distribution properties. As the third and fourth order statistics are Reynolds Number dependent this will enable the model to be applied to PIV data from physical scale models. In summary, this new data driven model is complementary to models based on the Navier-Stokes equations by providing better results in complicated design situations. Further research to develop the new model is viewed as an important step forward in the analysis of river control structures such as rock groynes that are prevalent on New Zealand Rivers protecting large cities.
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Hoffmann, Franca Karoline Olga. "Keller-Segel-type models and kinetic equations for interacting particles : long-time asymptotic analysis." Thesis, University of Cambridge, 2017. https://www.repository.cam.ac.uk/handle/1810/269646.
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This thesis consists of three parts: The first and second parts focus on long-time asymptotics of macroscopic and kinetic models respectively, while in the third part we connect these regimes using different scaling approaches. (1) Keller–Segel-type aggregation-diffusion equations: We study a Keller–Segel-type model with non-linear power-law diffusion and non-local particle interaction: Does the system admit equilibria? If yes, are they unique? Which solutions converge to them? Can we determine an explicit rate of convergence? To answer these questions, we make use of the special gradient flow structure of the equation and its associated free energy functional for which the overall convexity properties are not known. Special cases of this family of models have been investigated in previous works, and this part of the thesis represents a contribution towards a complete characterisation of the asymptotic behaviour of solutions. (2) Hypocoercivity techniques for a fibre lay-down model: We show existence and uniqueness of a stationary state for a kinetic Fokker-Planck equation modelling the fibre lay-down process in non-woven textile production. Further, we prove convergence to equilibrium with an explicit rate. This part of the thesis is an extension of previous work which considered the case of a stationary conveyor belt. Adding the movement of the belt, the global equilibrium state is not known explicitly and a more general hypocoercivity estimate is needed. Although we focus here on a particular application, this approach can be used for any equation with a similar structure as long as it can be understood as a certain perturbation of a system for which the global Gibbs state is known. (3) Scaling approaches for collective animal behaviour models: We study the multi-scale aspects of self-organised biological aggregations using various scaling techniques. Not many previous studies investigate how the dynamics of the initial models are preserved via these scalings. Firstly, we consider two scaling approaches (parabolic and grazing collision limits) that can be used to reduce a class of non-local kinetic 1D and 2D models to simpler models existing in the literature. Secondly, we investigate how some of the kinetic spatio-temporal patterns are preserved via these scalings using asymptotic preserving numerical methods.
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Pimentel, Edgard Almeida. "Um ensaio em teoria dos jogos." Universidade de São Paulo, 2010. http://www.teses.usp.br/teses/disponiveis/45/45132/tde-31082010-091851/.
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Esta dissertação aborda a teoria dos jogos diferenciais em sua estreita relação com a teoria das equações de Hamilton-Jacobi (HJ). Inicialmente, uma revisão da noção de solução em teoria dos jogos é empreendida. Discutem-se nesta ocasião as idéias de equilíbrio de Nash e alguns de seus refinamentos. Em seguida, tem lugar uma introdução à teoria dos jogos diferenciais, onde noções de solução como a função de valor de Isaacs e de Friedman são discutidas. É nesta altura do trabalho que fica evidente a conexão entre este conceito de solução e a teoria das equações de Hamilton-Jacobi. Por ocasião desta conexão, é explorada a noção de solução clássica e é exposta uma demonstração do fato de que se um jogo diferencial possuir uma função de valor pelo menos continuamente diferenciável, esta será uma solução da equação de Hamilton-Jacobi associada ao jogo. Este resultado faz uso do princípio da programação dinâmica, devido a Bellman, e cuja demonstração está presente no texto. No entanto, quando a função de valor do jogo é apenas contínua, então embora esta não seja uma solução clássica da equação HJ associada a jogo, vemos que ela será uma solução viscosa, ou solução no sentido da viscosidade - e a esta altura são discutidos os elementos e propriedades desta classe de soluções, um teorema de existência e unicidade e alguns exemplos. Por fim, retomamos o estudo dos jogos diferenciais à luz das soluções viscosas da equação de Hamilton-Jacobi e, assim, expomos uma demonstração de existência da função de valor e do princípio da programação dinâmica a partir das noções da viscosidadeThis dissertation aims to address the topic of Differential Game Theory in its connection with the Hamilton-Jacobi (HJ) equations framework. Firstly we introduce the idea of solution for a game, through the discussion of Nash equilibria and its refinements. Secondly, the solution concept is then translated to the context of Differential Games and the idea of value function is introduced in its Isaacs\'s as well as Friedman\'s version. As the value function is discussed, its relationship with the Hamilton-Jacobi equations theory becomes self-evident. Due to such relation, we investigate the HJ equation from two distinct points of view. First of all, we discuss a statement according to which if a differential game has a continuously differentiable value function, then such function is a classical solution of the HJ equation associated to the game. This result strongly relies on Bellman\'s Dynamic Programming Principle - and this is the reason why we devote an entire chapter to this theme. Furthermore, HJ is still at our sight from the PDE point of view. Our motivation is simple: under some lack of regularity - a value function which is continuous, but not continuously differentiable - a game may still have a value function represented as a solution of the associated HJ equation. In this case such a solution will be called a solution in the viscosity sense. We then discuss the properties of viscosity solutions as well as provide an existence and uniqueness theorem. Finally we turn our attention back to the theory of games and - through the notion of viscosity - establish the existence and uniqueness of value functions for a differential game within viscosity solution theory.
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Carrapatoso, Kléber. "Théorèmes asymptotiques pour les équations de Boltzmann et de Landau." Phd thesis, Université Paris Dauphine - Paris IX, 2013. http://tel.archives-ouvertes.fr/tel-00920455.
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Nous nous intéressons dans cette thèse à la théorie cinétique et aux systèmes de particules dans le cadre des équations de Boltzmann et Landau. Premièrement, nous étudions la dérivation des équations cinétiques comme des limites de champ moyen des systèmes de particules, en utilisant le concept de propagation du chaos. Plus précisément, nous étudions les probabilités chaotiques sur l'espace de phase de ces systèmes de particules : la sphère de Boltzmann, qui correspond à l'espace de phase d'un système de particules qui évolue conservant le moment et l'énergie ; et la sphère de Kac, correspondant à un système de particules qui conserve seulement l'énergie. Ensuite, nous nous intéressons à la propagation du chaos, avec des estimations quantitatives et uniforme en temps, pour les équations de Boltzmann et Landau. Deuxièmement, nous étudions le comportement asymptotique en temps grand des solutions de l'équation de Landau.
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Campos, Serrano Juan. "Modèles attractifs en astrophysique et biologie : points critiques et comportement en temps grand des solutions." Phd thesis, Université Paris Dauphine - Paris IX, 2012. http://tel.archives-ouvertes.fr/tel-00861568.
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Dans cette thèse, nous étudions l'ensemble des solutions d'équations aux dérivées partielles résultant de modèles d'astrophysique et de biologie. Nous répondons aux questions de l'existence, mais aussi nous essayons de décrire le comportement de certaines familles de solutions lorsque les paramètres varient. Tout d'abord, nous étudions deux problèmes issus de l'astrophysique, pour lesquels nous montrons l'existence d'ensembles particuliers de solutions dépendant d'un paramètre à l'aide de la méthode de réduction de Lyapunov-Schmidt. Ensuite un argument de perturbation et le théorème du Point xe de Banach réduisent le problème original à un problème de dimension finie, et qui peut être résolu, habituellement, par des techniques variationnelles. Le reste de la thèse est consacré à l'étude du modèle Keller-Segel, qui décrit le mouvement d'amibes unicellulaires. Dans sa version plus simple, le modèle de Keller-Segel est un système parabolique-elliptique qui partage avec certains modèles gravitationnels la propriété que l'interaction est calculée au moyen d'une équation de Poisson / Newton attractive. Une différence majeure réside dans le fait que le modèle est défini dans un espace bidimensionnel, qui est expérimentalement consistant, tandis que les modèles de gravitationnels sont ordinairement posés en trois dimensions. Pour ce problème, les questions de l'existence sont bien connues, mais le comportement des solutions au cours de l'évolution dans le temps est encore un domaine actif de recherche. Ici nous étendre les propriétés déjà connues dans des régimes particuliers à un intervalle plus large du paramètre de masse, et nous donnons une estimation précise de la vitesse de convergence de la solution vers un profil donné quand le temps tend vers l'infini. Ce résultat est obtenu à l'aide de divers outils tels que des techniques de symétrisation et des inégalités fonctionnelles optimales. Les derniers chapitres traitent de résultats numériques et de calculs formels liés au modèle Keller-Segel
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Dore-Hall, Skye. "Ramp function approximations of Michaelis-Menten functions in biochemical dynamical systems." Thesis, 2020. http://hdl.handle.net/1828/12485.
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In 2019, Adams, Ehlting, and Edwards developed a four-variable system of ordinary differential equations modelling phenylalanine metabolism in plants according to Michaelis-Menten kinetics. Analysis of the model suggested that when a series of reactions known as the Shikimate Ester Loop (SEL) is included, phenylalanine flux into primary metabolic pathways is prioritized over flux into secondary metabolic pathways when the availability of shikimate, a phenylalanine precursor, is low. Adams et al. called this mechanism of metabolic regulation the Precursor Shutoff Valve (PSV). Here, we attempt to simplify Adams and colleagues’ model by reducing the system to three variables and replacing the Michaelis-Menten terms with piecewise-defined approximations we call ramp functions. We examine equilibria and stability in this simplified model, and show that PSV-type regulation is still present in the version with the SEL. Then, we define a class of systems structurally similar to the simplified Adams model called biochemical ramp systems. We study the properties of the Jacobian matrices of these systems and then explore equilibria and stability in systems of n ≥ 2 variables. Finally, we make several suggestions regarding future work on biochemical ramp systems.Graduate
Books on the topic "Functional equilibrium equations"
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1973-, Warzel Simone, ed. Random operators: Disorder effects on quantum spectra and dynamics. Providence, Rhode Island: American Mathematical Society, 2015.
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Algebraic aspects of Darboux transformations, quantum integrable systems, and supersymmetric quantum mechanics. Providence, R.I: American Mathematical Society, 2012.
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Horing, Norman J. Morgenstern. Non-Equilibrium Green’s Functions: Variational Relations and Approximations for Particle Interactions. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198791942.003.0009.
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Chapter 09 Nonequilibrium Green’s functions (NEGF), including coupled-correlated (C) single- and multi-particle Green’s functions, are defined as averages weighted with the time-development operator U(t0+τ,t0). Linear conductivity is exhibited as a two-particle equilibrium Green’s function (Kubo-type formulation). Admitting particle sources (S:η,η+) and non-conservation of number, the non-equilibrium multi-particle Green’s functions are constructed with numbers of creation and annihilation operators that may differ, and they may be derived as variational derivatives with respect to sources η,η+ of a generating functional eW=TrU(t0+τ,t0)CS/TrU(t0+τ,t0)C. (In the non-interacting case this yields the n-particle Green’s function as a permanent/determinant of single-particle Green’s functions.) These variational relations yield a symmetric set of multi-particle Green’s function equations. Cumulants and the Linked Cluster Theorem are discussed and the Random Phase Approximation (RPA) is derived variationally. Schwinger’s variational differential formulation of perturbation theories for the Green’s function, self-energy, vertex operator, and also shielded potential perturbation theory, are reviewed. The Langreth Algebra arises from analytic continuation of integration of products of Green’s functions in imaginary time to the real-time axis with time-ordering along the integration contour in the complex time plane. An account of the Generalized Kadanoff-Baym Ansatz is presented.
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Morawetz, Klaus. Nonequilibrium Green’s Functions. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198797241.003.0007.
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The method of the equation of motion is used to derive the Martin–Schwinger hierarchy for the nonequilibrium Green’s functions. The formal closure of the hierarchy is reached by using the selfenergy which provides a recipe for how to construct selfenergies from approximations of the two-particle Green’s function. The Langreth–Wilkins rules for a diagrammatic technique are shown to be equivalent to the weakening of initial correlations. The quantum transport equations are derived in the general form of Kadanoff and Baym equations. The information contained in the Green’s function is discussed. In equilibrium this leads to the Matsubara diagrammatic technique.
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Mann, Peter. Near-Integrable Systems. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198822370.003.0024.
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This chapter extends the now familiar Lagrangian formulation to a field theory and covers elementary material in this new setting. The motion of systems with a very large number of degrees of freedom makes it necessary to specify an almost infinite number of discrete coordinates. It is possible to simplify the situation by taking the continuum limit, which replaces the individual coordinates with a continuous function that describes a displacement field, which assigns a displacement vector to each position the system could occupy relative to an equilibrium configuration. The field thus takes a point in the spacetime manifold and assigns it a value corresponding to whatever the field represents. In this chapter, many interdisciplinary examples are solved and pedagogical models are discussed. The chapter also discusses Lagrange density, the Lagrange field equation, instantons, the Klein–Gordon equation, Fourier transforms and the Korteweg–de Vries equation.
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Horing, Norman J. Morgenstern. Quantum Statistical Field Theory. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198791942.001.0001.
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The methods of coupled quantum field theory, which had great initial success in relativistic elementary particle physics and have subsequently played a major role in the extensive development of non-relativistic quantum many-particle theory and condensed matter physics, are at the core of this book. As an introduction to the subject, this presentation is intended to facilitate delivery of the material in an easily digestible form to students at a relatively early stage of their scientific development, specifically advanced undergraduates (rather than second or third year graduate students), who are mathematically strong physics majors. The mechanism to accomplish this is the early introduction of variational calculus with particle sources and the Schwinger Action Principle, accompanied by Green’s functions, and, in addition, a brief derivation of quantum mechanical ensemble theory introducing statistical thermodynamics. Important achievements of the theory in condensed matter and quantum statistical physics are reviewed in detail to help develop research capability. These include the derivation of coupled field Green’s function equations of motion for a model electron-hole-phonon system, extensive discussions of retarded, thermodynamic and non-equilibrium Green’s functions, and their associated spectral representations and approximation procedures. Phenomenology emerging in these discussions includes quantum plasma dynamic, nonlocal screening, plasmons, polaritons, linear electromagnetic response, excitons, polarons, phonons, magnetic Landau quantization, van der Waals interactions, chemisorption, etc. Considerable attention is also given to low-dimensional and nanostructured systems, including quantum wells, wires, dots and superlattices, as well as materials having exceptional conduction properties such as superconductors, superfluids and graphene.
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Batterman, Robert W. A Middle Way. Oxford University Press, 2021. http://dx.doi.org/10.1093/oso/9780197568613.001.0001.
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This book focuses on a method for exploring, explaining, and understanding the behavior of large many-body systems. It describes an approach to non-equilibrium behavior that focuses on structures (represented by correlation functions) that characterize mesoscale properties of the systems. In other words, rather than a fully bottom-up approach, starting with the components at the atomic or molecular scale, the “hydrodynamic approach” aims to describe and account for continuum behaviors by largely ignoring details at the “fundamental” level. This methodological approach has its origins in Einstein’s work on Brownian motion. He gave what may be the first instance of “upscaling” to determine an effective (continuum) value for a material parameter—the viscosity. His method is of a kind with much work in the science of materials. This connection and the wide-ranging interdisciplinary nature of these methods are stressed. Einstein also provided the first expression of a fundamental theorem of statistical mechanics called the Fluctuation-Dissipation theorem. This theorem provides the primary justification for the hydrodynamic, mesoscale methodology. Philosophical consequences include an argument to the effect that mesoscale parameters can be the natural variables for characterizing many-body systems. Further, the book offers a new argument for why continuum theories (fluid mechanics and equations for the bending of beams) are still justified despite completely ignoring the fact that fluids and materials have lower scale structure. The book argues for a middle way between continuum theories and atomic theories. A proper understanding of those connections can be had when mesoscales are taken seriously.
Book chapters on the topic "Functional equilibrium equations"
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Hale, Jack K., and Sjoerd M. Verduyn Lunel. "Near equilibrium and periodic orbits." In Introduction to Functional Differential Equations, 302–30. New York, NY: Springer New York, 1993. http://dx.doi.org/10.1007/978-1-4612-4342-7_11.
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Noor, Muhammad A., and Themistocles M. Rassias. "Some New Algorithms for Solving General Equilibrium Problems." In Handbook of Functional Equations, 407–17. New York, NY: Springer New York, 2014. http://dx.doi.org/10.1007/978-1-4939-1246-9_17.
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Shaikhet, Leonid. "Stability of SIR Epidemic Model Equilibrium Points." In Lyapunov Functionals and Stability of Stochastic Functional Differential Equations, 283–96. Heidelberg: Springer International Publishing, 2013. http://dx.doi.org/10.1007/978-3-319-00101-2_11.
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Shaikhet, Leonid. "Stability of Equilibrium Points of Nicholson’s Blowflies Equation with Stochastic Perturbations." In Lyapunov Functionals and Stability of Stochastic Functional Differential Equations, 251–56. Heidelberg: Springer International Publishing, 2013. http://dx.doi.org/10.1007/978-3-319-00101-2_9.
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Shaikhet, Leonid. "Stability of Positive Equilibrium Point of Nonlinear System of Type of Predator–Prey with Aftereffect and Stochastic Perturbations." In Lyapunov Functionals and Stability of Stochastic Functional Differential Equations, 257–82. Heidelberg: Springer International Publishing, 2013. http://dx.doi.org/10.1007/978-3-319-00101-2_10.
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Arrigoni, Enrico, and Antonius Dorda. "Master Equations Versus Keldysh Green’s Functions for Correlated Quantum Systems Out of Equilibrium." In Out-of-Equilibrium Physics of Correlated Electron Systems, 121–88. Cham: Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-319-94956-7_4.
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Garrett, Steven L. "Attenuation of Sound." In Understanding Acoustics, 673–98. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-44787-8_14.
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Abstract We will capitalize on our understanding of thermoviscous loss to develop an understanding of the attenuation of sound waves in fluids that are not influenced by proximity to solid surfaces. Such dissipation mechanisms are particularly important at very high frequencies and short distances (for ultrasound) or very low frequencies over geological distances (for infrasound). The Standard Linear Model of viscoelasticity introduced the nondimensional frequency, ωτR, that controlled the medium’s elastic (in-phase) and dissipative (quadrature) responses. Those response curves were “universal” in the sense that causality linked the elastic and dissipative responses through the Kramers-Kronig relations. That relaxation-time perspective is essential for attenuation of sound in media that can be characterized by one or more relaxation times related to those internal degrees of freedom that make their equation of state a function of frequency. Examples of these relaxation-time effects include the rate of collisions between different molecular species in a gas (e.g., nitrogen and water vapor in air), the pressure dependence of ionic association-dissociation of dissolved salts in sea water (e.g., MgSO4 and H3BO3), and evaporation-condensation effects when a fluid is oscillating about equilibrium with its vapor (e.g., fog droplets in air or gas bubbles in liquids).
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Anand, Lallit, and Sanjay Govindjee. "Principles of minimum potential energy and complementary energy." In Continuum Mechanics of Solids, 228–48. Oxford University Press, 2020. http://dx.doi.org/10.1093/oso/9780198864721.003.0012.
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With the displacement field taken as the only fundamental unknown field in a mixed-boundary-value problem for linear elastostatics, the principle of minimum potential energy asserts that a potential energy functional, which is defined as the difference between the free energy of the body and the work done by the prescribed surface tractions and the body forces --- assumes a smaller value for the actual solution of the mixed problem than for any other kinematically admissible displacement field which satisfies the displacement boundary condition. This principle provides a weak or variational method for solving mixed boundary-value-problems of elastostatics. In particular, instead of solving the governing Navier form of the partial differential equations of equilibrium, one can search for a displacement field such that the first variation of the potential energy functional vanishes. A similar principle of minimum complementary energy, which is phrased in terms of statically admissible stress fields which satisfy the equilibrium equation and the traction boundary condition, is also discussed. The principles of minimum potential energy and minimum complementary energy can also be applied to derive specialized principles which are particularly well-suited to solving structural problems; in this context the celebrated theorems of Castigliano are discussed.
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Aruğaslan-Çinçin, Duygu, and Nur Cengiz. "Stability Analysis of a Nonlinear Epidemic Model With Generalized Piecewise Constant Argument." In Emerging Applications of Differential Equations and Game Theory, 182–208. IGI Global, 2020. http://dx.doi.org/10.4018/978-1-7998-0134-4.ch009.
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The authors consider a nonlinear epidemic equation by modeling it with generalized piecewise constant argument (GPCA). The authors investigate invariance region for the considered model. Sufficient conditions guaranteeing the existence and uniqueness of the solutions of the model are given by creating integral equations. An important auxiliary result giving a relation between the values of the unknown function solutions at the deviation argument and at any time t is indicated. By using Lyapunov-Razumikhin method developed by Akhmet and Aruğaslan for the differential equations with generalized piecewise constant argument (EPCAG), the stability of the trivial equilibrium is investigated in addition to the stability examination of the positive equilibrium transformed into the trivial equilibrium. Then sufficient conditions for the uniform stability and the uniform asymptotic stability of trivial equilibrium and the positive equilibrium are given.
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"H-Functional Equation." In A Non-Equilibrium Statistical Mechanics, 33–67. WORLD SCIENTIFIC, 2003. http://dx.doi.org/10.1142/9789812795199_0003.
Full textConference papers on the topic "Functional equilibrium equations"
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Vallabhaneni, Ajit K., James Loy, Dhruv Singh, Xiulin Ruan, and Jayathi Murthy. "A Study of Spatially-Resolved Non-Equilibrium in Laser-Irradiated Graphene Using Boltzmann Transport Equation." In ASME 2013 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2013. http://dx.doi.org/10.1115/imece2013-66095.
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Raman spectroscopy is typically used to characterize graphene in experiments and also to measure properties like thermal conductivity and optical phonon lifetime. The laser-irradiation processes underlying this measurement technique include coupling between photons, electrons and phonons. Recent experimental studies have shown that e-ph scattering limits the performance of graphene-based electronic devices due to the difference in their timescales of relaxation resulting in various bottleneck effects. Furthermore, recently published thermal conductivity measurements on graphene are sensitive to the laser spot size which strengthens the possibility of non-equilibrium between various phonon groups. These studies point to the need to study the spatially-resolved non-equilibrium between various energy carriers in graphene. In this work, we demonstrate non-equilibrium in the e-ph interactions in graphene by solving the linearized electron and phonon Boltzmann transport equations (BTE) iteratively under steady state conditions. We start by assuming that all the electrons equilibrate rapidly to an elevated temperature under laser-irradiation and they gradually relax by phonon emission and reach a steady state. The electron and phonon BTEs are coupled because the e-ph scattering rate depends on the phonon population while the rate of phonon generation depends on the e-ph scattering rate. We used density-functional theory/density-functional perturbation theory (DFT/DFPT) to calculate the electronic eigen states, phonon frequencies and the e-ph coupling matrix elements. We calculated the rate of energy loss from the hot electrons in terms of the phonon generation rate (PGR) which serve as an input for solving the BTE. Likewise, ph-ph relaxation times are calculated from the anharmonic lattice dynamics (LD)/FGR. Through our work, we obtained the spatially resolved temperature profiles of all the relevant energy carriers throughout the entire domain; these are impossible to obtain through experiments.
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Yamada, Takayuki, Toshiro Matsumoto, and Shinji Nishiwaki. "Design of Mechanical Structures Considering Harmonic Loads Using Level Set-Based Topology Optimization." In ASME 2012 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2012. http://dx.doi.org/10.1115/detc2012-70235.
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This paper presents an optimum design method for mechanical structures considering harmonic loads using a level set-based topology optimization method and the Finite Element Method (FEM). First, we briefly discuss the level set-based topology optimization method. Second, a topology optimization problem is formulated for a dynamic elastic design problem using level set boundary expressions. The objective functional is set to minimize the displacement at specific boundaries. Based on this formulation, the topological sensitivities of the objective functional are derived. Next, a topology optimization algorithm is proposed that uses the FEM to solve the equilibrium and adjoint equations, and when updating the level set function. Finally, several numerical examples are provided to confirm the validity and utility of the proposed method.
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Ryckelynck, David. "Multidimensional Hyper-Reduction of Large Mechanical Models Involving Internal Variables." In ASME 2012 11th Biennial Conference on Engineering Systems Design and Analysis. American Society of Mechanical Engineers, 2012. http://dx.doi.org/10.1115/esda2012-82971.
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We propose to incorporate a Response Surface (RS) approximation of variables over a parametric domain into a weak form of parametric Partial Differential Equations (PDEs). Hence a multidimensional model-reduction can be achieved. We propose a multidimensional a priori model reduction method to generate or to enrich RSs. It is coined multidimensional because the fields to forecast are defined over an augmented domain in term of dimension. They are functions of both space variables and parameters that simultaneously evolve in time. This changes the functional space related to the weak form of the PDEs and the definition of the reduced bases. It has a significant impact on the proposed model reduction method. In particular, a new point of view on interpolation of variables has to be addressed. A Multidimensional Reduced Integration Domain (MRID) is proposed to reduce the complexity of the reduced formulation. A multidimensional Hyper-Reduction method extract from the MRID truncated equilibrium equations, truncated residuals and a truncated error indicator.
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Yang, Longxiang, and Stanley G. Hutton. "Numerical Formulation of Nonlinear Vibrations of Elastically-Constrained Rotating Disks." In ASME 1995 Design Engineering Technical Conferences collocated with the ASME 1995 15th International Computers in Engineering Conference and the ASME 1995 9th Annual Engineering Database Symposium. American Society of Mechanical Engineers, 1995. http://dx.doi.org/10.1115/detc1995-0322.
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Abstract An analysis of nonlinear vibrations of an elastically-constrained rotating disk is developed. The equations of motion, which are two coupled nonlinear partial differential equations corresponding to the transverse force equilibrium and to the deformation compatibility, are first developed by using von Karman thin plate theory. Then the stress function is analytically solved from the compatibility equation by assuming a multi-mode transverse displacement field. Galerkin’s method is applied to transform the force equilibrium equation into a set of coupled nonlinear ordinary differential equations in terms of time functions. Finally, numerical integration is used to solve the time governing equations, and the effects of nonlinearity on the vibrations of a rotating disk are discussed.
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Grover, Piyush. "Stability Analysis in Mean-Field Games via an Evans Function Approach." In ASME 2018 Dynamic Systems and Control Conference. American Society of Mechanical Engineers, 2018. http://dx.doi.org/10.1115/dscc2018-8926.
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This work is concerned with stability analysis of stationary and time-varying equilibria in a class of mean-field games that relate to multi-agent control problems of flocking and swarming. The mean-field game framework is a non-cooperative model of distributed optimal control in large populations, and characterizes the optimal control for a representative agent in Nash-equilibrium with the population. A mean-field game model is described by a coupled PDE system of forward-in-time Fokker-Planck (FP) equation for density of agents, and a backward-in-time Hamilton-Jacobi-Bellman (HJB) equation for control. The linear stability analysis of fixed points of these equations typically proceeds via numerical computation of spectrum of the linearized MFG operator. We explore the Evans function approach that provides a geometric alternative to solving the characteristic equation.
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Samsam Shariat, B., M. R. Eslami, and A. Bagri. "Thermoelastic Stability of Imperfect Functionally Graded Plates Based on the Third Order Shear Deformation Theory." In ASME 8th Biennial Conference on Engineering Systems Design and Analysis. ASMEDC, 2006. http://dx.doi.org/10.1115/esda2006-95018.
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Thermal buckling analysis of rectangular functionally graded plates with initial geometric imperfections is presented in this paper. It is assumed that the non-homogeneous mechanical properties vary linearly through the thickness of the plate. The plate is assumed to be under various types of thermal loadings, such as the uniform temperature rise and nonlinear temperature gradient through the thickness. A double-sine function for the geometric imperfection along the x and y-directions is considered. The equilibrium equations are derived using the third order shear deformation plate theory. Using a suitable method, equilibrium equations are reduced from 5 to 2 equations. The corresponding stability equations are established. Using these equations accompanied by the compatibility equation yield to the buckling loads in a closed form solution for each loading case. The results are compared with the known data in the literature.
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Wei, Zhigang, Fulun Yang, Shervin Maleki, and Kamran Nikbin. "Equilibrium Based Curve Fitting Method for Test Data With Nonuniform Variance." In ASME 2012 Pressure Vessels and Piping Conference. American Society of Mechanical Engineers, 2012. http://dx.doi.org/10.1115/pvp2012-78234.
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A recently developed equilibrium based curve/surface fitting method is extended to linear function with heteroscedastic data (variable variance). The concept of equilibrium weighted ‘force’ and ‘moment’ is proposed to derive curve fitting formulae, which are exactly the same as that obtained with the traditional weighted least squares (LS) method for a linear function. Furthermore, a system of four equations, i.e., a “force” equilibrium equation, a “moment” equilibrium equation, a “equivalency” equation, and a “moment balance” equation have been established to solve both mean curve and the standard deviation simultaneously. Finally, the application of these methods to data of fatigue and creep lives is presented.
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SMOLYANSKY, S. A., V. A. MIZERNY, D. V. VINNIK, A. V. PROZORKEVICH, and V. D. TONEEV. "THE NON-EQUILIBRIUM DISTRIBUTION FUNCTION OF PARTICLES AND ANTI-PARTICLES CREATED IN STRONG FIELDS." In Proceedings of the Conference “Kadanoff-Baym Equations: Progress and Perspectives for Many-Body Physics”. WORLD SCIENTIFIC, 2000. http://dx.doi.org/10.1142/9789812793812_0029.
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Kotchergenko, I. D. "The Areolar Strain Concept." In ASME 2008 International Mechanical Engineering Congress and Exposition. ASMEDC, 2008. http://dx.doi.org/10.1115/imece2008-66215.
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The fundamental equations of the isotropic and orthotropic plane theory of elasticity are reworked in the frame of the generalized analytic functions theory. The areolar strain concept and its equilibrium and compatibility equations are presented. In this conception, the plane strain is decomposed into two orthogonal complex strains. A canonic form for the equilibrium equation is provided, allowing achieving its general solution in a fairly straightforward fashion. For finite rotation, new equilibrium equations and boundary conditions, in terms of strains as well as in terms of stresses, are given. Orthogonal polynomial expansions, which fulfill both the equilibrium and the compatibility equations for isotropic and orthotropic planes, are provided. These polynomials exactly retain the drilling degrees of freedom in finite element models.
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Saidi, A. R., F. Hejripour, and E. Jomehzadeh. "On the Stress Singularities and Boundary Layer in Moderately Thick Functionally Graded Sectorial Plates." In ASME 2010 10th Biennial Conference on Engineering Systems Design and Analysis. ASMEDC, 2010. http://dx.doi.org/10.1115/esda2010-24395.
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In this paper, the stress analysis of moderately thick functionally graded (FG) sector plate is developed for studying the singularities in vicinity of the vertex. Based on the first-order shear deformation plate theory, the governing partial differential equations are obtained. Using an analytical method and defining some new functions, the stretching and bending equilibrium equations are decoupled. Also, introducing a function, called boundary layer function, the three bending equations are converted into two decoupled equations called edge-zone and interior equations. These equations are solved analytically for the sector plate with the simply supported radial edges and arbitrary boundary condition along the circular edge. The singularities of shear force and moment resultants are discussed for both salient and re-entrant sectorial plates. Also, the effects of power of the FGM, thickness to length ratio on the stress singularities of the FG sector plates are investigated.