Journal articles: 'Functional equilibrium equations'

1

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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10

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>

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11

VLADIMIROV, V. A., and K. I. ILIN. "On the energy instability of liquid crystals." European Journal of Applied Mathematics 9, no. 1 (February 1998): 23–36. http://dx.doi.org/10.1017/s0956792597003288.

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The direct Lyapunov method is used to investigate the stability of general equilibria of a nematic liquid crystal. First, we prove the converse Lagrange theorem stating that an equilibrium is unstable to small perturbations if the distortion energy has no minimum at this equilibrium (i.e. if the second variation of the distortion energy evaluated at the equilibrium is not positive definite). The proof is constructive rather than abstract: we explicitly construct a functional that grows exponentially with time by virtue of linearized equations of motion provided the condition of the theorem is satisfied. We obtain an explicit formula that gives the dependence of the perturbation growth rate upon the equilibrium considered and the initial data for the perturbation. Secondly, we obtain the upper and lower bounds for growing solutions of the linearized problem, and we identify the initial data corresponding to the most unstable mode (i.e. to the perturbation with maximal growth rate). All results are obtained in quite a general formulation: a nematic is inside a three-dimensional domain of an arbitrary shape and strong anchoring on the boundary is supposed; the standard equations of nematodynamics are employed as the governing equations.

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Zakharov, Anatoly Yu. "Manifestations of Short-Range and Long-Range Parts of Interatomic Potentials In Rearrangement Processes of Multicomponent Condensed Systems." Solid State Phenomena 138 (March 2008): 347–54. http://dx.doi.org/10.4028/www.scientific.net/ssp.138.347.

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Helmholtz free energy functional for generalized lattice model is reduced to Ginzburg- Landau-like form. Connections between interatomic potentials characteristics and parameters of Ginzburg-Landau-like functional are established. Equations for equilibrium distributions of species in multicomponent systems are derived. Equations of rearrangement kinetics of multicomponent systems are obtained. Description of the rearrangement processes via non-classical partial differential equations is proposed.

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13

Seguin, Brian, Yi-chao Chen, and Eliot Fried. "Closed Unstretchable Knotless Ribbons and the Wunderlich Functional." Journal of Nonlinear Science 30, no. 6 (May 22, 2020): 2577–611. http://dx.doi.org/10.1007/s00332-020-09630-z.

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Abstract In 1962, Wunderlich published the article “On a developable Möbius band,” in which he attempted to determine the equilibrium shape of a free standing Möbius band. In line with Sadowsky’s pioneering works on Möbius bands of infinitesimal width, Wunderlich used an energy minimization principle, which asserts that the equilibrium shape of the Möbius band has the lowest bending energy among all possible shapes of the band. By using the developability of the band, Wunderlich reduced the bending energy from a surface integral to a line integral without assuming that the width of the band is small. Although Wunderlich did not completely succeed in determining the equilibrium shape of the Möbius band, his dimensionally reduced energy integral is arguably one of the most important developments in the field. In this work, we provide a rigorous justification of the validity of the Wunderlich integral and fully formulate the energy minimization problem associated with finding the equilibrium shapes of closed bands, including both orientable and nonorientable bands with arbitrary number of twists. This includes characterizing the function space of the energy functional, dealing with the isometry and local injectivity constraints, and deriving the Euler–Lagrange equations. Special attention is given to connecting edge conditions, regularity properties of the deformed bands, determination of the parameter space needed to ensure that the deformation is surjective, reduction in isometry constraints, and deriving matching conditions and jump conditions associated with the Euler–Lagrange equations.

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14

Sun, Caixia, Lele Li, and Jianwen Jia. "Hopf bifurcation of an HIV-1 virus model with two delays and logistic growth." Mathematical Modelling of Natural Phenomena 15 (2020): 16. http://dx.doi.org/10.1051/mmnp/2019038.

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The paper establish and investigate an HIV-1 virus model with logistic growth, which also has intracellular delay and humoral immunity delay. The local stability of feasible equilibria are established by analyzing the characteristic equations. The globally stability of infection-free equilibrium and immunity-inactivated equilibrium are studied using the Lyapunov functional and LaSalles invariance principle. Besides, we prove that Hopf bifurcation will occur when the humoral immune delay pass through the critical value. And the stability of the positive equilibrium and Hopf bifurcations are investigated by using the normal form theory and the center manifold theorem. Finally, we confirm the theoretical results by numerical simulations.

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15

LI, ZHE, and RUI XU. "STABILITY ANALYSIS OF A RATIO-DEPENDENT CHEMOSTAT MODEL WITH TIME DELAY AND VARIABLE YIELD." International Journal of Biomathematics 03, no. 02 (June 2010): 243–53. http://dx.doi.org/10.1142/s1793524510000921.

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A chemostat model with time delay, variable yield and ratio-dependent functional response is investigated. By analyzing the corresponding characteristic equations, the local stability of a boundary equilibrium and a positive equilibrium is discussed and the existence of Hopf bifurcation is established. By using the comparison arguments, sufficient conditions are obtained for the global stability of the boundary equilibrium. By constructing a suitable Lyapunov functional, sufficient conditions are derived for the global asymptotic stability of the positive equilibrium. Finally, numerical simulations are carried out to illustrate the theoretical results.

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16

Yan, Caijuan, and Jianwen Jia. "Hopf Bifurcation of a Delayed Epidemic Model with Information Variable and Limited Medical Resources." Abstract and Applied Analysis 2014 (2014): 1–11. http://dx.doi.org/10.1155/2014/109372.

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We consider SIR epidemic model in which population growth is subject to logistic growth in absence of disease. We get the condition for Hopf bifurcation of a delayed epidemic model with information variable and limited medical resources. By analyzing the corresponding characteristic equations, the local stability of an endemic equilibrium and a disease-free equilibrium is discussed. If the basic reproduction ratioℛ0<1, we discuss the global asymptotical stability of the disease-free equilibrium by constructing a Lyapunov functional. Ifℛ0>1, we obtain sufficient conditions under which the endemic equilibriumE*of system is locally asymptotically stable. And we also have discussed the stability and direction of Hopf bifurcations. Numerical simulations are carried out to explain the mathematical conclusions.

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17

ZOU, WEI, JIEHUA XIE, and ZUOLIANG XIONG. "STABILITY AND HOPF BIFURCATION FOR AN ECO-EPIDEMIOLOGY MODEL WITH HOLLING-III FUNCTIONAL RESPONSE AND DELAYS." International Journal of Biomathematics 01, no. 03 (September 2008): 377–89. http://dx.doi.org/10.1142/s179352450800031x.

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In this paper, a system of retarded functional differential equations is proposed as a predator-prey model with disease in the prey. The invariance of non-negativity, nature of boundary equilibrium and global stability are analyzed. It also shows that positive equilibrium is locally asymptotically stable when time delay τ = τ1 + τ2 is suitable small, while a loss of stability by a Hopf bifurcation can occur around the positive equilibrium as the delays increase.

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Liu, Chao, and Qingling Zhang. "Dynamical Behavior and Stability Analysis in a Stage-Structured Prey Predator Model with Discrete Delay and Distributed Delay." Abstract and Applied Analysis 2014 (2014): 1–15. http://dx.doi.org/10.1155/2014/184174.

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We propose a prey predator model with stage structure for prey. A discrete delay and a distributed delay for predator described by an integral with a strong delay kernel are also considered. Existence of two feasible boundary equilibria and a unique interior equilibrium are analytically investigated. By analyzing associated characteristic equation, local stability analysis of boundary equilibrium and interior equilibrium is discussed, respectively. It reveals that interior equilibrium is locally stable when discrete delay is less than a critical value. According to Hopf bifurcation theorem for functional differential equations, it can be found that model undergoes Hopf bifurcation around the interior equilibrium when local stability switch occurs and corresponding stable limit cycle is observed. Furthermore, directions of Hopf bifurcation and stability of the bifurcating periodic solutions are studied based on normal form theory and center manifold theorem. Numerical simulations are carried out to show consistency with theoretical analysis.

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Saxena, Prashant, and Basant Lal Sharma. "On equilibrium equations and their perturbations using three different variational formulations of nonlinear electroelastostatics." Mathematics and Mechanics of Solids 25, no. 8 (April 27, 2020): 1589–609. http://dx.doi.org/10.1177/1081286520911073.

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We derive the equations of nonlinear electroelastostatics using three different variational formulations involving the deformation function and an independent field variable representing the electric character – considering either the electric field [Formula: see text], the electric displacement [Formula: see text] or the electric polarization [Formula: see text]. The first variation of the energy functional results in the set of Euler–Lagrange partial differential equations, which are the equilibrium equations, boundary conditions and certain constitutive equations for the electroelastic system. The partial differential equations for obtaining the bifurcation point have also been found using the bilinear functional based on the second variation. We show that the well-known Maxwell stress in a vacuum is a natural outcome of the derivation of equations from the variational principles and does not depend on the formulation used. As a result of careful analysis, it is found that there are certain terms in the bifurcation equation that appear to be difficult to obtain using ordinary perturbation-based analysis of the Euler–Lagrange equation. From a practical viewpoint, the formulations based on [Formula: see text] and [Formula: see text] result in simpler equations and are expected to be more suitable for analysing problems of stability as well as post-buckling behaviour.

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20

Iqbal, Naveed, and Ranchao Wu. "Pattern formation by fractional cross-diffusion in a predator–prey model with Beddington–DeAngelis type functional response." International Journal of Modern Physics B 33, no. 25 (October 10, 2019): 1950296. http://dx.doi.org/10.1142/s0217979219502965.

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In this paper, we explore the emergence of patterns in a fractional cross-diffusion model with Beddington–DeAngelis type functional response. First, we explore the stability of the equilibrium points with or without fractional cross-diffusion. Instability of equilibria can be induced by cross-diffusion. We perform the linear stability analysis to obtain the constraints for the Turing instability. It is found by theoretical analysis that cross-diffusion is an important mechanism for the appearance of Turing patterns. For the dynamics of pattern, the weakly nonlinear multi-scaling analysis has been performed to obtain the amplitude equations. Finally, we ensure the existence of Turing patterns such as squares, spots and stripes by using the stability analysis of the amplitude equations. Moreover, with the assistance of numerical simulations, we verify the theoretical results.

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Chen, Mengye, Liang You, Jie Tang, Shasha Su, and Ruiming Zhang. "Analysis of a Viral Infection Model with Delayed Nonlytic Immune Response." Discrete Dynamics in Nature and Society 2015 (2015): 1–11. http://dx.doi.org/10.1155/2015/235420.

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We investigate the dynamical behavior of a virus infection model with delayed nonlytic immune response. By analyzing corresponding characteristic equations, the local stabilities of two boundary equilibria are established. By using suitable Lyapunov functional and LaSalle’s invariance principle, we establish the global stability of the infection-free equilibrium. We find that the infection free equilibriumE0is globally asymptotically stable whenR0⩽1, and the infected equilibrium without immunityE1is local asymptotically stable when1<R0⩽1+bβ/cd. Under the conditionR0>1+bβ/cdwe obtain the sufficient conditions to the local stability of the infected equilibrium with immunityE2. We show that the time delay can change the stability ofE2and lead to the existence of Hopf bifurcations. The stabilities of bifurcating periodic solutions are studied and numerical simulations to our theorems are provided.

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Huang, T., and S. Chucheepsakul. "Large Displacement Analysis of a Marine Riser." Journal of Energy Resources Technology 107, no. 1 (March 1, 1985): 54–59. http://dx.doi.org/10.1115/1.3231163.

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A method of static analysis for a marine riser experiencing large displacements is presented. The method is suitable for analyzing a riser having a known top tension and a possible slippage at the top slip joint. Utilizing the stationary condition of a functional coupled with an equilibrium equation, one can conveniently obtain the equilibrium configuration numerically. The configuration is expressed in terms of the rectangular coordinates. The functional representing the energy and work of the riser system is expressed in terms of the horizontal coordinate which is parameterized in terms of the vertical depth instead of arc length. For a two-dimensional problem, two multipliers must be included in the functional. One of the two represents the variable axial force along the length of the riser and the other corresponds to the strain energy per unit riser length due to bending. Utilizing the finite element method, a numerical procedure to obtain the configuration of static equilibrium is given. The resulting algebraic equations are highly nonlinear and the Newton-Raphson iterative procedure is used to solve the equations. An example is given.

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23

Zhao, Huitao, Yiping Lin, and Yunxian Dai. "Stability and Global Hopf Bifurcation Analysis on a Ratio-Dependent Predator-Prey Model with Two Time Delays." Abstract and Applied Analysis 2013 (2013): 1–15. http://dx.doi.org/10.1155/2013/321930.

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A ratio-dependent predator-prey model with two time delays is studied. By means of an iteration technique, sufficient conditions are obtained for the global attractiveness of the positive equilibrium. By comparison arguments, the global stability of the semitrivial equilibrium is addressed. By using the theory of functional equation and Hopf bifurcation, the conditions on which positive equilibrium exists and the quality of Hopf bifurcation are given. Using a global Hopf bifurcation result of Wu (1998) for functional differential equations, the global existence of the periodic solutions is obtained. Finally, an example for numerical simulations is also included.

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Zuo, Wenjie, and Junjie Wei. "Stability and bifurcation in a ratio-dependent Holling-III system with diffusion and delay." Nonlinear Analysis: Modelling and Control 19, no. 1 (January 20, 2014): 132–53. http://dx.doi.org/10.15388/na.2014.1.9.

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A diffusive ratio-dependent predator-prey system with Holling-III functional response and delay effects is considered. Global stability of the boundary equilibrium and the stability of the unique positive steady state and the existence of spatially homogeneous and inhomogeneous periodic solutions are investigated in detail, by the maximum principle and the characteristic equations. Ratio-dependent functional response exhibits rich spatiotemporal patterns. It is found that, the system without delay is dissipative and uniformly permanent under certain conditions, the delay can destabilize the positive constant equilibrium and spatial Hopf bifurcations occur as the delay crosses through some critical values. Then, the direction and the stability of Hopf bifurcations are determined by applying the center manifold reduction and the normal form theory for partial functional differential equations. Some numerical simulations are carried out to illustrate the theoretical results.

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Mei, Hongwei, and Jiongmin Yong. "Equilibrium strategies for time-inconsistent stochastic switching systems." ESAIM: Control, Optimisation and Calculus of Variations 25 (2019): 64. http://dx.doi.org/10.1051/cocv/2018051.

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An optimal control problem is considered for a stochastic differential equation containing a state-dependent regime switching, with a recursive cost functional. Due to the non-exponential discounting in the cost functional, the problem is time-inconsistent in general. Therefore, instead of finding a global optimal control (which is not possible), we look for a time-consistent (approximately) locally optimal equilibrium strategy. Such a strategy can be represented through the solution to a system of partial differential equations, called an equilibrium Hamilton–Jacob–Bellman (HJB) equation which is constructed via a sequence of multi-person differential games. A verification theorem is proved and, under proper conditions, the well-posedness of the equilibrium HJB equation is established as well.

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Kolesnichenko, Aleksandr Vladimirovich. "On the construction of a family of anomalous-diffusion Fokker–Planck−Kolmogorov’s equations based on the Sharma–Taneja–Mittal entropy functional." Mathematica Montisnigri 51 (August 2021): 74–95. http://dx.doi.org/10.20948/mathmontis-2021-51-6.

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A logical scheme for constructing thermodynamics of anomalous stochastic systems based on the nonextensive two-parameter (κ, ς) -entropy of Sharma–Taneja–Mittal (SHTM) is considered. Thermodynamics within the framework (2 - q) -statistics of Tsallis was constructed, which belongs to the STM family of statistics. The approach of linear nonequilibrium thermodynamics to the construction of a family of nonlinear equations of Fokker−Planck−Kolmogorov (FPK), is used, correlated with the entropy of the STM, in which the stationary solution of the diffusion equation coincides with the corresponding generalized Gibbs distribution obtained from the extremality (κ, ς) - entropy condition of a non-additive stochastic system. Taking into account the convexity property of the Bregman divergence, it was shown that the principle of maximum equilibrium entropy is valid for (κ, ς) - systems, and also was proved the H - theorem determining the direction of the time evolution of the non-equilibrium state of the system. This result is extended also to non-equilibrium systems that evolve to a stationary state in accordance with the nonlinear FPK equation. The method of the ansatz- approach for solving non-stationary FPK equations is considered, which allows us to find the time dependence of the probability density distribution function for non-equilibrium anomalous systems. Received diffusive equations FPК can be used, in particular, at the analysis of diffusion of every possible epidemics and pandemics. The obtained diffusion equations of the FPK can be used, in particular, in the analysis of the spread of various epidemics and pandemics.

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An, Qiguang, and Qingfeng Zhu. "Partially Observed Nonzero-Sum Differential Game of BSDEs with Delay and Applications." Mathematical Problems in Engineering 2020 (June 19, 2020): 1–10. http://dx.doi.org/10.1155/2020/3518961.

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A class of partially observed nonzero-sum differential games for backward stochastic differential equations with time delays is studied, in which both game system and cost functional involve the time delays of state variables and control variables under each participant with different observation equations. A necessary condition (maximum principle) for the Nash equilibrium point to this kind of partially observed game is established, and a sufficient condition (verification theorem) for the Nash equilibrium point is given. A partially observed linear quadratic game is taken as an example to illustrate the application of the maximum principle.

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Wu, Jianhong, and H. I. Freedman. "Monotone Semiflows Generated by Neutral Functional Differential Equations With Application to Compartmental Systems." Canadian Journal of Mathematics 43, no. 5 (October 1, 1991): 1098–120. http://dx.doi.org/10.4153/cjm-1991-064-1.

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AbstractThis paper is devoted to the machinery necessary to apply the general theory of monotone dynamical systems to neutral functional differential equations. We introduce an ordering structure for the phase space, investigate its compatibility with the usual uniform convergence topology, and develop several sufficient conditions of strong monotonicity of the solution semiflows to neutral equations. By applying some general results due to Hirsch and Matano for monotone dynamical systems to neutral equations, we establish several (generic) convergence results and an equivalence theorem of the order stability and convergence of precompact orbits. These results are applied to show that each orbit of a closed biological compartmental system is convergent to a single equilibrium.

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Montanino, Andrea, Gianluca Alaimo, and Ettore Lanzarone. "A gradient-based optimization method with functional principal component analysis for efficient structural topology optimization." Structural and Multidisciplinary Optimization 64, no. 1 (March 25, 2021): 177–88. http://dx.doi.org/10.1007/s00158-021-02872-9.

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AbstractStructural topology optimization (STO) is usually treated as a constrained minimization problem, which is iteratively addressed by solving the equilibrium equations for the problem under consideration. To reduce the computational effort, several reduced basis approaches that solve the equilibrium equations in a reduced space have been proposed. In this work, we apply functional principal component analysis (FPCA) to generate the reduced basis, and we couple FPCA with a gradient-based optimization method for the first time in the literature. The proposed algorithm has been tested on a large STO problem with 4.8 million degrees of freedom. Results show that the proposed algorithm achieves significant computational time savings with negligible loss of accuracy. Indeed, the density maps obtained with the proposed algorithm capture the larger features of maps obtained without reduced basis, but in significantly lower computational times, and are associated with similar values of the minimized compliance.

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Moon, Jun, and Wonhee Kim. "Explicit Characterization of Feedback Nash Equilibria for Indefinite, Linear-Quadratic, Mean-Field-Type Stochastic Zero-Sum Differential Games with Jump-Diffusion Models." Mathematics 8, no. 10 (September 28, 2020): 1669. http://dx.doi.org/10.3390/math8101669.

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We consider the indefinite, linear-quadratic, mean-field-type stochastic zero-sum differential game for jump-diffusion models (I-LQ-MF-SZSDG-JD). Specifically, there are two players in the I-LQ-MF-SZSDG-JD, where Player 1 minimizes the objective functional, while Player 2 maximizes the same objective functional. In the I-LQ-MF-SZSDG-JD, the jump-diffusion-type state dynamics controlled by the two players and the objective functional include the mean-field variables, i.e., the expected values of state and control variables, and the parameters of the objective functional do not need to be (positive) definite matrices. These general settings of the I-LQ-MF-SZSDG-JD make the problem challenging, compared with the existing literature. By considering the interaction between two players and using the completion of the squares approach, we obtain the explicit feedback Nash equilibrium, which is linear in state and its expected value, and expressed as the coupled integro-Riccati differential equations (CIRDEs). Note that the interaction between the players is analyzed via a class of nonanticipative strategies and the “ordered interchangeability” property of multiple Nash equilibria in zero-sum games. We obtain explicit conditions to obtain the Nash equilibrium in terms of the CIRDEs. We also discuss the different solvability conditions of the CIRDEs, which lead to characterization of the Nash equilibrium for the I-LQ-MF-SZSDG-JD. Finally, our results are applied to the mean-field-type stochastic mean-variance differential game, for which the explicit Nash equilibrium is obtained and the simulation results are provided.

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31

Wang, Lingshu, and Guanghui Feng. "Global Stability and Hopf Bifurcation of a Predator-Prey Model with Time Delay and Stage Structure." Journal of Applied Mathematics 2014 (2014): 1–10. http://dx.doi.org/10.1155/2014/431671.

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A delayed predator-prey system with Holling type II functional response and stage structure for both the predator and the prey is investigated. By analyzing the corresponding characteristic equations, the local stability of each of the feasible equilibria of the system is addressed and the existence of a Hopf bifurcation at the coexistence equilibrium is established. By means of persistence theory on infinite dimensional systems, it is proved that the system is permanent. By using Lyapunov functions and the LaSalle invariant principle, the global stability of each of the feasible equilibria of the model is discussed. Numerical simulations are carried out to illustrate the main theoretical results.

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32

Fruman, Mark D., and Theodore G. Shepherd. "Symmetric Stability of Compressible Zonal Flows on a Generalized Equatorial β Plane." Journal of the Atmospheric Sciences 65, no. 6 (June 1, 2008): 1927–40. http://dx.doi.org/10.1175/2007jas2582.1.

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Abstract Sufficient conditions are derived for the linear stability with respect to zonally symmetric perturbations of a steady zonal solution to the nonhydrostatic compressible Euler equations on an equatorial β plane, including a leading order representation of the Coriolis force terms due to the poleward component of the planetary rotation vector. A version of the energy–Casimir method of stability proof is applied: an invariant functional of the Euler equations linearized about the equilibrium zonal flow is found, and positive definiteness of the functional is shown to imply linear stability of the equilibrium. It is shown that an equilibrium is stable if the potential vorticity has the same sign as latitude and the Rayleigh centrifugal stability condition that absolute angular momentum increase toward the equator on surfaces of constant pressure is satisfied. The result generalizes earlier results for hydrostatic and incompressible systems and for systems that do not account for the nontraditional Coriolis force terms. The stability of particular equilibrium zonal velocity, entropy, and density fields is assessed. A notable case in which the effect of the nontraditional Coriolis force is decisive is the instability of an angular momentum profile that decreases away from the equator but is flatter than quadratic in latitude, despite its satisfying both the centrifugal and convective stability conditions.

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33

Peng, Miao, and Zhengdi Zhang. "Bifurcation analysis and control of a delayed stage-structured predator–prey model with ratio-dependent Holling type III functional response." Journal of Vibration and Control 26, no. 13-14 (December 30, 2019): 1232–45. http://dx.doi.org/10.1177/1077546319892144.

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A delayed stage-structured predator–prey model with ratio-dependent Holling type III functional response is proposed and explored in this study. We discuss the positivity and the existence of equilibrium points. By choosing time delay as the bifurcation parameter and analyzing the relevant characteristic equations, the local stability of the trivial equilibrium, the predator-extinction equilibrium, and the coexistence equilibrium of the system is investigated. In accordance with the normal form method and center manifold theorem, the property analysis of Hopf bifurcation of the system is obtained. Furthermore, for the purpose of protecting the stability of such a biological system, a hybrid control method is presented to control the Hopf bifurcation. Finally, numerical examples are given to verify the theoretical findings.

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34

HU, GUANG-PING, WAN-TONG LI, and XIANG-PING YAN. "HOPF BIFURCATION AND STABILITY OF PERIODIC SOLUTIONS IN THE DELAYED LIÉNARD EQUATION." International Journal of Bifurcation and Chaos 18, no. 10 (October 2008): 3147–57. http://dx.doi.org/10.1142/s0218127408022317.

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In this paper, the classical Liénard equation with a discrete delay is considered. Under the assumption that the classical Liénard equation without delay has a unique stable trivial equilibrium, we consider the effect of the delay on the stability of zero equilibrium. It is found that the increase of delay not only can change the stability of zero equilibrium but can also lead to the occurrence of periodic solutions near the zero equilibrium. Furthermore, the stability of bifurcated periodic solutions is investigated by applying the normal form theory and center manifold reduction for functional differential equations. Finally, in order to verify these theoretical conclusions, some numerical simulations are given.

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35

Jiang, Zhichao, Wenzhi Zhang, Jing Zhang, and Tongqian Zhang. "Dynamical Analysis of a Phytoplankton–Zooplankton System with Harvesting Term and Holling III Functional Response." International Journal of Bifurcation and Chaos 28, no. 13 (December 12, 2018): 1850162. http://dx.doi.org/10.1142/s0218127418501626.

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A toxin-producing phytoplankton and zooplankton system is investigated. Considering that zooplankton can be harvested for food in some bodies of water, the harvesting term is introduced to zooplankton population. Firstly, from the ordinary differential equation (ODE) system, we obtain the global asymptotic stability of equilibrium and optimal capture problem. Secondly, based on the ODE system, the diffusion term is introduced and the global asymptotic stability of the steady state solution is obtained. As a result, the diffusion cannot affect the global asymptotic stability of equilibrium, and Turing instability cannot occur. Once again, a delayed differential equation (DDE) system is put forward. The global asymptotic stability of boundary equilibrium and the existence of local Hopf bifurcation at positive equilibrium are discussed. Furthermore, it is proved that there exists at least one positive periodic solution as delay varies in some region by using the global Hopf result of Wu for functional differential equations. Lastly, some numerical simulations are carried out for supporting the theoretical analyses and the positive impacts of harvesting effort, and the release rate of toxin is given. The unstable interval of the positive equilibrium becomes smaller and smaller with the increase of harvesting effort or the release rate of toxin.

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36

Vorotnikov, V. I. "On Problem of Partial Stability for Functional Differential Systems with Holdover." Mekhatronika, Avtomatizatsiya, Upravlenie 20, no. 7 (July 4, 2019): 398–404. http://dx.doi.org/10.17587/mau.20.398-404.

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The theory of systems of functional differential equations is a significant and rapidly developing sphere of modern mathematics which finds extensive application in complex systems of automatic control and also in economic, modern technical, ecological, and biological models. Naturally, the problems arises of stability and partial stability of the processes described by the class of the equation. The article studies the problem of partial stability which arise in applications either from the requirement of proper performance of a system or in assessing system capability. Also very effective is the approach to the problem of stability with respect to all variables based on preliminary analysis of partial stability. We suppose that the system have the zero equilibrium position. A conditions are obtained under which the uniform stability (uniform asymptotic stability) of the zero equilibrium position with respect to the part of the variables implies the uniform stability (uniform asymptotic stability) of this equilibrium position with respect to the other, larger part of the variables, which include an additional group of coordinates of the phase vector. These conditions include: 1) the condition for uniform asymptotic stability of the zero equilibrium position of the "reduced" subsystem of the original system with respect to the additional group of variables; 2) the restriction on the coupling between the "reduced" subsystem and the rest parts of the system. Application of the obtained results to a problem of stabilization with respect to a part of the variables for nonlinear controlled systems is discussed.

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Liu, Junli, and Tailei Zhang. "Stability and Hopf Bifurcation Analysis of a Plant Virus Propagation Model with Two Delays." Discrete Dynamics in Nature and Society 2018 (2018): 1–12. http://dx.doi.org/10.1155/2018/7126135.

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To understand the interaction between the insects and the plants, a system of delay differential equations is proposed and studied. We prove that if R0≤1, the disease-free equilibrium is globally asymptotically stable for any length of time delays by constructing a Lyapunov functional, and the system admits a unique endemic equilibrium if R0>1. We establish the sufficient conditions for the stability of the endemic equilibrium and existence of Hopf bifurcation. Using the normal form theory and center manifold theorem, the explicit formulae which determine the stability, direction, and other properties of bifurcating periodic solutions are derived. Some numerical simulations are given to confirm our analytic results.

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Zheng, ZuoHuan, and XiLiang Li. "Necessary and sufficient conditions for the existence of equilibrium in abstract non-autonomous functional differential equations." Science China Mathematics 53, no. 8 (June 16, 2010): 2045–59. http://dx.doi.org/10.1007/s11425-010-3012-0.

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39

Xie, Xiaoliang, and Wen Zhang. "Hopf bifurcations in a three-species food chain system with multiple delays." Open Mathematics 15, no. 1 (April 26, 2017): 508–19. http://dx.doi.org/10.1515/math-2017-0039.

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Abstract This paper is concerned with a three-species Lotka-Volterra food chain system with multiple delays. By linearizing the system at the positive equilibrium and analyzing the associated characteristic equation, the stability of the positive equilibrium and existence of Hopf bifurcations are investigated. Furthermore, the direction of bifurcations and the stability of bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem for functional differential equations. Finally, some numerical simulations are carried out for illustrating the theoretical results.

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40

Kanatnikov, A. N., and A. P. Krishchenko. "Functional Method of Localization and LaSalle Invariance Principle." Mathematics and Mathematical Modeling, no. 1 (May 4, 2021): 1–12. http://dx.doi.org/10.24108/mathm.0121.0000256.

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A functional method of localization has proved to be good in solving the qualitative analysis problems of dynamic systems. Proposed in the 90s, it was intensively used when studying a number of well-known systems of differential equations, both of autonomous and of non-autonomous discrete systems, including systems that involve control and / or disturbances.The method essence is to construct a set containing all invariant compact sets in the phase space of a dynamical system. A concept of the invariant compact set includes equilibrium positions, limit cycles, attractors, repellers, and other structures in the phase space of a system that play an important role in describing the behavior of a dynamical system. The constructed set is called localizing and represents an external assessment of the appropriate structures in the phase space.Relatively recently, it was found that the functional localization method allows one to analyze a behavior of the dynamical system trajectories. In particular, the localization method can be used to check the stability of the equilibrium positions.Here naturally emerges an issue of the relationship between the functional localization method and the well-known La Salle invariance principle, which can be regarded as a further development of the method of Lyapunov functions for establishing stability. The article discusses this issue.

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41

Yang, Peng, and Yuanshi Wang. "Periodic Solutions of a Delayed Eco-Epidemiological Model with Infection-Age Structure and Holling Type II Functional Response." International Journal of Bifurcation and Chaos 30, no. 01 (January 2020): 2050011. http://dx.doi.org/10.1142/s021812742050011x.

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This paper is devoted to the study of a new delayed eco-epidemiological model with infection-age structure and Holling type II functional response. Firstly, the disease transmission rate function among the predator population is treated as the piecewise function concerning the incubation period [Formula: see text] of the epidemic disease and the model is rewritten as an abstract nondensely defined Cauchy problem. Besides, the prerequisite which guarantees the presence of the coexistence equilibrium is achieved. Secondly, via utilizing the theory of integrated semigroup and the Hopf bifurcation theorem for semilinear equations with nondense domain, it is found that the model exhibits a Hopf bifurcation near the coexistence equilibrium, which suggests that this model has a nontrivial periodic solution that bifurcates from the coexistence equilibrium as the bifurcation parameter [Formula: see text] crosses the bifurcation critical value [Formula: see text]. That is, there is a continuous periodic oscillation phenomenon. Finally, some numerical simulations are shown to support and extend the analytical results and visualize the interesting phenomenon.

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42

Bai, Yuzhen, and Xiaopeng Zhang. "Stability and Hopf Bifurcation in a Diffusive Predator-Prey System with Beddington-DeAngelis Functional Response and Time Delay." Abstract and Applied Analysis 2011 (2011): 1–22. http://dx.doi.org/10.1155/2011/463721.

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This paper is concerned with a diffusive predator-prey system with Beddington-DeAngelis functional response and delay effect. By analyzing the distribution of the eigenvalues, the stability of the positive equilibrium and the existence of spatially homogeneous and spatially inhomogeneous periodic solutions are investigated. Also, it is shown that the small diffusion can affect the Hopf bifurcations. Finally, the direction and stability of Hopf bifurcations are determined by normal form theory and center manifold reduction for partial functional differential equations.

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43

Masoumi, S., M. Akhlaghi, and M. Salehi. "Multi-scale analysis of viscoelastic–viscoplastic laminated composite plates using generalized differential quadrature method." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 227, no. 7 (October 25, 2012): 1406–16. http://dx.doi.org/10.1177/0954406212464929.

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Multi-scale analysis of laminated composite plates with viscoelastic–viscoplastic behavior of matrix is studied. Simplified unit cell method is developed to derive a new formulation for analysis of composite materials, including viscoelastic–viscoplastic matrix. The viscoelastic behavior of the matrix is modeled using Boltzmann superposition principle and the creep compliance is modeled using Prony series. Zapas–Crissman functional model is applied to obtain viscoplastic behavior of the matrix. In structural level, equations of equilibrium of laminated composite plate in terms of displacements have been derived using first order shear deformation theory with von Karman kinematic nonlinearity type. The nonlinear equations of equilibrium of plate are solved using generalized differential quadrature method. The details of the multi-scale analysis process have been discussed. Results include the effect of different parameters on creep behavior of composite materials in microscale and also micro-macro analysis.

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44

Zhang, Xuebing, and Hongyong Zhao. "Harvest control for a delayed stage-structured diffusive predator–prey model." International Journal of Biomathematics 10, no. 01 (November 15, 2016): 1750004. http://dx.doi.org/10.1142/s1793524517500048.

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In this paper, we have considered a delayed stage-structured diffusive prey–predator model, in which predator is assumed to undergo exploitation. By using the theory of partial functional differential equations, the local stability of an interior equilibrium is established and the existence of Hopf bifurcations at the interior equilibrium is also discussed. By applying the normal form and the center manifold theory, an explicit algorithm to determine the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is derived. Finally, the complex dynamics are obtained and numerical simulations substantiate the analytical results.

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45

Xiao, Zaowang, Zhong Li, Zhenliang Zhu, and Fengde Chen. "Hopf bifurcation and stability in a Beddington-DeAngelis predator-prey model with stage structure for predator and time delay incorporating prey refuge." Open Mathematics 17, no. 1 (March 26, 2019): 141–59. http://dx.doi.org/10.1515/math-2019-0014.

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Abstract In this paper, we consider a Beddington-DeAngelis predator-prey system with stage structure for predator and time delay incorporating prey refuge. By analyzing the characteristic equations, we study the local stability of the equilibrium of the system. Using the delay as a bifurcation parameter, the model undergoes a Hopf bifurcation at the coexistence equilibrium when the delay crosses some critical values. After that, by constructing a suitable Lyapunov functional, sufficient conditions are derived for the global stability of the system. Finally, the influence of prey refuge on densities of prey species and predator species is discussed.

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46

Bracken, Paul. "Shape equations for two-dimensional manifolds with nonempty boundary based on a variational method." International Journal of Geometric Methods in Modern Physics 17, no. 06 (April 30, 2020): 2050082. http://dx.doi.org/10.1142/s0219887820500826.

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A smooth surface is considered which has a curved boundary. A system of exterior differential forms is introduced which describes the surface and boundary curves completely in the moving frame approach. A total free energy functional is defined based on these forms for which an equilibrium equation and boundary conditions of the surface are derived by calculating the variation of the total free energy. These results can be applied to a surface with several freely exposed edges.

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47

BERETTA, E., P. FERGOLA, and C. TENNERIELLO. "CHEMOSTAT EQUATIONS FOR A PREDATOR-PREY CHAIN WITH DELAYED NUTRIENT RECYCLING." Journal of Biological Systems 03, no. 02 (June 1995): 483–94. http://dx.doi.org/10.1142/s0218339095000459.

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The problem of the equilibrium stability for the chemostat equation is considered in the special case of a biotic species feeding on one limiting nutrient and predated by another biotic species. Both the biotic species through the decomposition process can return with delay to the chemostat fraction of dead biomass as new nutrient. The delay kernels of nutrient recycling are assumed to be general L2(0, +∞) non-negative functions which admit up to second order finite moments. Two approaches are adopted: the first one applies when both the biotic species have a self-regulating term in their evolution equations and can be worked out without linearizing the equations. This approach does not require any further constraint on delay kernels, whereas it introduces constraints on the remaining parameters of the model. The second approach applies to the linearized equations when the predator self-regulating term is set equal to zero. In this case the stability condition requires a constraint on the average time delays in the recycling processes. Both the approaches are performed by constructing suitable Krasovskii-Lyapunov functionals for the related functional retarded differential equations.

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48

ZHANG, CUN-HUA, and XIANG-PING YAN. "STABILITY AND HOPF BIFURCATIONS IN A DELAYED PREDATOR–PREY SYSTEM WITH A DISTRIBUTED DELAY." International Journal of Bifurcation and Chaos 19, no. 07 (July 2009): 2283–94. http://dx.doi.org/10.1142/s0218127409024062.

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This paper is concerned with a delayed Lotka–Volterra two-species predator–prey system with a distributed delay. By linearizing the system at the positive equilibrium and analyzing the associated characteristic equation, the asymptotic stability of positive equilibrium is investigated and Hopf bifurcations are demonstrated. It is found that the positive equilibrium of the system is always locally asymptotically stable when the delay kernel is the weak kernel while there is a stability switch of positive equilibrium when the delay kernel is the strong kernel and the system can undergo a Hopf bifurcation at the positive equilibrium when the average time delay in the delay kernel crosses certain critical values. In particular, by applying the normal form theory and center manifold reduction to functional differential equations (FDEs), the explicit formula determining the direction of Hopf bifurcations and the stability of bifurcated periodic solutions is given. Finally, some numerical simulations are also included to support the analytical results obtained.

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SAMANTA, G. P., A. Mondal, D. Sahoo, and P. Dolai. "A prey-predator system with herd behaviour of prey in a rapidly fluctuating environment." Mathematics in Applied Sciences and Engineering 1, no. 1 (December 6, 2019): 16–26. http://dx.doi.org/10.5206/mase/8196.

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A statistical theory of non-equilibrium fluctuation in damped Volterra-Lotka prey-predator system where prey population lives in herd in a rapidly fluctuating random environment has been presented. The method is based on the technique of perturbation approximation of non-linear coupled stochastic differential equations. The characteristic of group-living of prey population has been emphasized using square root of prey density in the functional response.

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Di Francesco, Marco, Klemens Fellner, and Peter A. Markowich. "The entropy dissipation method for spatially inhomogeneous reaction–diffusion-type systems." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 464, no. 2100 (August 21, 2008): 3273–300. http://dx.doi.org/10.1098/rspa.2008.0214.

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We study the long-time asymptotics of reaction–diffusion-type systems that feature a monotone decaying entropy (Lyapunov, free energy) functional. We consider both bounded domains and confining potentials on the whole space for arbitrary space dimensions. Our aim is to derive quantitative expressions for (or estimates of) the rates of convergence towards an (entropy minimizing) equilibrium state in terms of the constants of diffusion and reaction and with respect to conserved quantities. Our method, the so-called entropy approach, seeks to quantify convergence to equilibrium by using functional inequalities, which relate quantitatively the entropy and its dissipation in time. The entropy approach is well suited to nonlinear problems and known to be quite robust with respect to model variations. It has already been widely applied to scalar diffusion–convection equations, and the main goal of this paper is to study its generalization to systems of partial differential equations that contain diffusion and reaction terms and admit fewer conservation laws than the size of the system. In particular, we successfully apply the entropy approach to general linear systems and to a nonlinear example of a reaction–diffusion–convection system arising in solid-state physics as a paradigm for general nonlinear systems.

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Journal articles on the topic 'Functional equilibrium equations'

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