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Автор: Grafiati

Опубліковано: 8 лютого 2024

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1

Cioranescu, D., A. Damlamian, and G. Griso. "The Periodic Unfolding Method in Homogenization." SIAM Journal on Mathematical Analysis 40, no. 4 (January 2008): 1585–620. http://dx.doi.org/10.1137/080713148.

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2

Cioranescu, D., A. Damlamian, P. Donato, G. Griso, and R. Zaki. "The Periodic Unfolding Method in Domains with Holes." SIAM Journal on Mathematical Analysis 44, no. 2 (January 2012): 718–60. http://dx.doi.org/10.1137/100817942.

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3

DIMINNIE, DAVID C., and RICHARD HABERMAN. "ACTION AND PERIOD OF HOMOCLINIC AND PERIODIC ORBITS FOR THE UNFOLDING OF A SADDLE-CENTER BIFURCATION." International Journal of Bifurcation and Chaos 13, no. 11 (November 2003): 3519–30. http://dx.doi.org/10.1142/s0218127403008569.

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Анотація:

At a saddle-center bifurcation for Hamiltonian systems, the homoclinic orbit is cusp shaped at the nonlinear nonhyperbolic saddle point. Near but before the bifurcation, orbits are periodic corresponding to the unfolding of the homoclinic orbit, while after the bifurcation a double homoclinic orbit is formed with a local and global branch. The saddle-center bifurcation is dynamically unfolded due to a slowly varying potential. Near the unfolding of the homoclinic orbit, the period and action are analyzed. Asymptotic expansions for the action, period and dissipation are obtained in an overlap region near the homoclinic orbit of the saddle-center bifurcation. In addition, the unfoldings of the action and dissipation functions associated with zero energy orbits (periodic and homoclinic) near the saddle-center bifurcation are determined using the method of matched asymptotic expansions for integrals.

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4

Cioranescu, Doina, Alain Damlamian, and Riccardo De Arcangelis. "Homogenization of Quasiconvex Integrals via the Periodic Unfolding Method." SIAM Journal on Mathematical Analysis 37, no. 5 (January 2006): 1435–53. http://dx.doi.org/10.1137/040620898.

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5

Cioranescu, Doina, Alain Damlamian, and Riccardo De Arcangelis. "Homogenization of nonlinear integrals via the periodic unfolding method." Comptes Rendus Mathematique 339, no. 1 (July 2004): 77–82. http://dx.doi.org/10.1016/j.crma.2004.03.028.

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6

Avila, Jake, and Bituin Cabarrubias. "Periodic unfolding method for domains with very small inclusions." Electronic Journal of Differential Equations 2023, no. 01-?? (December 20, 2023): 85. http://dx.doi.org/10.58997/ejde.2023.85.

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Анотація:

This work creates a version of the periodic unfolding method suitable for domains with very small inclusions in \(\mathbb{R}^N\) for \(N\geq 3\). In the first part, we explore the properties of the associated operators. The second part involves the application of the method in obtaining the asymptotic behavior of a stationary heat dissipation problem depending on the parameter \( \gamma < 0\). In particular, we consider the cases when \(\gamma \in (-1,0)\), \( \gamma < -1\) and \(\gamma = -1\). We also include here the corresponding corrector results for the solution of the problem, to complete the homogenization process. For more information see https://ejde.math.txstate.edu/Volumes/2023/85/abstr.html

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7

Sánchez-Ochoa, F., Francisco Hidalgo, Miguel Pruneda, and Cecilia Noguez. "Unfolding method for periodic twisted systems with commensurate Moiré patterns." Journal of Physics: Condensed Matter 32, no. 2 (October 17, 2019): 025501. http://dx.doi.org/10.1088/1361-648x/ab44f0.

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8

Ptashnyk, Mariya. "Locally Periodic Unfolding Method and Two-Scale Convergence on Surfaces of Locally Periodic Microstructures." Multiscale Modeling & Simulation 13, no. 3 (January 2015): 1061–105. http://dx.doi.org/10.1137/140978405.

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9

Cioranescu, D., A. Damlamian, G. Griso, and D. Onofrei. "The periodic unfolding method for perforated domains and Neumann sieve models." Journal de Mathématiques Pures et Appliquées 89, no. 3 (March 2008): 248–77. http://dx.doi.org/10.1016/j.matpur.2007.12.008.

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10

Donato, P., K. H. Le Nguyen, and R. Tardieu. "The periodic unfolding method for a class of imperfect transmission problems." Journal of Mathematical Sciences 176, no. 6 (July 13, 2011): 891–927. http://dx.doi.org/10.1007/s10958-011-0443-2.

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11

Donato, Patrizia, and ZhanYing Yang. "The periodic unfolding method for the heat equation in perforated domains." Science China Mathematics 59, no. 5 (December 5, 2015): 891–906. http://dx.doi.org/10.1007/s11425-015-5103-4.

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12

Ganghoffer, Jean-François, Gérard Maurice, and Yosra Rahali. "Determination of closed form expressions of the second-gradient elastic moduli of multi-layer composites using the periodic unfolding method." Mathematics and Mechanics of Solids 24, no. 5 (November 9, 2018): 1475–502. http://dx.doi.org/10.1177/1081286518798873.

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Анотація:

The present paper aims at introducing a homogenization scheme for the identification of strain–gradient elastic moduli of composite materials, based on the unfolding mathematical method. We expose in the first part of this paper the necessary mathematical apparatus in view of the derivation of the effective first- and second-gradient mechanical properties of two-phase composite materials, focusing on a one-dimensional situation. Each of the two phases is supposed to obey a second-gradient linear elastic constitutive law. Application of the unfolding method to the homogenization of multi-layer materials provides closed form expressions of all effective first- and second-gradient elastic moduli as well as coupling moduli between first- and second-gradient elasticity. A comparison between the unfolding method and the method of oscillating functions shows that both methods, despite their differences, deliver the same effective second-gradient elastic constitutive law for stratified materials.

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13

Ene, Horia, and Claudia Timofte. "Microstructure models for composites with imperfect interface via the periodic unfolding method." Asymptotic Analysis 89, no. 1-2 (2014): 111–22. http://dx.doi.org/10.3233/asy-141239.

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14

Cabarrubias, Bituin. "Homogenization of optimal control problems in perforated domains via periodic unfolding method." Applicable Analysis 95, no. 11 (October 13, 2015): 2517–34. http://dx.doi.org/10.1080/00036811.2015.1094799.

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15

Cioranescu, Doina, Alain Damlamian, and Riccardo De Arcangelis. "Homogenization of integrals with pointwise gradient constraints via the periodic unfolding method." Ricerche di Matematica 55, no. 1 (July 2006): 31–54. http://dx.doi.org/10.1007/s11587-006-0003-0.

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16

Mohammed, Mogtaba. "Homogenization of nonlinear hyperbolic problem with a dynamical boundary condition." AIMS Mathematics 8, no. 5 (2023): 12093–108. http://dx.doi.org/10.3934/math.2023609.

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Анотація:

<abstract><p>In this work, we look at homogenization results for nonlinear hyperbolic problem with a non-local boundary condition. We use the periodic unfolding method to obtain a homogenized nonlinear hyperbolic equation in a fixed domain. Due to the investigation's peculiarity, the unfolding technique must be developed with special attention, creating an unusual two-scale model. We note that the non-local boundary condition caused a damping on the homogenized model.</p></abstract>

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17

Coatléven, Julien. "Mathematical justification of macroscopic models for diffusion MRI through the periodic unfolding method." Asymptotic Analysis 93, no. 3 (June 30, 2015): 219–58. http://dx.doi.org/10.3233/asy-151294.

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18

Arrieta, José M., and Manuel Villanueva-Pesqueira. "Unfolding Operator Method for Thin Domains with a Locally Periodic Highly Oscillatory Boundary." SIAM Journal on Mathematical Analysis 48, no. 3 (January 2016): 1634–71. http://dx.doi.org/10.1137/15m101600x.

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19

Yang, Zhanying. "The periodic unfolding method for a class of parabolic problems with imperfect interfaces." ESAIM: Mathematical Modelling and Numerical Analysis 48, no. 5 (July 28, 2014): 1279–302. http://dx.doi.org/10.1051/m2an/2013139.

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20

Mohammed, Mogtaba. "Homogenization and correctors for linear stochastic equations via the periodic unfolding methods." Stochastics and Dynamics 19, no. 05 (August 19, 2019): 1950040. http://dx.doi.org/10.1142/s0219493719500400.

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Анотація:

In this paper, we use the periodic unfolding method and Prokhorov’s and Skorokhod’s probabilistic compactness results to obtain homogenization and corrector results for stochastic partial differential equations (PDEs) with periodically oscillating coefficients. We show the convergence of the solutions of the original problems to the solutions of the homogenized problems. In contrast to the two-scale convergence method, the corrector results obtained in this paper are without any additional regularity assumptions on the solutions of the original problems

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21

OULD-HAMMOUDA, AMAR, and RACHAD ZAKI. "Homogenization of a class of elliptic problems with nonlinear boundary conditions in domains with small holes." Carpathian Journal of Mathematics 31, no. 1 (2015): 77–88. http://dx.doi.org/10.37193/cjm.2015.01.09.

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Анотація:

We consider a class of second order elliptic problems in a domain of RN , N > 2, ε-periodically perforated by holes of size r(ε) , with r(ε)/ε → 0 as ε → 0. A nonlinear Robin-type condition is prescribed on the boundary of some holes while on the boundary of the others as well as on the external boundary of the domain, a Dirichlet condition is imposed. We are interested in the asymptotic behavior of the solutions as ε → 0. We use the periodic unfolding method introduced in [Cioranescu, D., Damlamian, A. and Griso, G., Periodic unfolding and homogenization, C. R. Acad. Sci. Paris, Ser. I, 335 (2002), 99–104]. The method allows us to consider second order operators with highly oscillating coefficients and so, to generalize the results of [Cioranescu, D., Donato, P. and Zaki, R., Asymptotic behavior of elliptic problems in perforated domains with nonlinear boundary conditions, Asymptot. Anal., Vol. 53 (2007), No. 4, 209–235].

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22

Li, Qunhong, Pu Chen, and Jieqiong Xu. "Codimension-Two Grazing Bifurcations in Three-Degree-of-Freedom Impact Oscillator with Symmetrical Constraints." Discrete Dynamics in Nature and Society 2015 (2015): 1–15. http://dx.doi.org/10.1155/2015/353581.

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This paper investigates the codimension-two grazing bifurcations of a three-degree-of-freedom vibroimpact system with symmetrical rigid stops since little research can be found on this important issue. The criterion for existence of double grazing periodic motion is presented. Using the classical discontinuity mapping method, the Poincaré mapping of double grazing periodic motion is obtained. Based on it, the sufficient condition of codimension-two bifurcation of double grazing periodic motion is formulated, which is simplified further using the Jacobian matrix of smooth Poincaré mapping. At the end, the existence regions of different types of periodic-impact motions in the vicinity of the codimension-two grazing bifurcation point are displayed numerically by unfolding diagram and phase diagrams.

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23

Aiyappan, Srinivasan, Giuseppe Cardone, Carmen Perugia, and Ravi Prakash. "Homogenization of a nonlinear monotone problem in a locally periodic domain via unfolding method." Nonlinear Analysis: Real World Applications 66 (August 2022): 103537. http://dx.doi.org/10.1016/j.nonrwa.2022.103537.

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24

Zaki, Rachad. "Homogenization of a Stokes problem in a porous medium by the periodic unfolding method." Asymptotic Analysis 79, no. 3-4 (2012): 229–50. http://dx.doi.org/10.3233/asy-2012-1094.

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25

Graf, Isabell, and Malte A. Peter. "A convergence result for the periodic unfolding method related to fast diffusion on manifolds." Comptes Rendus Mathematique 352, no. 6 (June 2014): 485–90. http://dx.doi.org/10.1016/j.crma.2014.03.002.

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26

Yang, Zhanying. "Homogenization and correctors for the hyperbolic problems with imperfect interfaces via the periodic unfolding method." Communications on Pure & Applied Analysis 13, no. 1 (2014): 249–72. http://dx.doi.org/10.3934/cpaa.2014.13.249.

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27

CAPATINA, ANCA, and HORIA ENE. "Homogenisation of the Stokes problem with a pure non-homogeneous slip boundary condition by the periodic unfolding method." European Journal of Applied Mathematics 22, no. 4 (February 21, 2011): 333–45. http://dx.doi.org/10.1017/s0956792511000088.

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Анотація:

We study the homogenisation of the Stokes system with a non-homogeneous Fourier boundary condition on the boundary of the holes, depending on a parameter γ. Such systems arise in the modelling of the flow of an incompressible viscous fluid through a porous medium under the influence of body forces. At the limit, by using the periodic unfolding method in perforated domains, we obtain, following the values of γ, different Darcy's laws of typeMu= −N∇p+Fwith suitable matricesMandNwithFdepending on the right-hand side in the bulk term and in the boundary condition.

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28

TORRESI, A. M., G. L. CALANDRINI, P. A. BONFILI, and J. L. MOIOLA. "GENERALIZED HOPF BIFURCATION IN A FREQUENCY DOMAIN FORMULATION." International Journal of Bifurcation and Chaos 22, no. 08 (August 2012): 1250197. http://dx.doi.org/10.1142/s0218127412501970.

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The multiplicity problem of limit cycles arising from a weak focus is addressed. The proposed methodology is a combination of the frequency domain method to handle some degenerate Hopf bifurcations with the powerful tools of the singularity theory. The frequency domain approach uses the harmonic balance method to study the existence of periodic solutions. On the other hand, the singularity theory provides the conditions and formulas for the classification problem of the unfolding of the singularity in terms of the distinguished and auxiliary parameters. A classical example introduced by Bautin is shown in which the multiplicity of limit cycles is recovered by using this type of hybrid methodology and standard software in the continuation of periodic solutions (LOCBIF and XPPAUT). For small amplitude limit cycles, the proposed methodology gives accurate results.

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29

Zappale, Elvira. "A note on dimension reduction for unbounded integrals with periodic microstructure via the unfolding method for slender domains." Evolution Equations & Control Theory 6, no. 2 (2017): 299–318. http://dx.doi.org/10.3934/eect.2017016.

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30

Mohammed, Mogtaba, and Noor Ahmed. "Homogenization and correctors of Robin problem for linear stochastic equations in periodically perforated domains." Asymptotic Analysis 120, no. 1-2 (October 6, 2020): 123–49. http://dx.doi.org/10.3233/asy-191582.

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Анотація:

In this paper, we present homogenization and corrector results for stochastic linear parabolic equations in periodically perforated domains with non-homogeneous Robin conditions on the holes. We use the periodic unfolding method and probabilistic compactness results. Homogenization results presented in this paper are stochastic counterparts of some fundamental work given in [Cioranescu, Donato and Zaki in Port. Math. (N.S.) 63 (2006), 467–496]. We show that the sequence of solutions of the original problem converges in suitable topologies to the solution of a homogenized problem, which is a parabolic stochastic equation in fixed domain with Dirichlet condition on the boundary. In contrast to the two scale convergence method, the corrector results obtained in this paper are without any additional regularity assumptions on the solutions of the original problems.

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31

Gentile, Franco S., and Jorge L. Moiola. "Hopf Bifurcation Analysis of Distributed Delay Equations with Applications to Neural Networks." International Journal of Bifurcation and Chaos 25, no. 11 (October 2015): 1550156. http://dx.doi.org/10.1142/s0218127415501564.

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In this paper, we study how to capture smooth oscillations arising from delay-differential equations with distributed delays. For this purpose, we introduce a modified version of the frequency-domain method based on the Graphical Hopf Bifurcation Theorem. Our approach takes advantage of a simple interpretation of the distributed delay effect by means of some Laplace-transformed properties. Our theoretical results are illustrated through an example of two coupled neurons with distributed delay in their communication channel. For this system, we compute several bifurcation diagrams and approximations of the amplitudes of periodic solutions. In addition, we establish analytical conditions for the appearance of a double zero bifurcation and investigate the unfolding by the proposed methodology.

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32

Belyamoun, M. H., and S. Zouhdi. "On the modeling of effective constitutive parameters of bianisotropic media by a periodic unfolding method in time and frequency domains." Applied Physics A 103, no. 3 (January 22, 2011): 881–87. http://dx.doi.org/10.1007/s00339-011-6250-2.

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33

Griso, Georges, Larysa Khilkova, Julia Orlik, and Olena Sivak. "Homogenization of Perforated Elastic Structures." Journal of Elasticity 141, no. 2 (June 5, 2020): 181–225. http://dx.doi.org/10.1007/s10659-020-09781-w.

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Анотація:

Abstract The paper is dedicated to the asymptotic behavior of $\varepsilon$ ε -periodically perforated elastic (3-dimensional, plate-like or beam-like) structures as $\varepsilon \to 0$ ε → 0 . In case of plate-like or beam-like structures the asymptotic reduction of dimension from $3D$ 3 D to $2D$ 2 D or $1D$ 1 D respectively takes place. An example of the structure under consideration can be obtained by a periodic repetition of an elementary “flattened” ball or cylinder for plate-like or beam-like structures in such a way that the contact surface between two neighboring balls/cylinders has a non-zero measure. Since the domain occupied by the structure might have a non-Lipschitz boundary, the classical homogenization approach based on the extension cannot be used. Therefore, for obtaining Korn’s inequalities, which are used for the derivation of a priori estimates, we use the approach based on interpolation. In case of plate-like and beam-like structures the proof of Korn’s inequalities is based on the displacement decomposition for a plate or a beam, respectively. In order to pass to the limit as $\varepsilon \to 0$ ε → 0 we use the periodic unfolding method.

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34

Nassar, H., A. Lebée, and L. Monasse. "Curvature, metric and parametrization of origami tessellations: theory and application to the eggbox pattern." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 473, no. 2197 (January 2017): 20160705. http://dx.doi.org/10.1098/rspa.2016.0705.

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Анотація:

Origami tessellations are particular textured morphing shell structures. Their unique folding and unfolding mechanisms on a local scale aggregate and bring on large changes in shape, curvature and elongation on a global scale. The existence of these global deformation modes allows for origami tessellations to fit non-trivial surfaces thus inspiring applications across a wide range of domains including structural engineering, architectural design and aerospace engineering. The present paper suggests a homogenization-type two-scale asymptotic method which, combined with standard tools from differential geometry of surfaces, yields a macroscopic continuous characterization of the global deformation modes of origami tessellations and other similar periodic pin-jointed trusses. The outcome of the method is a set of nonlinear differential equations governing the parametrization, metric and curvature of surfaces that the initially discrete structure can fit. The theory is presented through a case study of a fairly generic example: the eggbox pattern. The proposed continuous model predicts correctly the existence of various fittings that are subsequently constructed and illustrated.

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35

Wang, Meiqi, Wenli Ma, Enli Chen, and Yujian Chang. "Study on a Class of Piecewise Nonlinear Systems with Fractional Delay." Shock and Vibration 2021 (October 7, 2021): 1–13. http://dx.doi.org/10.1155/2021/3411390.

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In this paper, a dynamic model of piecewise nonlinear system with fractional-order time delay is simplified. The amplitude frequency response equation of the dynamic model of piecewise nonlinear system with fractional-order time delay under periodic excitation is obtained by using the average method. It is found that the amplitude of the system changes when the external excitation frequency changes. At the same time, the amplitude frequency response characteristics of the system under different time delay parameters, different fractional-order parameters, and coefficient are studied. By analyzing the amplitude frequency response characteristics, the influence of time delay and fractional-order parameters on the stability of the system is analyzed in this paper, and the bifurcation equations of the system are studied by using the theory of continuity. The transition sets under different piecewise states and the constrained bifurcation behaviors under the corresponding unfolding parameters are obtained. The variation of the bifurcation topology of the system with the change of system parameters is given.

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36

Li, Songtao, Qunhong Li, and Zhongchuan Meng. "Dynamic Behaviors of a Two-Degree-of-Freedom Impact Oscillator with Two-Sided Constraints." Shock and Vibration 2021 (April 1, 2021): 1–14. http://dx.doi.org/10.1155/2021/8854115.

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Анотація:

The dynamic model of a vibroimpact system subjected to harmonic excitation with symmetric elastic constraints is investigated with analytical and numerical methods. The codimension-one bifurcation diagrams with respect to frequency of the excitation are obtained by means of the continuation technique, and the different types of bifurcations are detected, such as grazing bifurcation, saddle-node bifurcation, and period-doubling bifurcation, which predicts the complexity of the system considered. Based on the grazing phenomenon obtained, the zero-time-discontinuity mapping is extended from the single constraint system presented in the literature to the two-sided elastic constraint system discussed in this paper. The Poincare mapping of double grazing periodic motion is derived, and this compound mapping is applied to obtain the existence conditions of codimension-two grazing bifurcation point of the system. According to the deduced theoretical result, the grazing curve and the codimension-two grazing bifurcation points are validated by numerical simulation. Finally, various types of periodic-impact motions near the codimension-two grazing bifurcation point are illustrated through the unfolding diagram and phase diagrams.

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37

Martin, Sébastien. "Influence of Multiscale Roughness Patterns in Cavitated Flows: Applications to Journal Bearings." Mathematical Problems in Engineering 2008 (2008): 1–26. http://dx.doi.org/10.1155/2008/439319.

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Анотація:

This paper deals with the coupling of two major problems in lubrication theory: cavitation phenomena and roughness of the surfaces in relative motion. Cavitation is defined as the rupture of the continuous film due to the formation of air bubbles, leading to the presence of a liquid-gas mixture. For this, the Elrod-Adams model (which is a pressure-saturation model) is classically used to describe the behavior of a cavitated thin film flow. In addition, in practical situations, the surfaces of the devices are rough, due to manufacturing processes which induce defaults. Thus, we study the behavior of the solution, when highly oscillating roughness effects on the rigid surfaces occur. In particular, we deal with the reiterated homogenization of this Elrod-Adams problem, using periodic unfolding methods. A numerical simulation illustrates the behavior of the solution. Although the pressure tends to a smooth one, the saturation oscillations are not damped. This does not prevent us from defining an equivalent homogenized saturation and highlights the anisotropic effects on the saturation function in cavitated areas.

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38

CIORANESCU, D. "Homogenization of nonlinear integrals via the periodic unfolding method." Comptes Rendus Mathematique, May 2004. http://dx.doi.org/10.1016/s1631-073x(04)00165-7.

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39

Mohammed, Mogtaba, and Waseem Asghar Khan. "Homogenization and Correctors for Stochastic Hyperbolic Equations in Domains with Periodically Distributed Holes." Journal of Multiscale Modelling 12, no. 03 (September 2021). http://dx.doi.org/10.1142/s1756973721500086.

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Анотація:

The goal of this paper is to present new results on homogenization and correctors for stochastic linear hyperbolic equations in periodically perforated domains with homogeneous Neumann conditions on the holes. The main tools are the periodic unfolding method, energy estimates, probabilistic and deterministic compactness results. The findings of this paper are stochastic counterparts of the celebrated work [D. Cioranescu, P. Donato and R. Zaki, The periodic unfolding method in perforated domains, Port. Math. (N.S.) 63 (2006) 467–496]. The convergence of the solution of the original problem to a homogenized problem with Dirichlet condition has been shown in suitable topologies. Homogenization and convergence of the associated energies results recover the work in [M. Mohammed and M. Sango, Homogenization of Neumann problem for hyperbolic stochastic partial differential equations in perforated domains, Asymptot. Anal. 97 (2016) 301–327]. In addition to that, we obtain corrector results.

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40

Li, Yanqiu, and Lei Zhang. "Bifurcations in a General Delay Sel’kov–Schnakenberg Reaction–Diffusion System." International Journal of Bifurcation and Chaos 33, no. 16 (December 30, 2023). http://dx.doi.org/10.1142/s021812742350195x.

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Анотація:

The dynamics of a delay Sel’kov–Schnakenberg reaction–diffusion system are explored. The existence and the occurrence conditions of the Turing and the Hopf bifurcations of the system are found by taking the diffusion coefficient and the time delay as the bifurcation parameters. Based on that, the existence of codimension-2 bifurcations including Turing–Turing, Hopf–Hopf and Turing–Hopf bifurcations are given. Using the center manifold theory and the normal form method, the universal unfolding of the Turing–Hopf bifurcation at the positive constant steady-state is demonstrated. According to the universal unfolding, a Turing–Hopf bifurcation diagram is shown under a set of specific parameters. Furthermore, in different parameter regions, we find the existence of the spatially inhomogeneous steady-state, the spatially homogeneous and inhomogeneous periodic solutions. Discretization of time and space visualizes these spatio-temporal solutions. In particular, near the critical point of Hopf–Hopf bifurcation, the spatially homogeneous periodic and inhomogeneous quasi-periodic solutions are found numerically.

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41

Bader, Fakhrielddine, Mostafa Bendahmane, Mazen Saad, and Raafat Talhouk. "Microscopic tridomain model of electrical activity in the heart with dynamical gap junctions. Part 2 – Derivation of the macroscopic tridomain model by unfolding homogenization method." Asymptotic Analysis, September 8, 2022, 1–32. http://dx.doi.org/10.3233/asy-221804.

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We study the homogenization of a novel microscopic tridomain system, allowing for a more detailed analysis of the properties of cardiac conduction than the classical bidomain and monodomain models. In (Acta Appl.Math. 179 (2022) 1–35), we detail this model in which gap junctions are considered as the connections between adjacent cells in cardiac muscle and could serve as alternative or supporting pathways for cell-to-cell electrical signal propagation. Departing from this microscopic cellular model, we apply the periodic unfolding method to derive the macroscopic tridomain model. Several difficulties prevent the application of unfolding homogenization results, including the degenerate temporal structure of the tridomain equations and a nonlinear dynamic boundary condition on the cellular membrane. To prove the convergence of the nonlinear terms, especially those defined on the microscopic interface, we use the boundary unfolding operator and a Kolmogorov–Riesz compactness’s result.

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42

Raimondi, Federica. "Homogenization of a class of singular elliptic problems in two-component domains." Asymptotic Analysis, June 6, 2022, 1–27. http://dx.doi.org/10.3233/asy-221783.

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This paper deals with the homogenization of a quasilinear elliptic problem having a singular lower order term and posed in a two-component domain with an ε-periodic imperfect interface. We prescribe a Dirichlet condition on the exterior boundary, while we assume that the continuous heat flux is proportional to the jump of the solution on the interface via a function of order ε γ . We prove an homogenization result for − 1 < γ < 1 by means of the periodic unfolding method (see SIAM J. Math. Anal. 40 (2008) 1585–1620 and The Periodic Unfolding Method (2018) Springer), adapted to two-component domains in (J. Math. Sci. 176 (2011) 891–927). One of the main tools in the homogenization process is a convergence result for a suitable auxiliary linear problem, associated with the weak limit of the sequence { u ε } of the solutions, as ε → 0. More precisely, our result shows that the gradient of u ε behaves like that of the solution of the auxiliary problem, which allows us to pass to the limit in the quasilinear term, and to study the singular term near its singularity, via an accurate a priori estimate.

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43

Neukamm, Stefan, Mario Varga, and Marcus Waurick. "Two-scale homogenization of abstract linear time-dependent PDEs." Asymptotic Analysis, November 10, 2020, 1–41. http://dx.doi.org/10.3233/asy-201654.

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Many time-dependent linear partial differential equations of mathematical physics and continuum mechanics can be phrased in the form of an abstract evolutionary system defined on a Hilbert space. In this paper we discuss a general framework for homogenization (periodic and stochastic) of such systems. The method combines a unified Hilbert space approach to evolutionary systems with an operator theoretic reformulation of the well-established periodic unfolding method in homogenization. Regarding the latter, we introduce a well-structured family of unitary operators on a Hilbert space that allows to describe and analyze differential operators with rapidly oscillating (possibly random) coefficients. We illustrate the approach by establishing periodic and stochastic homogenization results for elliptic partial differential equations, Maxwell’s equations, and the wave equation.

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44

Donato, Patrizia, and Iulian Ţenţea. "Homogenization of an elastic double-porosity medium with imperfect interface via the periodic unfolding method." Boundary Value Problems 2013, no. 1 (December 2013). http://dx.doi.org/10.1186/1687-2770-2013-265.

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45

Pei, Lijun, and Chenyu Wang. "Periodic, Quasi-Periodic and Phase-Locked Oscillations and Stability in the Fiscal Dynamical Model with Tax Collection and Decision-Making Delays." International Journal of Bifurcation and Chaos 31, no. 16 (December 20, 2021). http://dx.doi.org/10.1142/s0218127421502473.

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In this paper, we consider the complex dynamics of a fiscal dynamical model, which was improved from Wolfstetter classical growth cycle model by Sportelli et al. The main work of the present paper is to study the impact of fiscal policy delays on the national income adjustment processes using a dynamical method, such as double Hopf bifurcation analysis. We first use DDE-BIFTOOL to find the double Hopf bifurcation points of the system, and draw the bifurcation diagrams with two bifurcation parameters, i.e. the tax collection delay [Formula: see text] and the public expenditure decision-making delay [Formula: see text]. Then we employ the method of multiple scales to obtain two amplitude equations. By analyzing these amplitude equations, we derive the classification and unfolding of these double Hopf bifurcation points. And three types of double Hopf bifurcations are found. Finally, we verify the results by numerical simulations. We find complex dynamic behaviors of the system via the analytical method, such as stable equilibrium, stable periodic, quasi-periodic and phase-locked solutions in respective regions. The dynamical phenomena can help policy makers to choose a proper range of the delays so that they could effectively formulate fiscal policies to stabilize the economy.

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46

Ma, Hongru, and Yanbin Tang. "Homogenization of a semilinear elliptic problem in a thin composite domain with an imperfect interface." Mathematical Methods in the Applied Sciences, August 19, 2023. http://dx.doi.org/10.1002/mma.9628.

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In this paper, we consider the asymptotic behavior of a semilinear elliptic problem in a thin two‐composite domain with an imperfect interface, where the flux is discontinuous. For this thin domain, both the height and the period are of order . We first use Minty–Browder theorem to prove the well‐posedness of the problem and then apply the periodic unfolding method to obtain the limit problems and some corrector results for three cases of a real parameter , and , respectively. To deal with the semilinear terms, the extension operator and the averaged function are used.

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47

Amar, M., D. Andreucci, and C. Timofte. "Interface potential in composites with general imperfect transmission conditions." Zeitschrift für angewandte Mathematik und Physik 74, no. 5 (September 20, 2023). http://dx.doi.org/10.1007/s00033-023-02094-7.

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AbstractThe model analyzed in this paper has its origins in the description of composites made by a hosting medium containing a periodic array of inclusions coated by a thin layer consisting of sublayers of two different materials. This two-phase coating material is such that the external part has a low diffusivity in the orthogonal direction, while the internal one has high diffusivity along the tangential direction. In a previous paper (Amar in IFB 21:41–59, 2019), by means of a concentration procedure, the internal layer was replaced by an imperfect interface. The present paper is concerned with the concentration of the external coating layer and the homogenization, via the periodic unfolding method, of the resulting model, which is far from being a standard one. Despite the fact that the limit problem looks like a classical Dirichlet problem for an elliptic equation, in the construction of the homogenized matrix and of the source term, a very delicate analysis is required.

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48

Grant Kirkland, W., and S. C. Sinha. "Symbolic Computation of Quantities Associated With Time-Periodic Dynamical Systems1." Journal of Computational and Nonlinear Dynamics 11, no. 4 (May 13, 2016). http://dx.doi.org/10.1115/1.4033382.

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Many dynamical systems can be modeled by a set of linear/nonlinear ordinary differential equations with periodic time-varying coefficients. The state transition matrix (STM) Φ(t,α), associated with the linear part of the equation, can be expressed in terms of the periodic Lyapunov–Floquét (L-F) transformation matrix Q(t,α) and a time-invariant matrix R(α) containing a set of symbolic system parameters α. Computation of Q(t,α) and R(α) in symbolic form as a function of α is of paramount importance in stability, bifurcation analysis, and control system design. In earlier studies, since Q(t,α) and R(α) were available only in numerical forms, general results for parameter unfolding and control system design could not be obtained in the entire parameter space. In 2009, an attempt was made by Butcher et al. (2009, “Magnus' Expansion for Time-Periodic Systems: Parameter Dependent Approximations,” Commun. Nonlinear Sci. Numer. Simul., 14(12), pp. 4226–4245) to compute the Q(t,α) matrix in a symbolic form using the Magnus expansions with some success. In this work, an efficient technique for symbolic computation of Q(t,α) and R(α) matrices is presented. First, Φ(t,α) is computed symbolically using the shifted Chebyshev polynomials and Picard iteration method as suggested in the literature. Then, R(α) is computed using a Gaussian quadrature integral formula. Finally, Q(t,α) is computed using the matrix exponential summation method. Using mathematica, this approach has successfully been applied to the well-known Mathieu equation and a four-dimensional time-periodic system in order to demonstrate the applications of the proposed method to linear as well as nonlinear problems.

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49

Nandakumaran, Akambadath, and Abu Sufian. "Oscillating PDE in a rough domain with a curved interface: homogenization of an optimal control problem." ESAIM: Control, Optimisation and Calculus of Variations, July 21, 2020. http://dx.doi.org/10.1051/cocv/2020045.

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Homogenization of an elliptic PDE with periodic oscillating coefficients and an associated optimal control problems with energy type cost functional is considered. The domain is a 3-dimensional region (method applies to any $n$ dimensional region) with oscillating boundary, where the base of the oscillation is curved and it is given by a Lipschitz function. Further, we consider a general elliptic PDE with oscillating coefficients. We also include very general type cost functional of Dirichlet type given with oscillating coefficients which can be different from the coefficient matrix of the equation. We introduce appropriate unfolding operators and approximate unfolded domain to study the limiting analysis. The present article is new in this generality.

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50

Yu, Guodong, Zewen Wu, Zhen Zhan, Mikhail I. Katsnelson, and Shengjun Yuan. "Dodecagonal bilayer graphene quasicrystal and its approximants." npj Computational Materials 5, no. 1 (December 2019). http://dx.doi.org/10.1038/s41524-019-0258-0.

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AbstractDodecagonal bilayer graphene quasicrystal has 12-fold rotational order but lacks translational symmetry which prevents the application of band theory. In this paper, we study the electronic and optical properties of graphene quasicrystal with large-scale tight-binding calculations involving more than ten million atoms. We propose a series of periodic approximants which reproduce accurately the properties of quasicrystal within a finite unit cell. By utilizing the band-unfolding method on the smallest approximant with only 2702 atoms, the effective band structure of graphene quasicrystal is derived. The features, such as the emergence of new Dirac points (especially the mirrored ones), the band gap at $$M$$M point and the Fermi velocity are all in agreement with recent experiments. The properties of quasicrystal states are identified in the Landau level spectrum and optical excitations. Importantly, our results show that the lattice mismatch is the dominant factor determining the accuracy of layered approximants. The proposed approximants can be used directly for other layered materials in honeycomb lattice, and the design principles can be applied for any quasi-periodic incommensurate structures.

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