
Previous Article
Observable optimal state points of subadditive potentials
 DCDS Home
 This Issue

Next Article
Entropy of endomorphisms of Lie groups
Attractors for differential equations with multiple variable delays
1.  Dpto. Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, Campus Reina Mercedes, Apdo. de Correos 1160, 41080 Sevilla 
2.  Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom 
References:
[1] 
T. Caraballo, G. Lukaszewicz and J. Real, Pullback attractors for asymptotically compact nonautonomous dynamical systems, Nonlinear Anal., 64 (2006), 484498. Google Scholar 
[2] 
T. Caraballo, P. MarínRubio and J. Valero, Autonomous and nonautonomous attractors for differential equations with delays, J. Differential Equations, 208 (2005), 941. Google Scholar 
[3] 
Tomás Caraballo, José A. Langa and James C. Robinson, Attractors for differential equations with variable delays, J. Math. Anal. Appl., 260 (2001), 421438. Google Scholar 
[4] 
Huabin Chen, Impulsiveintegral inequality and exponential stability for stochastic partial differential equations with delays, Statist. Probab. Lett., 80 (2010), 5056. Google Scholar 
[5] 
Hans Crauel and Franco Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365393. Google Scholar 
[6] 
Jack K. Hale, "Asymptotic Behavior of Dissipative Systems," Mathematical Surveys and Monographs, 25, American Mathematical Society, Providence, RI, 1988. Google Scholar 
[7] 
Jack K. Hale and Sjoerd M. Verduyn Lunel, "Introduction to FunctionalDifferential Equations," volume 99 of Applied Mathematical Sciences. SpringerVerlag, New York, 1993. Google Scholar 
[8] 
Gábor Kiss and Bernd Krauskopf, Stability implications of delay distribution for firstorder and secondorder systems, Discrete Contin. Dyn. Syst. Ser. B, 13 (2010), 327345. Google Scholar 
[9] 
Gábor Kiss and Bernd Krauskopf, Stabilizing effect of delay distribution for a class of secondorder systems without instantaneous feedback, Dynamical Systems: An International Journal, 26 (2011), 85101. Google Scholar 
[10] 
Gábor Kiss and JeanPhilippe Lessard, Computational fixed point theory for differential delay equations with multiple time lags, J. Differential Equations, 252 (2012), 30933115. Google Scholar 
[11] 
P. E. Kloeden, Pullback attractors of nonautonomous semidynamical systems, Stoch. Dyn., 3 (2003), 101112. Google Scholar 
[12] 
Tibor Krisztin, Global dynamics of delay differential equations, Periodica Mathematica Hungarica, 56 (2008) 8395. Google Scholar 
[13] 
Tibor Krisztin, HansOtto Walther and Jianhong Wu, "Shape, Smoothness and Invariant Stratification of an Attracting Set for Delayed Monotone Positive Feedback," volume 11 of Fields Institute Monographs. American Mathematical Society, Providence, RI, 1999. Google Scholar 
[14] 
Michal Křížek, Numerical experience with the finite speed of gravitational interaction, Math. Comput. Simulation, 50 (1999), 237245. Modelling '98 (Prague). Google Scholar 
[15] 
Yang Kuang, "Delay Differential Equations with Applications in Population Dynamics," volume 191 of Mathematics in Science and Engineering. Academic Press Inc., Boston, MA, 1993. Google Scholar 
[16] 
Pedro MarínRubio and José Real, On the relation between two different concepts of pullback attractors for nonautonomous dynamical systems, Nonlinear Anal., 71 (2009), 39563963. Google Scholar 
[17] 
Roger D. Nussbaum, Differentialdelay equations with two time lags, Mem. Amer. Math. Soc., 16 (1978), vi+62. Google Scholar 
[18] 
Roger D. Nussbaum, Functional differential equations, in "Handbook of Dynamical Systems," 2, 461499. NorthHolland, Amsterdam, 2002. Google Scholar 
[19] 
Martin Rasmussen, "Attractivity and Bifurcation for Nonautonomous Dynamical Systems," volume 1907 of Lecture Notes in Mathematics. Springer, Berlin, 2007. Google Scholar 
[20] 
B. Schmalfuss, Backward cocycles and attractors of stochastic differential equations, in "International Seminar on Applied MathematicsNonlinear Dynamics: Attractor Approximation and Global Behaviour," 185192. Dresden, 1992. Google Scholar 
[21] 
George R. Sell, Nonautonomous differential equations and topological dynamics. I. The basic theory, Trans. Amer. Math. Soc., 127 (1967), 241262. Google Scholar 
[22] 
George R. Sell, Nonautonomous differential equations and topological dynamics. II. Limiting equations, Trans. Amer. Math. Soc., 127 (1967), 263283. Google Scholar 
[23] 
H. O. Walther, Dynamics of delay differential equations, in "Delay Differential Equations and Applications," 205 of NATO Sci. Ser. II Math. Phys. Chem., 411476. Springer, Dordrecht, 2006. Google Scholar 
show all references
References:
[1] 
T. Caraballo, G. Lukaszewicz and J. Real, Pullback attractors for asymptotically compact nonautonomous dynamical systems, Nonlinear Anal., 64 (2006), 484498. Google Scholar 
[2] 
T. Caraballo, P. MarínRubio and J. Valero, Autonomous and nonautonomous attractors for differential equations with delays, J. Differential Equations, 208 (2005), 941. Google Scholar 
[3] 
Tomás Caraballo, José A. Langa and James C. Robinson, Attractors for differential equations with variable delays, J. Math. Anal. Appl., 260 (2001), 421438. Google Scholar 
[4] 
Huabin Chen, Impulsiveintegral inequality and exponential stability for stochastic partial differential equations with delays, Statist. Probab. Lett., 80 (2010), 5056. Google Scholar 
[5] 
Hans Crauel and Franco Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365393. Google Scholar 
[6] 
Jack K. Hale, "Asymptotic Behavior of Dissipative Systems," Mathematical Surveys and Monographs, 25, American Mathematical Society, Providence, RI, 1988. Google Scholar 
[7] 
Jack K. Hale and Sjoerd M. Verduyn Lunel, "Introduction to FunctionalDifferential Equations," volume 99 of Applied Mathematical Sciences. SpringerVerlag, New York, 1993. Google Scholar 
[8] 
Gábor Kiss and Bernd Krauskopf, Stability implications of delay distribution for firstorder and secondorder systems, Discrete Contin. Dyn. Syst. Ser. B, 13 (2010), 327345. Google Scholar 
[9] 
Gábor Kiss and Bernd Krauskopf, Stabilizing effect of delay distribution for a class of secondorder systems without instantaneous feedback, Dynamical Systems: An International Journal, 26 (2011), 85101. Google Scholar 
[10] 
Gábor Kiss and JeanPhilippe Lessard, Computational fixed point theory for differential delay equations with multiple time lags, J. Differential Equations, 252 (2012), 30933115. Google Scholar 
[11] 
P. E. Kloeden, Pullback attractors of nonautonomous semidynamical systems, Stoch. Dyn., 3 (2003), 101112. Google Scholar 
[12] 
Tibor Krisztin, Global dynamics of delay differential equations, Periodica Mathematica Hungarica, 56 (2008) 8395. Google Scholar 
[13] 
Tibor Krisztin, HansOtto Walther and Jianhong Wu, "Shape, Smoothness and Invariant Stratification of an Attracting Set for Delayed Monotone Positive Feedback," volume 11 of Fields Institute Monographs. American Mathematical Society, Providence, RI, 1999. Google Scholar 
[14] 
Michal Křížek, Numerical experience with the finite speed of gravitational interaction, Math. Comput. Simulation, 50 (1999), 237245. Modelling '98 (Prague). Google Scholar 
[15] 
Yang Kuang, "Delay Differential Equations with Applications in Population Dynamics," volume 191 of Mathematics in Science and Engineering. Academic Press Inc., Boston, MA, 1993. Google Scholar 
[16] 
Pedro MarínRubio and José Real, On the relation between two different concepts of pullback attractors for nonautonomous dynamical systems, Nonlinear Anal., 71 (2009), 39563963. Google Scholar 
[17] 
Roger D. Nussbaum, Differentialdelay equations with two time lags, Mem. Amer. Math. Soc., 16 (1978), vi+62. Google Scholar 
[18] 
Roger D. Nussbaum, Functional differential equations, in "Handbook of Dynamical Systems," 2, 461499. NorthHolland, Amsterdam, 2002. Google Scholar 
[19] 
Martin Rasmussen, "Attractivity and Bifurcation for Nonautonomous Dynamical Systems," volume 1907 of Lecture Notes in Mathematics. Springer, Berlin, 2007. Google Scholar 
[20] 
B. Schmalfuss, Backward cocycles and attractors of stochastic differential equations, in "International Seminar on Applied MathematicsNonlinear Dynamics: Attractor Approximation and Global Behaviour," 185192. Dresden, 1992. Google Scholar 
[21] 
George R. Sell, Nonautonomous differential equations and topological dynamics. I. The basic theory, Trans. Amer. Math. Soc., 127 (1967), 241262. Google Scholar 
[22] 
George R. Sell, Nonautonomous differential equations and topological dynamics. II. Limiting equations, Trans. Amer. Math. Soc., 127 (1967), 263283. Google Scholar 
[23] 
H. O. Walther, Dynamics of delay differential equations, in "Delay Differential Equations and Applications," 205 of NATO Sci. Ser. II Math. Phys. Chem., 411476. Springer, Dordrecht, 2006. Google Scholar 
[1] 
Suping Wang, Qiaozhen Ma. Existence of pullback attractors for the nonautonomous suspension bridge equation with time delay. Discrete & Continuous Dynamical Systems  B, 2020, 25 (4) : 12991316. doi: 10.3934/dcdsb.2019221 
[2] 
Mirelson M. Freitas, Anderson J. A. Ramos, Manoel J. Dos Santos, Eraldo R. N. Fonseca. Attractors and pullback dynamics for nonautonomous piezoelectric system with magnetic and thermal effects. Communications on Pure & Applied Analysis, , () : . doi: 10.3934/cpaa.2021129 
[3] 
XueLi Song, YanRen Hou. Pullback $\mathcal{D}$attractors for the nonautonomous NewtonBoussinesq equation in twodimensional bounded domain. Discrete & Continuous Dynamical Systems, 2012, 32 (3) : 9911009. doi: 10.3934/dcds.2012.32.991 
[4] 
Bo You, Yanren Hou, Fang Li, Jinping Jiang. Pullback attractors for the nonautonomous quasilinear complex GinzburgLandau equation with $p$Laplacian. Discrete & Continuous Dynamical Systems  B, 2014, 19 (6) : 18011814. doi: 10.3934/dcdsb.2014.19.1801 
[5] 
Flank D. M. Bezerra, Vera L. Carbone, Marcelo J. D. Nascimento, Karina Schiabel. Pullback attractors for a class of nonautonomous thermoelastic plate systems. Discrete & Continuous Dynamical Systems  B, 2018, 23 (9) : 35533571. doi: 10.3934/dcdsb.2017214 
[6] 
Zhijian Yang, Yanan Li. Upper semicontinuity of pullback attractors for nonautonomous Kirchhoff wave equations. Discrete & Continuous Dynamical Systems  B, 2019, 24 (9) : 48994912. doi: 10.3934/dcdsb.2019036 
[7] 
Peter E. Kloeden, Jacson Simsen. Pullback attractors for nonautonomous evolution equations with spatially variable exponents. Communications on Pure & Applied Analysis, 2014, 13 (6) : 25432557. doi: 10.3934/cpaa.2014.13.2543 
[8] 
Rodrigo Samprogna, Tomás Caraballo. Pullback attractor for a dynamic boundary nonautonomous problem with Infinite Delay. Discrete & Continuous Dynamical Systems  B, 2018, 23 (2) : 509523. doi: 10.3934/dcdsb.2017195 
[9] 
T. Caraballo, J. A. Langa, J. Valero. Structure of the pullback attractor for a nonautonomous scalar differential inclusion. Discrete & Continuous Dynamical Systems  S, 2016, 9 (4) : 979994. doi: 10.3934/dcdss.2016037 
[10] 
Iacopo P. Longo, Sylvia Novo, Rafael Obaya. Topologies of continuity for Carathéodory delay differential equations with applications in nonautonomous dynamics. Discrete & Continuous Dynamical Systems, 2019, 39 (9) : 54915520. doi: 10.3934/dcds.2019224 
[11] 
Tomás Caraballo, P.E. Kloeden. Nonautonomous attractors for integrodifferential evolution equations. Discrete & Continuous Dynamical Systems  S, 2009, 2 (1) : 1736. doi: 10.3934/dcdss.2009.2.17 
[12] 
Jong Yeoul Park, Jae Ug Jeong. Pullback attractors for a $2D$nonautonomous incompressible nonNewtonian fluid with variable delays. Discrete & Continuous Dynamical Systems  B, 2016, 21 (8) : 26872702. doi: 10.3934/dcdsb.2016068 
[13] 
Zhijian Yang, Yanan Li. Criteria on the existence and stability of pullback exponential attractors and their application to nonautonomous kirchhoff wave models. Discrete & Continuous Dynamical Systems, 2018, 38 (5) : 26292653. doi: 10.3934/dcds.2018111 
[14] 
Anhui Gu. Weak pullback mean random attractors for nonautonomous $ p $Laplacian equations. Discrete & Continuous Dynamical Systems  B, 2021, 26 (7) : 38633878. doi: 10.3934/dcdsb.2020266 
[15] 
Julia GarcíaLuengo, Pedro MarínRubio, José Real, James C. Robinson. Pullback attractors for the nonautonomous 2D NavierStokes equations for minimally regular forcing. Discrete & Continuous Dynamical Systems, 2014, 34 (1) : 203227. doi: 10.3934/dcds.2014.34.203 
[16] 
Bo You, Chengkui Zhong, Fang Li. Pullback attractors for three dimensional nonautonomous planetary geostrophic viscous equations of largescale ocean circulation. Discrete & Continuous Dynamical Systems  B, 2014, 19 (4) : 12131226. doi: 10.3934/dcdsb.2014.19.1213 
[17] 
Ting Li. Pullback attractors for asymptotically upper semicompact nonautonomous multivalued semiflows. Communications on Pure & Applied Analysis, 2007, 6 (1) : 279285. doi: 10.3934/cpaa.2007.6.279 
[18] 
Fang Li, Bo You. Pullback exponential attractors for the three dimensional nonautonomous NavierStokes equations with nonlinear damping. Discrete & Continuous Dynamical Systems  B, 2020, 25 (1) : 5580. doi: 10.3934/dcdsb.2019172 
[19] 
Lu Yang, Meihua Yang, Peter E. Kloeden. Pullback attractors for nonautonomous quasilinear parabolic equations with dynamical boundary conditions. Discrete & Continuous Dynamical Systems  B, 2012, 17 (7) : 26352651. doi: 10.3934/dcdsb.2012.17.2635 
[20] 
Na Lei, Shengfan Zhou. Upper semicontinuity of pullback attractors for nonautonomous lattice systems under singular perturbations. Discrete & Continuous Dynamical Systems, 2021 doi: 10.3934/dcds.2021108 
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]