Figure 2.2.
Global attractor for vector feld (2.4) in its positive hull Σ+.
-
Previous Article
Dynamics of fermentation models for the production of dry and sweet wine
- CPAA Home
- This Issue
-
Next Article
Stability and forward attractors for non-autonomous impulsive semidynamical systems
April
2020, 19(4): 1997-2013.
doi: 10.3934/cpaa.2020088
Forwards dynamics of non-autonomous dynamical systems: Driving semigroups without backwards uniqueness and structure of the attractor
| 1. | Instituto de Ciências Matemáticas e de Computaçao, Universidade de São Paulo-Campus de São Carlos, Caixa Postal 668, 13560-970 São Carlos SP, Brazil |
| 2. | Departamento de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, Sevilla 41012, Spain |
| 3. | Mathematical Institute, University of Warwick, Coventry CV4 7AL, UK |
We investigate the forwards asymptotic dynamics of non-autonomous differential equations. Our approach is centred on those models for which the vector field is only defined for non-negative times, that is, the laws of evolution are not given, or simply not known, for times before a given time (say time $ t = 0 $). We will be interested in the cases for which the 'driving' (time shift) semigroup has a global attractor in which backwards solutions are not necessarily unique. Considering vector fields in the global attractor of the driving semigroup allows for a natural way to extend vector fields, defined only for non-negative times, to the whole real line. These objects play a crucial role in the description of the asymptotic dynamics of our non-autonomous differential equation. We will study, in some particular cases, the isolated invariant sets of the associated skew-product semigroup with the aim of characterising the global attractor. We develop an example for which we derive decomposition for the global attractor of skew-product semigroup from the characterisation of the attractor of the associated driving semigroup.
Keywords:
Non-autonomous dynamical systems,
skew-product semiflow,
global attractor,
driving system.
Mathematics Subject Classification:
Primary: 37B55, 37L30, 35B40, 35B41, 37L05.
Citation:
Alexandre N. Carvalho, José A. Langa, James C. Robinson. Forwards dynamics of non-autonomous dynamical systems: Driving semigroups without backwards uniqueness and structure of the attractor. Communications on Pure and Applied Analysis,
2020, 19
(4)
: 1997-2013.
doi: 10.3934/cpaa.2020088
![]()
References:
| [1] |
E. R. Aragão-Costa, T. Caraballo, A. N. Carvalho and J. A. Langa, Stability of gradient semigroups under perturbations, Nonlinearity, 24 (2011), 2099117.
doi: 10.1088/0951-7715/24/7/010. |
| [2] |
A. V. Babin and M. I. Vishik, Attractors in Evolutionary Equations, Studies in Mathematics and its Applications 25, North-Holland Publishing Co., Amsterdam, 1992. |
| [3] |
T. Caraballo, J. C. Jara, J. A. Langa and Z. Liu, Morse decomposition of attractors for non-autonomous dynamical systems, Advanced Nonlinear Studies, to appear.
doi: 10.1515/ans-2013-0204. |
| [4] |
A. N. Carvalho and J. A. Langa,
An extension of the concept of gradient semigroups which is stable under perturbation, J. Differential Equations, 246 (2009), 2646-2668.
doi: 10.1016/j.jde.2009.01.007. |
| [5] |
A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-dimensional Non-autonomous Dynamical Systems, Applied Mathematical Sciences 182, Springer-Verlag, New York, 2013.
doi: 10.1007/978-1-4614-4581-4. |
| [6] |
D. N. Cheban, Global Attractors of Non-autonomous Dissipative Dynamical Systems, World Scientific, New Jersey, 2004.
doi: 10.1142/9789812563088. |
| [7] |
V. V. Chepyzhov and M.I. Vishik,
Vishik, Attractors of nonautonomous dynamical systems and their dimension, J. Math. Pures Appl., 73 (1994), 279-333.
|
| [8] |
V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Colloquium Publications, vol.49. American Mathematical Society, Providence, RI, 2001. |
| [9] |
C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Regional Conference Series in Mathematics, 38. American Mathematical Society, Providence, R.I., 1978. |
| [10] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs Number 25 (American Mathematical Society, Providence, RI), 1988. |
| [11] |
P. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Mathematical Surveys and Monographs 176, AMS, Providence RI, 2011.
doi: 10.1090/surv/176. |
| [12] |
O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, Cambridge, 1991.
doi: 10.1017/CBO9780511569418. |
| [13] |
J. A. Langa, J. C. Robinson, A. Rodríguez-Bernal, A. Suárez and A. Vidal-López,
Existence and nonexistence of unbounded forwards attractor for a class of non-autonomous reaction diffusion equations, Discrete Contin. Dyn. Syst., 18 (2007), 483-497.
doi: 10.3934/dcds.2007.18.483. |
| [14] |
J. A. Langa, J. C. Robinson, A. Suárez and A. Vidal-López,
The stability of attractors for non-autonomous perturbations of gradient-like systems, J. Differential Equations, 234 (2007), 607-625.
doi: 10.1016/j.jde.2006.11.016. |
| [15] |
K. Mischaikow, H. Smith and H. R. Thieme,
Asymptotically autonomous semiflows: chain recurrent and Lyapunov functions, Trans. Amer. Math. Soc., 347 (1995), 1669-1685.
doi: 10.2307/2154964. |
| [16] |
C. Pötzsche, Geometric Theory of Discrete Nonautonomous Dynamical Systems, Lecture Notes in Mathemathics 2002, Springer-Verlag, New York, 2010.
doi: 10.1007/978-3-642-14258-1. |
| [17] |
M. Rasmussen, Attractivity and Bifurcation for Nonautonomous Dynamical Systems, Lecture Notes in Mathemathics 1907, Springer-Verlag, New York, 2007. |
| [18] |
K. P. Rybakowski, The Homotopy Index and Partial Differential Equations, Universitext, Springer-Verlag, 1987.
doi: 10.1007/978-3-642-72833-4. |
| [19] |
G. R. Sell,
Nonautonomous differential equations and dynamical systems, Trans. Amer. Math. Soc., 127 (1967), 241-283.
doi: 10.2307/1994645. |
| [20] |
G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Applied Mathematical Sciences, 143. Springer-Verlag, New York, 2002.
doi: 10.1007/978-1-4757-5037-9. |
show all references
References:
| [1] |
E. R. Aragão-Costa, T. Caraballo, A. N. Carvalho and J. A. Langa, Stability of gradient semigroups under perturbations, Nonlinearity, 24 (2011), 2099117.
doi: 10.1088/0951-7715/24/7/010. |
| [2] |
A. V. Babin and M. I. Vishik, Attractors in Evolutionary Equations, Studies in Mathematics and its Applications 25, North-Holland Publishing Co., Amsterdam, 1992. |
| [3] |
T. Caraballo, J. C. Jara, J. A. Langa and Z. Liu, Morse decomposition of attractors for non-autonomous dynamical systems, Advanced Nonlinear Studies, to appear.
doi: 10.1515/ans-2013-0204. |
| [4] |
A. N. Carvalho and J. A. Langa,
An extension of the concept of gradient semigroups which is stable under perturbation, J. Differential Equations, 246 (2009), 2646-2668.
doi: 10.1016/j.jde.2009.01.007. |
| [5] |
A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-dimensional Non-autonomous Dynamical Systems, Applied Mathematical Sciences 182, Springer-Verlag, New York, 2013.
doi: 10.1007/978-1-4614-4581-4. |
| [6] |
D. N. Cheban, Global Attractors of Non-autonomous Dissipative Dynamical Systems, World Scientific, New Jersey, 2004.
doi: 10.1142/9789812563088. |
| [7] |
V. V. Chepyzhov and M.I. Vishik,
Vishik, Attractors of nonautonomous dynamical systems and their dimension, J. Math. Pures Appl., 73 (1994), 279-333.
|
| [8] |
V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Colloquium Publications, vol.49. American Mathematical Society, Providence, RI, 2001. |
| [9] |
C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Regional Conference Series in Mathematics, 38. American Mathematical Society, Providence, R.I., 1978. |
| [10] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs Number 25 (American Mathematical Society, Providence, RI), 1988. |
| [11] |
P. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Mathematical Surveys and Monographs 176, AMS, Providence RI, 2011.
doi: 10.1090/surv/176. |
| [12] |
O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, Cambridge, 1991.
doi: 10.1017/CBO9780511569418. |
| [13] |
J. A. Langa, J. C. Robinson, A. Rodríguez-Bernal, A. Suárez and A. Vidal-López,
Existence and nonexistence of unbounded forwards attractor for a class of non-autonomous reaction diffusion equations, Discrete Contin. Dyn. Syst., 18 (2007), 483-497.
doi: 10.3934/dcds.2007.18.483. |
| [14] |
J. A. Langa, J. C. Robinson, A. Suárez and A. Vidal-López,
The stability of attractors for non-autonomous perturbations of gradient-like systems, J. Differential Equations, 234 (2007), 607-625.
doi: 10.1016/j.jde.2006.11.016. |
| [15] |
K. Mischaikow, H. Smith and H. R. Thieme,
Asymptotically autonomous semiflows: chain recurrent and Lyapunov functions, Trans. Amer. Math. Soc., 347 (1995), 1669-1685.
doi: 10.2307/2154964. |
| [16] |
C. Pötzsche, Geometric Theory of Discrete Nonautonomous Dynamical Systems, Lecture Notes in Mathemathics 2002, Springer-Verlag, New York, 2010.
doi: 10.1007/978-3-642-14258-1. |
| [17] |
M. Rasmussen, Attractivity and Bifurcation for Nonautonomous Dynamical Systems, Lecture Notes in Mathemathics 1907, Springer-Verlag, New York, 2007. |
| [18] |
K. P. Rybakowski, The Homotopy Index and Partial Differential Equations, Universitext, Springer-Verlag, 1987.
doi: 10.1007/978-3-642-72833-4. |
| [19] |
G. R. Sell,
Nonautonomous differential equations and dynamical systems, Trans. Amer. Math. Soc., 127 (1967), 241-283.
doi: 10.2307/1994645. |
| [20] |
G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Applied Mathematical Sciences, 143. Springer-Verlag, New York, 2002.
doi: 10.1007/978-1-4757-5037-9. |
| [1] |
P.E. Kloeden, Victor S. Kozyakin. The perturbation of attractors of skew-product flows with a shadowing driving system. Discrete and Continuous Dynamical Systems, 2001, 7 (4) : 883-893. doi: 10.3934/dcds.2001.7.883 |
| [2] |
Sylvia Novo, Carmen Núñez, Rafael Obaya, Ana M. Sanz. Skew-product semiflows for non-autonomous partial functional differential equations with delay. Discrete and Continuous Dynamical Systems, 2014, 34 (10) : 4291-4321. doi: 10.3934/dcds.2014.34.4291 |
| [3] |
Tomás Caraballo, David Cheban. On the structure of the global attractor for non-autonomous dynamical systems with weak convergence. Communications on Pure and Applied Analysis, 2012, 11 (2) : 809-828. doi: 10.3934/cpaa.2012.11.809 |
| [4] |
Tomás Caraballo, David Cheban. On the structure of the global attractor for infinite-dimensional non-autonomous dynamical systems with weak convergence. Communications on Pure and Applied Analysis, 2013, 12 (1) : 281-302. doi: 10.3934/cpaa.2013.12.281 |
| [5] |
Alexandre N. Carvalho, José A. Langa, James C. Robinson. Non-autonomous dynamical systems. Discrete and Continuous Dynamical Systems - B, 2015, 20 (3) : 703-747. doi: 10.3934/dcdsb.2015.20.703 |
| [6] |
Michael Zgurovsky, Mark Gluzman, Nataliia Gorban, Pavlo Kasyanov, Liliia Paliichuk, Olha Khomenko. Uniform global attractors for non-autonomous dissipative dynamical systems. Discrete and Continuous Dynamical Systems - B, 2017, 22 (5) : 2053-2065. doi: 10.3934/dcdsb.2017120 |
| [7] |
Saša Kocić. Reducibility of skew-product systems with multidimensional Brjuno base flows. Discrete and Continuous Dynamical Systems, 2011, 29 (1) : 261-283. doi: 10.3934/dcds.2011.29.261 |
| [8] |
Noriaki Yamazaki. Global attractors for non-autonomous multivalued dynamical systems associated with double obstacle problems. Conference Publications, 2003, 2003 (Special) : 935-944. doi: 10.3934/proc.2003.2003.935 |
| [9] |
Xinyuan Liao, Caidi Zhao, Shengfan Zhou. Compact uniform attractors for dissipative non-autonomous lattice dynamical systems. Communications on Pure and Applied Analysis, 2007, 6 (4) : 1087-1111. doi: 10.3934/cpaa.2007.6.1087 |
| [10] |
Grzegorz Łukaszewicz, James C. Robinson. Invariant measures for non-autonomous dissipative dynamical systems. Discrete and Continuous Dynamical Systems, 2014, 34 (10) : 4211-4222. doi: 10.3934/dcds.2014.34.4211 |
| [11] |
Michael Dellnitz, Christian Horenkamp. The efficient approximation of coherent pairs in non-autonomous dynamical systems. Discrete and Continuous Dynamical Systems, 2012, 32 (9) : 3029-3042. doi: 10.3934/dcds.2012.32.3029 |
| [12] |
Wen Tan. The regularity of pullback attractor for a non-autonomous p-Laplacian equation with dynamical boundary condition. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 529-546. doi: 10.3934/dcdsb.2018194 |
| [13] |
V. V. Chepyzhov, M. I. Vishik, W. L. Wendland. On non-autonomous sine-Gordon type equations with a simple global attractor and some averaging. Discrete and Continuous Dynamical Systems, 2005, 12 (1) : 27-38. doi: 10.3934/dcds.2005.12.27 |
| [14] |
T. Tachim Medjo. Non-autonomous 3D primitive equations with oscillating external force and its global attractor. Discrete and Continuous Dynamical Systems, 2012, 32 (1) : 265-291. doi: 10.3934/dcds.2012.32.265 |
| [15] |
Tomás Caraballo, Alexandre N. Carvalho, Henrique B. da Costa, José A. Langa. Equi-attraction and continuity of attractors for skew-product semiflows. Discrete and Continuous Dynamical Systems - B, 2016, 21 (9) : 2949-2967. doi: 10.3934/dcdsb.2016081 |
| [16] |
David Cheban, Cristiana Mammana. Continuous dependence of attractors on parameters of non-autonomous dynamical systems and infinite iterated function systems. Discrete and Continuous Dynamical Systems, 2007, 18 (2&3) : 499-515. doi: 10.3934/dcds.2007.18.499 |
| [17] |
Xiaoyue Li, Xuerong Mao. Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation. Discrete and Continuous Dynamical Systems, 2009, 24 (2) : 523-545. doi: 10.3934/dcds.2009.24.523 |
| [18] |
Emma D'Aniello, Saber Elaydi. The structure of $ \omega $-limit sets of asymptotically non-autonomous discrete dynamical systems. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 903-915. doi: 10.3934/dcdsb.2019195 |
| [19] |
Mikhail B. Sevryuk. Invariant tori in quasi-periodic non-autonomous dynamical systems via Herman's method. Discrete and Continuous Dynamical Systems, 2007, 18 (2&3) : 569-595. doi: 10.3934/dcds.2007.18.569 |
| [20] |
Bixiang Wang. Multivalued non-autonomous random dynamical systems for wave equations without uniqueness. Discrete and Continuous Dynamical Systems - B, 2017, 22 (5) : 2011-2051. doi: 10.3934/dcdsb.2017119 |
2021 Impact Factor: 1.273
Tools
Metrics
Other articles
by authors
[Back to Top]