-
Previous Article
A functional CLT for nonconventional polynomial arrays
- DCDS Home
- This Issue
-
Next Article
Measure theoretic pressure and dimension formula for non-ergodic measures
Pullback attractors to impulsive evolution processes: Applications to differential equations and tube conditions
1. | Departamento de Matemática, Universidade Federal de Santa Catarina, Florianópolis - Brazil |
2. | Departamento de Estatística, Análise Matemática e Optimización & Instituto de Matemáticas, Universidade de Santiago de Compostela, Santiago de Compostela - Spain |
We define the notions of impulsive evolution processes and their pullback attractors, and exhibit conditions under which a given impulsive evolution process has a pullback attractor. We apply our results to a nonautonomous ordinary differential equation describing an integrate-and-fire model of neuron membrane, as well as to a heat equation with nonautonomous impulse and a nonautonomous 2D Navier-Stokes equation. Finally, we introduce the notion of tube conditions to impulsive evolution processes, and use them as an alternative way to obtain pullback attractors.
References:
[1] |
R. Ambrosino, F. Calabrese, C. Cosentino and G. De Tommasi,
Sufficient conditions for finite-time stability of impulsive dynamical systems, IEEE Trans. Autom. Control, 54 (2009), 861-865.
doi: 10.1109/TAC.2008.2010965. |
[2] |
L. Barreira and C. Valls,
Lyapunov regularity of impulsive differential equations, J. Differ. Equations, 249 (2010), 1596-1619.
doi: 10.1016/j.jde.2010.07.016. |
[3] |
E. M. Bonotto, M. C. Bortolan, A. N. Carvalho and R. Czaja, Global attractors for impulsive dynamical systems - a precompact approach, J. Differ. Equations,, 259 (2015), 2602–2625.
doi: 10.1016/j.jde.2015.03.033. |
[4] |
E. M. Bonotto, M. C. Bortolan, T. Caraballo and R. Collegari,
Impulsive non-autonomous dynamical systems and impulsive cocycle attractors, Math. Method. Appl. Sci., 40 (2017), 1095-1113.
doi: 10.1002/mma.4038. |
[5] |
E. M. Bonotto, M. C. Bortolan, T. Caraballo and R. Collegari,
Attractors for impulsive non-autonomous dynamical systems and their relations, J. Differ. Equations, 262 (2017), 3524-3550.
doi: 10.1016/j.jde.2016.11.036. |
[6] |
E. M. Bonotto, M. C. Bortolan, R. Czaja and R. Collegari,
Semicontinuity of attractors for impulsive dynamical systems, J. Differ. Equations, 261 (2016), 4338-4367.
doi: 10.1016/j.jde.2016.06.024. |
[7] |
E. M. Bonotto and D. P. Demuner,
Attractors of impulsive dissipative semidynamical systems, Bull. Sci. Math., 137 (2013), 617-642.
doi: 10.1016/j.bulsci.2012.12.005. |
[8] |
E. M. Bonotto and D. P. Demuner,
Autonomous dissipative semidynamical systems with impulses, Topol. Method. Nonl. An., 41 (2013), 1-38.
|
[9] |
E. M. Bonotto and M. Federson,
Topological conjugation and asymptotic stability in impulsive semidynamical systems, J. Math. Anal. Appl., 326 (2007), 869-881.
doi: 10.1016/j.jmaa.2006.03.042. |
[10] |
E. M. Bonotto and G. M. Souto,
On the Lyapunov stability theory for impulsive dynamical systems, Topol. Method. Nonl. An., 53 (2019), 127-150.
doi: 10.12775/TMNA.2018.042. |
[11] |
E. M. Bonotto, L. P. Gimenes and G. M. Souto,
Asymptotically almost periodic motions in impulsive semidynamical systems, Topol. Method. Nonl. An., 49 (2017), 133-163.
doi: 10.12775/tmna.2016.065. |
[12] |
E. M. Bonotto and P. Kalita, On attractors of generalized semiflows with impulses, J. Geom. Anal. (online), (2019), 1–38.
doi: 10.1007/s12220-019-00143-0. |
[13] |
R. Brette and W. Gerstner,
Adaptive exponential integrate-and-fire model as an effective description of neuronal activity, J. Neurophysiol., 94 (2005), 3637-3642.
doi: 10.1152/jn.00686.2005. |
[14] |
T. Caraballo, G. Lukaszewicz and J. Real,
Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonl. Anal., 64 (2006), 484-498.
doi: 10.1016/j.na.2005.03.111. |
[15] |
T. Cardinali and R. Servadei,
Periodic solutions of nonlinear impulsive differential inclusions with constraints, Proc. Am. Math. Soc., 132 (2004), 2339-2349.
doi: 10.1090/S0002-9939-04-07343-5. |
[16] |
J. F. Chu and J. J. Nieto,
Impulsive periodic solutions of first-order singular differential equations, Bull. Lond. Math. Soc., 40 (2008), 143-150.
doi: 10.1112/blms/bdm110. |
[17] |
K. Ciesielski,
On semicontinuity in impulsive dynamical systems, Bull. Polish Acad. Sci. Math., 52 (2004), 71-80.
doi: 10.4064/ba52-1-8. |
[18] |
K. Ciesielski,
On stability in impulsive dynamical systems, Bull. Polish Acad. Sci. Math., 52 (2004), 81-91.
doi: 10.4064/ba52-1-9. |
[19] |
K. Ciesielski,
Sections in semidynamical systems, Bull. Polish Acad. Sci. Math., 40 (1992), 297-307.
|
[20] |
S. Kaul,
On impulsive semidynamical systems, J. Math. Anal. Appl., 150 (1990), 120-128.
doi: 10.1016/0022-247X(90)90199-P. |
[21] |
S. K. Kaul,
On impulsive semidynamical systems. II. Recursive properties, Nonl. Anal., 16 (1991), 635-645.
doi: 10.1016/0362-546X(91)90171-V. |
[22] |
S. K. Kaul,
Stability and asymptotic stability in impulsive semidynamical systems, J. Appl. Math. Stochastic Anal., 7 (1994), 509-523.
doi: 10.1155/S1048953394000390. |
[23] |
J. P. Keener, F. C. Hoppensteadt and J. Rinzel,
Integrate-and-fire models of nerve membrane response to oscillatory input, SIAM J. Appl. Math., 41 (1981), 503-517.
doi: 10.1137/0141042. |
[24] |
K. H. Li, C. M. Ding, F. Y. Wang and J. J. Hu,
Limit set maps in impulsive semidynamical systems, J. Dyn. Control. Syst., 20 (2014), 47-58.
doi: 10.1007/s10883-013-9204-5. |
[25] |
V. Rozko,
A certain class of almost periodic motions in systems with pulses, Diff. Uravn., 8 (1972), 2012-2022.
|
[26] |
V. Rozko,
Lyapunov stability in discontinuous dynamical systems, Diff. Uravn., 11 (1975), 1005-1012.
|
show all references
References:
[1] |
R. Ambrosino, F. Calabrese, C. Cosentino and G. De Tommasi,
Sufficient conditions for finite-time stability of impulsive dynamical systems, IEEE Trans. Autom. Control, 54 (2009), 861-865.
doi: 10.1109/TAC.2008.2010965. |
[2] |
L. Barreira and C. Valls,
Lyapunov regularity of impulsive differential equations, J. Differ. Equations, 249 (2010), 1596-1619.
doi: 10.1016/j.jde.2010.07.016. |
[3] |
E. M. Bonotto, M. C. Bortolan, A. N. Carvalho and R. Czaja, Global attractors for impulsive dynamical systems - a precompact approach, J. Differ. Equations,, 259 (2015), 2602–2625.
doi: 10.1016/j.jde.2015.03.033. |
[4] |
E. M. Bonotto, M. C. Bortolan, T. Caraballo and R. Collegari,
Impulsive non-autonomous dynamical systems and impulsive cocycle attractors, Math. Method. Appl. Sci., 40 (2017), 1095-1113.
doi: 10.1002/mma.4038. |
[5] |
E. M. Bonotto, M. C. Bortolan, T. Caraballo and R. Collegari,
Attractors for impulsive non-autonomous dynamical systems and their relations, J. Differ. Equations, 262 (2017), 3524-3550.
doi: 10.1016/j.jde.2016.11.036. |
[6] |
E. M. Bonotto, M. C. Bortolan, R. Czaja and R. Collegari,
Semicontinuity of attractors for impulsive dynamical systems, J. Differ. Equations, 261 (2016), 4338-4367.
doi: 10.1016/j.jde.2016.06.024. |
[7] |
E. M. Bonotto and D. P. Demuner,
Attractors of impulsive dissipative semidynamical systems, Bull. Sci. Math., 137 (2013), 617-642.
doi: 10.1016/j.bulsci.2012.12.005. |
[8] |
E. M. Bonotto and D. P. Demuner,
Autonomous dissipative semidynamical systems with impulses, Topol. Method. Nonl. An., 41 (2013), 1-38.
|
[9] |
E. M. Bonotto and M. Federson,
Topological conjugation and asymptotic stability in impulsive semidynamical systems, J. Math. Anal. Appl., 326 (2007), 869-881.
doi: 10.1016/j.jmaa.2006.03.042. |
[10] |
E. M. Bonotto and G. M. Souto,
On the Lyapunov stability theory for impulsive dynamical systems, Topol. Method. Nonl. An., 53 (2019), 127-150.
doi: 10.12775/TMNA.2018.042. |
[11] |
E. M. Bonotto, L. P. Gimenes and G. M. Souto,
Asymptotically almost periodic motions in impulsive semidynamical systems, Topol. Method. Nonl. An., 49 (2017), 133-163.
doi: 10.12775/tmna.2016.065. |
[12] |
E. M. Bonotto and P. Kalita, On attractors of generalized semiflows with impulses, J. Geom. Anal. (online), (2019), 1–38.
doi: 10.1007/s12220-019-00143-0. |
[13] |
R. Brette and W. Gerstner,
Adaptive exponential integrate-and-fire model as an effective description of neuronal activity, J. Neurophysiol., 94 (2005), 3637-3642.
doi: 10.1152/jn.00686.2005. |
[14] |
T. Caraballo, G. Lukaszewicz and J. Real,
Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonl. Anal., 64 (2006), 484-498.
doi: 10.1016/j.na.2005.03.111. |
[15] |
T. Cardinali and R. Servadei,
Periodic solutions of nonlinear impulsive differential inclusions with constraints, Proc. Am. Math. Soc., 132 (2004), 2339-2349.
doi: 10.1090/S0002-9939-04-07343-5. |
[16] |
J. F. Chu and J. J. Nieto,
Impulsive periodic solutions of first-order singular differential equations, Bull. Lond. Math. Soc., 40 (2008), 143-150.
doi: 10.1112/blms/bdm110. |
[17] |
K. Ciesielski,
On semicontinuity in impulsive dynamical systems, Bull. Polish Acad. Sci. Math., 52 (2004), 71-80.
doi: 10.4064/ba52-1-8. |
[18] |
K. Ciesielski,
On stability in impulsive dynamical systems, Bull. Polish Acad. Sci. Math., 52 (2004), 81-91.
doi: 10.4064/ba52-1-9. |
[19] |
K. Ciesielski,
Sections in semidynamical systems, Bull. Polish Acad. Sci. Math., 40 (1992), 297-307.
|
[20] |
S. Kaul,
On impulsive semidynamical systems, J. Math. Anal. Appl., 150 (1990), 120-128.
doi: 10.1016/0022-247X(90)90199-P. |
[21] |
S. K. Kaul,
On impulsive semidynamical systems. II. Recursive properties, Nonl. Anal., 16 (1991), 635-645.
doi: 10.1016/0362-546X(91)90171-V. |
[22] |
S. K. Kaul,
Stability and asymptotic stability in impulsive semidynamical systems, J. Appl. Math. Stochastic Anal., 7 (1994), 509-523.
doi: 10.1155/S1048953394000390. |
[23] |
J. P. Keener, F. C. Hoppensteadt and J. Rinzel,
Integrate-and-fire models of nerve membrane response to oscillatory input, SIAM J. Appl. Math., 41 (1981), 503-517.
doi: 10.1137/0141042. |
[24] |
K. H. Li, C. M. Ding, F. Y. Wang and J. J. Hu,
Limit set maps in impulsive semidynamical systems, J. Dyn. Control. Syst., 20 (2014), 47-58.
doi: 10.1007/s10883-013-9204-5. |
[25] |
V. Rozko,
A certain class of almost periodic motions in systems with pulses, Diff. Uravn., 8 (1972), 2012-2022.
|
[26] |
V. Rozko,
Lyapunov stability in discontinuous dynamical systems, Diff. Uravn., 11 (1975), 1005-1012.
|
[1] |
Matheus C. Bortolan, José Manuel Uzal. Upper and weak-lower semicontinuity of pullback attractors to impulsive evolution processes. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3667-3692. doi: 10.3934/dcdsb.2020252 |
[2] |
Alexandre Nolasco de Carvalho, Stefanie Sonner. Pullback exponential attractors for evolution processes in Banach spaces: Theoretical results. Communications on Pure and Applied Analysis, 2013, 12 (6) : 3047-3071. doi: 10.3934/cpaa.2013.12.3047 |
[3] |
Alexandre Nolasco de Carvalho, Stefanie Sonner. Pullback exponential attractors for evolution processes in Banach spaces: Properties and applications. Communications on Pure and Applied Analysis, 2014, 13 (3) : 1141-1165. doi: 10.3934/cpaa.2014.13.1141 |
[4] |
Michele Barbi, Angelo Di Garbo, Rita Balocchi. Improved integrate-and-fire model for RSA. Mathematical Biosciences & Engineering, 2007, 4 (4) : 609-615. doi: 10.3934/mbe.2007.4.609 |
[5] |
Aniello Buonocore, Luigia Caputo, Enrica Pirozzi, Maria Francesca Carfora. A leaky integrate-and-fire model with adaptation for the generation of a spike train. Mathematical Biosciences & Engineering, 2016, 13 (3) : 483-493. doi: 10.3934/mbe.2016002 |
[6] |
José A. Langa, Alain Miranville, José Real. Pullback exponential attractors. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1329-1357. doi: 10.3934/dcds.2010.26.1329 |
[7] |
Peter E. Kloeden, Jacson Simsen. Pullback attractors for non-autonomous evolution equations with spatially variable exponents. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2543-2557. doi: 10.3934/cpaa.2014.13.2543 |
[8] |
Aniello Buonocore, Luigia Caputo, Enrica Pirozzi, Maria Francesca Carfora. A simple algorithm to generate firing times for leaky integrate-and-fire neuronal model. Mathematical Biosciences & Engineering, 2014, 11 (1) : 1-10. doi: 10.3934/mbe.2014.11.1 |
[9] |
Yejuan Wang, Chengkui Zhong, Shengfan Zhou. Pullback attractors of nonautonomous dynamical systems. Discrete and Continuous Dynamical Systems, 2006, 16 (3) : 587-614. doi: 10.3934/dcds.2006.16.587 |
[10] |
Alexey Cheskidov, Landon Kavlie. Pullback attractors for generalized evolutionary systems. Discrete and Continuous Dynamical Systems - B, 2015, 20 (3) : 749-779. doi: 10.3934/dcdsb.2015.20.749 |
[11] |
Tomás Caraballo, Antonio M. Márquez-Durán, José Real. Pullback and forward attractors for a 3D LANS$-\alpha$ model with delay. Discrete and Continuous Dynamical Systems, 2006, 15 (2) : 559-578. doi: 10.3934/dcds.2006.15.559 |
[12] |
Mikhail Turbin, Anastasiia Ustiuzhaninova. Pullback attractors for weak solution to modified Kelvin-Voigt model. Evolution Equations and Control Theory, 2022 doi: 10.3934/eect.2022011 |
[13] |
Chang-Yuan Cheng, Shyan-Shiou Chen, Rui-Hua Chen. Delay-induced spiking dynamics in integrate-and-fire neurons. Discrete and Continuous Dynamical Systems - B, 2021, 26 (4) : 1867-1887. doi: 10.3934/dcdsb.2020363 |
[14] |
Sana Netchaoui, Mohamed Ali Hammami, Tomás Caraballo. Pullback exponential attractors for differential equations with delay. Discrete and Continuous Dynamical Systems - S, 2021, 14 (4) : 1345-1358. doi: 10.3934/dcdss.2020367 |
[15] |
Yonghai Wang. On the upper semicontinuity of pullback attractors with applications to plate equations. Communications on Pure and Applied Analysis, 2010, 9 (6) : 1653-1673. doi: 10.3934/cpaa.2010.9.1653 |
[16] |
Bernd Aulbach, Martin Rasmussen, Stefan Siegmund. Invariant manifolds as pullback attractors of nonautonomous differential equations. Discrete and Continuous Dynamical Systems, 2006, 15 (2) : 579-596. doi: 10.3934/dcds.2006.15.579 |
[17] |
Yejuan Wang, Kuang Bai. Pullback attractors for a class of nonlinear lattices with delays. Discrete and Continuous Dynamical Systems - B, 2015, 20 (4) : 1213-1230. doi: 10.3934/dcdsb.2015.20.1213 |
[18] |
Tomás Caraballo, José Real, I. D. Chueshov. Pullback attractors for stochastic heat equations in materials with memory. Discrete and Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 525-539. doi: 10.3934/dcdsb.2008.9.525 |
[19] |
Mohamed Ali Hammami, Lassaad Mchiri, Sana Netchaoui, Stefanie Sonner. Pullback exponential attractors for differential equations with variable delays. Discrete and Continuous Dynamical Systems - B, 2020, 25 (1) : 301-319. doi: 10.3934/dcdsb.2019183 |
[20] |
Xiaolei Dong, Yuming Qin. Strong pullback attractors for a nonclassical diffusion equation. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2021313 |
2021 Impact Factor: 1.588
Tools
Metrics
Other articles
by authors
[Back to Top]