American Institute of Mathematical Sciences

Advanced
May  2020, 40(5): 2791-2826. doi: 10.3934/dcds.2020150

Pullback attractors to impulsive evolution processes: Applications to differential equations and tube conditions

Matheus C. Bortolan 1,, and  José Manuel Uzal 2,

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

* Corresponding author: Matheus C. Bortolan

Received  July 2019 Published  March 2020

Fund Project: The first author is by supported by CNPq, project # 407635/2016-5. The second author was partially supported by the predoctoral contact BES-2017-082334

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.

Keywords: Impulsive evolution processes, pullback semi attractors, pullback attractors, integrate-and-fire model, tube conditions.
Mathematics Subject Classification: Primary: 37L05, 37L30. Secondary: 34H05.
Citation: Matheus C. Bortolan, José Manuel Uzal. Pullback attractors to impulsive evolution processes: Applications to differential equations and tube conditions. Discrete & Continuous Dynamical Systems - A, 2020, 40 (5) : 2791-2826. doi: 10.3934/dcds.2020150
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

E. M. Bonotto and D. P. Demuner, Autonomous dissipative semidynamical systems with impulses, Topol. Method. Nonl. An., 41 (2013), 1-38.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

K. Ciesielski, On semicontinuity in impulsive dynamical systems, Bull. Polish Acad. Sci. Math., 52 (2004), 71-80.  doi: 10.4064/ba52-1-8.  Google Scholar

[18]

K. Ciesielski, On stability in impulsive dynamical systems, Bull. Polish Acad. Sci. Math., 52 (2004), 81-91.  doi: 10.4064/ba52-1-9.  Google Scholar

[19]

K. Ciesielski, Sections in semidynamical systems, Bull. Polish Acad. Sci. Math., 40 (1992), 297-307.   Google Scholar

[20]

S. Kaul, On impulsive semidynamical systems, J. Math. Anal. Appl., 150 (1990), 120-128.  doi: 10.1016/0022-247X(90)90199-P.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

V. Rozko, A certain class of almost periodic motions in systems with pulses, Diff. Uravn., 8 (1972), 2012-2022.   Google Scholar

[26]

V. Rozko, Lyapunov stability in discontinuous dynamical systems, Diff. Uravn., 11 (1975), 1005-1012.   Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

E. M. Bonotto and D. P. Demuner, Autonomous dissipative semidynamical systems with impulses, Topol. Method. Nonl. An., 41 (2013), 1-38.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

K. Ciesielski, On semicontinuity in impulsive dynamical systems, Bull. Polish Acad. Sci. Math., 52 (2004), 71-80.  doi: 10.4064/ba52-1-8.  Google Scholar

[18]

K. Ciesielski, On stability in impulsive dynamical systems, Bull. Polish Acad. Sci. Math., 52 (2004), 81-91.  doi: 10.4064/ba52-1-9.  Google Scholar

[19]

K. Ciesielski, Sections in semidynamical systems, Bull. Polish Acad. Sci. Math., 40 (1992), 297-307.   Google Scholar

[20]

S. Kaul, On impulsive semidynamical systems, J. Math. Anal. Appl., 150 (1990), 120-128.  doi: 10.1016/0022-247X(90)90199-P.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

V. Rozko, A certain class of almost periodic motions in systems with pulses, Diff. Uravn., 8 (1972), 2012-2022.   Google Scholar

[26]

V. Rozko, Lyapunov stability in discontinuous dynamical systems, Diff. Uravn., 11 (1975), 1005-1012.   Google Scholar

[1]

Alexandre Nolasco de Carvalho, Stefanie Sonner. Pullback exponential attractors for evolution processes in Banach spaces: Theoretical results. Communications on Pure & Applied Analysis, 2013, 12 (6) : 3047-3071. doi: 10.3934/cpaa.2013.12.3047
[2]

Alexandre Nolasco de Carvalho, Stefanie Sonner. Pullback exponential attractors for evolution processes in Banach spaces: Properties and applications. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1141-1165. doi: 10.3934/cpaa.2014.13.1141
[3]

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
[4]

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
[5]

José A. Langa, Alain Miranville, José Real. Pullback exponential attractors. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1329-1357. doi: 10.3934/dcds.2010.26.1329
[6]

Peter E. Kloeden, Jacson Simsen. Pullback attractors for non-autonomous evolution equations with spatially variable exponents. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2543-2557. doi: 10.3934/cpaa.2014.13.2543
[7]

Yejuan Wang, Chengkui Zhong, Shengfan Zhou. Pullback attractors of nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems - A, 2006, 16 (3) : 587-614. doi: 10.3934/dcds.2006.16.587
[8]

Alexey Cheskidov, Landon Kavlie. Pullback attractors for generalized evolutionary systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 749-779. doi: 10.3934/dcdsb.2015.20.749
[9]

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
[10]

Tomás Caraballo, Antonio M. Márquez-Durán, José Real. Pullback and forward attractors for a 3D LANS$-\alpha$ model with delay. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 559-578. doi: 10.3934/dcds.2006.15.559
[11]

Yonghai Wang. On the upper semicontinuity of pullback attractors with applications to plate equations. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1653-1673. doi: 10.3934/cpaa.2010.9.1653
[12]

Bernd Aulbach, Martin Rasmussen, Stefan Siegmund. Invariant manifolds as pullback attractors of nonautonomous differential equations. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 579-596. doi: 10.3934/dcds.2006.15.579
[13]

Yejuan Wang, Kuang Bai. Pullback attractors for a class of nonlinear lattices with delays. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1213-1230. doi: 10.3934/dcdsb.2015.20.1213
[14]

Tomás Caraballo, José Real, I. D. Chueshov. Pullback attractors for stochastic heat equations in materials with memory. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 525-539. doi: 10.3934/dcdsb.2008.9.525
[15]

Mohamed Ali Hammami, Lassaad Mchiri, Sana Netchaoui, Stefanie Sonner. Pullback exponential attractors for differential equations with variable delays. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 301-319. doi: 10.3934/dcdsb.2019183
[16]

Sana Netchaoui, Mohamed Ali Hammami, Tomás Caraballo. Pullback exponential attractors for differential equations with delay. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020367
[17]

Lu Yang, Meihua Yang, Peter E. Kloeden. Pullback attractors for non-autonomous quasi-linear parabolic equations with dynamical boundary conditions. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2635-2651. doi: 10.3934/dcdsb.2012.17.2635
[18]

Hongyong Cui. Convergences of asymptotically autonomous pullback attractors towards semigroup attractors. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3525-3535. doi: 10.3934/dcdsb.2018276
[19]

Yonghai Wang, Chengkui Zhong. Upper semicontinuity of pullback attractors for nonautonomous Kirchhoff wave models. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 3189-3209. doi: 10.3934/dcds.2013.33.3189
[20]

Zhen Zhang, Jianhua Huang, Xueke Pu. Pullback attractors of FitzHugh-Nagumo system on the time-varying domains. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3691-3706. doi: 10.3934/dcdsb.2017150

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (55)
  • HTML views (62)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences