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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 and Continuous Dynamical Systems, 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.

[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. 

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