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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.
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