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Invariant Probability Measures and Non-wandering Sets for Impulsive Semiflows

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Abstract

We consider impulsive dynamical systems defined on compact metric spaces and their respective impulsive semiflows. We establish sufficient conditions for the existence of probability measures which are invariant under such impulsive semiflows. Under these conditions we also deduce the forward invariance of their non-wandering sets except the discontinuity points.

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References

  1. Baĭnov D. D., Simeonov P. S.: Systems with impulse effect. Stability, Theory and Applications. Ellis Horwood Series: Mathematics and its Applications. Ellis Horwood Ltd., Chichester (1989)

  2. Bonotto, E.M.: Flows of characteristic \(0^+\) in impulsive semidynamical systems. J. Math. Anal. Appl. 332(1), 81–96 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bonotto, E.M., Federson, M.: Topological conjugation and asymptotic stability in impulsive semidynamical systems. J. Math. Anal. Appl. 326(2), 869–881 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bonotto, E.M., Federson, M.: Limit sets and the Poincaré–Bendixson theorem in impulsive semidynamical systems. J. Differ. Equ. 244(9), 2334–2349 (2008)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  5. Bowen, R., Ruelle, D.: The ergodic theory of Axiom A flows. Invent. Math. 29, 181–202 (1975)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  6. Choisy, M., Guégan, J.-F., Rohani, P.: Dynamics of infectious diseases and pulse vaccination: teasing apart the embedded resonance effects. Physica D 223(1), 26–35 (2006)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  7. Ciesielski, K.: On semicontinuity in impulsive dynamical systems. Bull. Pol. Acad. Sci. Math. 52(1), 71–80 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  8. Ciesielski, K.: On stability in impulsive dynamical systems. Bull. Pol. Acad. Sci. Math. 52(1), 81–91 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  9. Dishliev, A.B., Baĭnov, D.D.: Dependence upon initial conditions and parameter of solutions of impulsive differential equations with variable structure. Int. J. Theor. Phys. 29(6), 655–675 (1990)

    Article  MATH  Google Scholar 

  10. d’Onofrio, A.: On pulse vaccination strategy in the SIR epidemic model with vertical transmission. Appl. Math. Lett. 18(7), 729–732 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  11. Erbe, L.H., Freedman, H.I., Liu, X., Wu, J.H.: Comparison principles for impulsive parabolic equations with applications to models of single species growth. J. Austral. Math. Soc. Ser. B 32(4), 382–400 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  12. Halmos P. R.: Lectures on ergodic theory. Publications of the Mathematical Society of Japan, no. 3. The Mathematical Society of Japan, 1956.

  13. Jiang, G., Lu, Q.: Impulsive state feedback control of a predator–prey model. J. Comput. Appl. Math. 200(1), 193–207 (2007)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  14. Kaul, S.K.: Stability and asymptotic stability in impulsive semidynamical systems. J. Appl. Math. Stoch. Anal. 7(4), 509–523 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  15. Kryloff, N., Bogoliouboff, N.: La théorie générale de la mesure dans son application à l’étude des systèmes dynamiques de la mécanique non linéaire. Ann. of Math. (2), 38(1):65–113 (1937)

  16. Lakshmikantham, V., Baĭnov, D.D., Simeonov, P.S.: Theory of Impulsive Differential Equations. Modern Applied Mathematics, vol. 6. World Scientific Publishing Co., Inc., Teaneck (1989)

  17. Liu, J.H.: Nonlinear impulsive evolution equations. Dyn. Contin. Discret. Impuls. Syst. 6(1), 77–85 (1999)

    MATH  Google Scholar 

  18. Nenov S.I.: Impulsive controllability and optimization problems in population dynamics. Nonlinear Anal. 36(7, Ser. A: Theory Methods):881–890 (1999)

  19. Palis, J.: On the \(C^1\) \(\Omega \)-stability conjecture. Inst. Hautes Études Sci. Publ. Math. 66, 211–215 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  20. Purves, R.: Bimeasurable functions. Fundam. Math. 58, 149–157 (1966)

    MATH  MathSciNet  Google Scholar 

  21. Rogovchenko, Y.V.: Impulsive evolution systems: main results and new trends. Dyn. Contin. Discret. Impuls. Syst. 3(1), 57–88 (1997)

    MATH  MathSciNet  Google Scholar 

  22. Smale S.: The \(\Omega \)-stability theorem. In Global Analysis (Proceedings of the Symposium on Pure Mathematics, Vol. XIV, Berkeley, Calif., 1968), pp. 289–297. American Mathematical Society, Providence (1970)

  23. Thorne, K.: Black Holes and Time Warps: Einstein’s Outrageous Legacy. W.W Norton, Commonwealth Fund Book Program (1994)

  24. Visser, M.: Lorentzian Wormholes - From Einstein to Hawking. American Institute of Physics Press, Springer-Verlag, New York (1996)

  25. Walters, P.: An Introduction to Ergodic Theory. Graduate Texts in Mathematics, vol. 79. Springer, New York (1982)

    MATH  Google Scholar 

  26. Willard, S.: General Topology. Series in Mathematics. Addison-Wesley, Reading (1970)

    MATH  Google Scholar 

  27. Zavalishchin, S.T.: Impulse dynamic systems and applications to mathematical economics. Dyn. Syst. Appl. 3(3), 443–449 (1994)

    MATH  MathSciNet  Google Scholar 

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Acknowledgments

The authors were partially funded by the European Regional Development Fund through the program COMPETE and by the Portuguese Government through the FCT - Fundação para a Ciência e a Tecnologia under the project PEst-C/MAT/UI0144/2013. JFA was also partially supported by Fundação Calouste Gulbenkian and the project PTDC/MAT/120346/2010.

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  1. Centro de Matemática da Universidade do Porto, Rua do Campo Alegre 687, 4169-007, Porto, Portugal

    José F. Alves & Maria Carvalho

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  1. José F. Alves
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  2. Maria Carvalho
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Correspondence to José F. Alves.

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Alves, J.F., Carvalho, M. Invariant Probability Measures and Non-wandering Sets for Impulsive Semiflows. J Stat Phys 157, 1097–1113 (2014). https://doi.org/10.1007/s10955-014-1101-0

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