# American Institute of Mathematical Sciences

March  2013, 18(2): 483-493. doi: 10.3934/dcdsb.2013.18.483

## Periodic solutions of isotone hybrid systems

 1 Department of Mathematics and Statistics, University of Maryland Baltimore County (UMBC), Baltimore, MD 21250, United States 2 Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Mohrenstr. 39, 10117 Berlin, Germany

Received  December 2011 Revised  August 2012 Published  November 2012

Suggested by conversations in 1991 (Mark Krasnosel'skiĭ and Aleksei Pokrovskiĭ with TIS), this paper generalizes earlier work [7] of theirs by defining a setting of hybrid systems with isotone switching rules for a partially ordered set of modes and then obtaining a periodicity result in that context. An application is given to a partial differential equation modeling calcium release and diffusion in cardiac cells.
Citation: Thomas I. Seidman, Olaf Klein. Periodic solutions of isotone hybrid systems. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 483-493. doi: 10.3934/dcdsb.2013.18.483
##### References:
 [1] G. Birkhoff, "Lattice Theory,", $2^{nd}$ rev. ed., 25 (1948). Google Scholar [2] A. Friedman and L.-S. Jiang, Periodic solutions for a thermostat control problem,, Comm. PDE, 13 (1988), 515. Google Scholar [3] K. Glashoff and J. Sprekels, The regulation of temperature by thermostats and set-valued integral equations,, J. Int. Eqns., 4 (1982), 95. Google Scholar [4] G. Gripenberg, On periodic solutions of a thermostat equation,, SIAM J. Math. Anal., 18 (1987), 694. Google Scholar [5] F. Hante, G. Leugering and T. I. Seidman, An augmented BV setting for feedback switching control,, J. Systems Sci. & Comp., 23 (2010), 456. doi: 10.1007/s11424-010-0140-0. Google Scholar [6] L. T. Izu, W. G. Wier, and C. W. Balke, Evolution of cardiac calcium waves from stochastic calcium sparks,, Biophysical Journal, 80 (2001), 103. doi: 10.1016/S0006-3495(01)75998-X. Google Scholar [7] M. A. Krasnosel'skiĭ and A. V. Pokrovskiĭ, Periodic oscillations in systems with relay nonlinearities,, (transl. from (MR0355210), 216 (1974), 733. Google Scholar [8] M. A. Krasnosel'skiĭ and A .V. Pokrovskiĭ, "Systems with Hysteresis,", (transl. of, (1983). Google Scholar [9] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,", Translations of Math. Monographs, (1968). Google Scholar [10] T. I. Seidman, Switching systems: thermostats and periodicity,, Report MRR-83-07, MRR-83-07 (1983), 83. Google Scholar [11] T. I. Seidman, Switching systems I,, Control and Cybernetics, 19 (1990), 63. Google Scholar [12] T. I. Seidman, Switching systems and periodicity,, in, (1989), 199. Google Scholar [13] B. Stoth, "Periodische Lösungen Von Linearen Thermostat Problemen,", Diplomthesis: (Report SFB 256), SFB 256 (1987). Google Scholar [14] W. Szczechla, "Periodicity for Certain Switching Systems,", Ph.D thesis, (1993). Google Scholar

show all references

##### References:
 [1] G. Birkhoff, "Lattice Theory,", $2^{nd}$ rev. ed., 25 (1948). Google Scholar [2] A. Friedman and L.-S. Jiang, Periodic solutions for a thermostat control problem,, Comm. PDE, 13 (1988), 515. Google Scholar [3] K. Glashoff and J. Sprekels, The regulation of temperature by thermostats and set-valued integral equations,, J. Int. Eqns., 4 (1982), 95. Google Scholar [4] G. Gripenberg, On periodic solutions of a thermostat equation,, SIAM J. Math. Anal., 18 (1987), 694. Google Scholar [5] F. Hante, G. Leugering and T. I. Seidman, An augmented BV setting for feedback switching control,, J. Systems Sci. & Comp., 23 (2010), 456. doi: 10.1007/s11424-010-0140-0. Google Scholar [6] L. T. Izu, W. G. Wier, and C. W. Balke, Evolution of cardiac calcium waves from stochastic calcium sparks,, Biophysical Journal, 80 (2001), 103. doi: 10.1016/S0006-3495(01)75998-X. Google Scholar [7] M. A. Krasnosel'skiĭ and A. V. Pokrovskiĭ, Periodic oscillations in systems with relay nonlinearities,, (transl. from (MR0355210), 216 (1974), 733. Google Scholar [8] M. A. Krasnosel'skiĭ and A .V. Pokrovskiĭ, "Systems with Hysteresis,", (transl. of, (1983). Google Scholar [9] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,", Translations of Math. Monographs, (1968). Google Scholar [10] T. I. Seidman, Switching systems: thermostats and periodicity,, Report MRR-83-07, MRR-83-07 (1983), 83. Google Scholar [11] T. I. Seidman, Switching systems I,, Control and Cybernetics, 19 (1990), 63. Google Scholar [12] T. I. Seidman, Switching systems and periodicity,, in, (1989), 199. Google Scholar [13] B. Stoth, "Periodische Lösungen Von Linearen Thermostat Problemen,", Diplomthesis: (Report SFB 256), SFB 256 (1987). Google Scholar [14] W. Szczechla, "Periodicity for Certain Switching Systems,", Ph.D thesis, (1993). Google Scholar
 [1] Teck-Cheong Lim. On the largest common fixed point of a commuting family of isotone maps. Conference Publications, 2005, 2005 (Special) : 621-623. doi: 10.3934/proc.2005.2005.621 [2] Silvia Sastre-Gomez. Equivalent formulations for steady periodic water waves of fixed mean-depth with discontinuous vorticity. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2669-2680. doi: 10.3934/dcds.2017114 [3] Sorin Micu, Ademir F. Pazoto. Almost periodic solutions for a weakly dissipated hybrid system. Mathematical Control & Related Fields, 2014, 4 (1) : 101-113. doi: 10.3934/mcrf.2014.4.101 [4] Luca Dieci, Timo Eirola, Cinzia Elia. Periodic orbits of planar discontinuous system under discretization. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2743-2762. doi: 10.3934/dcdsb.2018103 [5] Jerry L. Bona, Thierry Colin, Colette Guillopé. Propagation of long-crested water waves. Ⅱ. Bore propagation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5543-5569. doi: 10.3934/dcds.2019244 [6] Adrian Constantin. Dispersion relations for periodic traveling water waves in flows with discontinuous vorticity. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1397-1406. doi: 10.3934/cpaa.2012.11.1397 [7] Calin Iulian Martin. Dispersion relations for periodic water waves with surface tension and discontinuous vorticity. Discrete & Continuous Dynamical Systems - A, 2014, 34 (8) : 3109-3123. doi: 10.3934/dcds.2014.34.3109 [8] Bogdan Kazmierczak, Zbigniew Peradzynski. Calcium waves with mechano-chemical couplings. Mathematical Biosciences & Engineering, 2013, 10 (3) : 743-759. doi: 10.3934/mbe.2013.10.743 [9] Nicholas Long. Fixed point shifts of inert involutions. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1297-1317. doi: 10.3934/dcds.2009.25.1297 [10] Calin Iulian Martin, Adrián Rodríguez-Sanjurjo. Dispersion relations for steady periodic water waves of fixed mean-depth with two rotational layers. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5149-5169. doi: 10.3934/dcds.2019209 [11] Yakov Krasnov, Alexander Kononovich, Grigory Osharovich. On a structure of the fixed point set of homogeneous maps. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 1017-1027. doi: 10.3934/dcdss.2013.6.1017 [12] Jorge Groisman. Expansive and fixed point free homeomorphisms of the plane. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1709-1721. doi: 10.3934/dcds.2012.32.1709 [13] Shui-Hung Hou. On an application of fixed point theorem to nonlinear inclusions. Conference Publications, 2011, 2011 (Special) : 692-697. doi: 10.3934/proc.2011.2011.692 [14] Luis Hernández-Corbato, Francisco R. Ruiz del Portal. Fixed point indices of planar continuous maps. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 2979-2995. doi: 10.3934/dcds.2015.35.2979 [15] Antonio Garcia. Transition tori near an elliptic-fixed point. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 381-392. doi: 10.3934/dcds.2000.6.381 [16] Pavel Gurevich. Periodic solutions of parabolic problems with hysteresis on the boundary. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1041-1083. doi: 10.3934/dcds.2011.29.1041 [17] Fábio Natali, Ademir Pastor. Orbital stability of periodic waves for the Klein-Gordon-Schrödinger system. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 221-238. doi: 10.3934/dcds.2011.31.221 [18] Hany A. Hosham, Eman D Abou Elela. Discontinuous phenomena in bioreactor system. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2955-2969. doi: 10.3934/dcdsb.2018294 [19] Shin-Ichiro Ei, Kei Nishi, Yasumasa Nishiura, Takashi Teramoto. Annihilation of two interfaces in a hybrid system. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 857-869. doi: 10.3934/dcdss.2015.8.857 [20] Wenhua Qiu, Jianguo Si. On small perturbation of four-dimensional quasi-periodic system with degenerate equilibrium point. Communications on Pure & Applied Analysis, 2015, 14 (2) : 421-437. doi: 10.3934/cpaa.2015.14.421

2018 Impact Factor: 1.008