Periodic solutions of isotone hybrid systems

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

Periodic solutions of isotone hybrid systems

Thomas I. Seidman ^1, and Olaf Klein ^2,

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

Abstract
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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.

Keywords: fixed point, isotone, Periodic, discontinuous, hybrid system, calcium waves., hysteresis.

Mathematics Subject Classification: Primary: 34K34, 47J40; Secondary: 34A38, 34C55, 93C3.

Citation: Thomas I. Seidman, Olaf Klein. Periodic solutions of isotone hybrid systems. Discrete and 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., Amer. Math. Soc. Colloq. Publ., 25 AMS, Providence, 1948.
[2]	A. Friedman and L.-S. Jiang, Periodic solutions for a thermostat control problem, Comm. PDE, 13 (1988), 515-550.
[3]	K. Glashoff and J. Sprekels, The regulation of temperature by thermostats and set-valued integral equations, J. Int. Eqns., 4 (1982), 95-112.
[4]	G. Gripenberg, On periodic solutions of a thermostat equation, SIAM J. Math. Anal., 18 (1987), 694-702.
[5]	F. Hante, G. Leugering and T. I. Seidman, An augmented BV setting for feedback switching control, J. Systems Sci. & Comp., 23 (2010), 456-466. doi: 10.1007/s11424-010-0140-0.
[6]	L. T. Izu, W. G. Wier, and C. W. Balke, Evolution of cardiac calcium waves from stochastic calcium sparks, Biophysical Journal, 80 (2001), 103-120. doi: 10.1016/S0006-3495(01)75998-X.
[7]	M. A. Krasnosel'skiĭ and A. V. Pokrovskiĭ, Periodic oscillations in systems with relay nonlinearities, (transl. from (MR0355210), DAN USSR, 216 (1974), 733-736). Soviet Math Doklady, 15 (1974), 873-877.
[8]	M. A. Krasnosel'skiĭ and A .V. Pokrovskiĭ, "Systems with Hysteresis," (transl. of "Sistemy s Gisterezisom'', Nauka, Moscow, 1983) Springer-Verlag, Berlin, 1989.
[9]	O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," Translations of Math. Monographs, vol. 23, Amer. Math. Soc., Providence, 1968.
[10]	T. I. Seidman, Switching systems: thermostats and periodicity, Report MRR-83-07, UMBC, 1983, Available from: http://www.math.umbc.edu/~seidman/papers.html
[11]	T. I. Seidman, Switching systems I, Control and Cybernetics, 19 (1990), 63-92.
[12]	T. I. Seidman, Switching systems and periodicity, in "Nonlinear Semigroups, PDE, and Attractors'' (eds. T. E. Gill and W. W. Zachary), LCM \#1394; Springer-Verlag, New York, (1989), 199-210.
[13]	B. Stoth, "Periodische Lösungen Von Linearen Thermostat Problemen," Diplomthesis: (Report SFB 256), Univ. Bonn, 1987.
[14]	W. Szczechla, "Periodicity for Certain Switching Systems," Ph.D thesis, UMBC, 1993.

show all references

References:

[1]	G. Birkhoff, "Lattice Theory," $2^{nd}$ rev. ed., Amer. Math. Soc. Colloq. Publ., 25 AMS, Providence, 1948.
[2]	A. Friedman and L.-S. Jiang, Periodic solutions for a thermostat control problem, Comm. PDE, 13 (1988), 515-550.
[3]	K. Glashoff and J. Sprekels, The regulation of temperature by thermostats and set-valued integral equations, J. Int. Eqns., 4 (1982), 95-112.
[4]	G. Gripenberg, On periodic solutions of a thermostat equation, SIAM J. Math. Anal., 18 (1987), 694-702.
[5]	F. Hante, G. Leugering and T. I. Seidman, An augmented BV setting for feedback switching control, J. Systems Sci. & Comp., 23 (2010), 456-466. doi: 10.1007/s11424-010-0140-0.
[6]	L. T. Izu, W. G. Wier, and C. W. Balke, Evolution of cardiac calcium waves from stochastic calcium sparks, Biophysical Journal, 80 (2001), 103-120. doi: 10.1016/S0006-3495(01)75998-X.
[7]	M. A. Krasnosel'skiĭ and A. V. Pokrovskiĭ, Periodic oscillations in systems with relay nonlinearities, (transl. from (MR0355210), DAN USSR, 216 (1974), 733-736). Soviet Math Doklady, 15 (1974), 873-877.
[8]	M. A. Krasnosel'skiĭ and A .V. Pokrovskiĭ, "Systems with Hysteresis," (transl. of "Sistemy s Gisterezisom'', Nauka, Moscow, 1983) Springer-Verlag, Berlin, 1989.
[9]	O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," Translations of Math. Monographs, vol. 23, Amer. Math. Soc., Providence, 1968.
[10]	T. I. Seidman, Switching systems: thermostats and periodicity, Report MRR-83-07, UMBC, 1983, Available from: http://www.math.umbc.edu/~seidman/papers.html
[11]	T. I. Seidman, Switching systems I, Control and Cybernetics, 19 (1990), 63-92.
[12]	T. I. Seidman, Switching systems and periodicity, in "Nonlinear Semigroups, PDE, and Attractors'' (eds. T. E. Gill and W. W. Zachary), LCM \#1394; Springer-Verlag, New York, (1989), 199-210.
[13]	B. Stoth, "Periodische Lösungen Von Linearen Thermostat Problemen," Diplomthesis: (Report SFB 256), Univ. Bonn, 1987.
[14]	W. Szczechla, "Periodicity for Certain Switching Systems," Ph.D thesis, UMBC, 1993.

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