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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

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 & 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. Google Scholar

[2]

A. Friedman and L.-S. Jiang, Periodic solutions for a thermostat control problem, Comm. PDE, 13 (1988), 515-550.  Google Scholar

[3]

K. Glashoff and J. Sprekels, The regulation of temperature by thermostats and set-valued integral equations, J. Int. Eqns., 4 (1982), 95-112.  Google Scholar

[4]

G. Gripenberg, On periodic solutions of a thermostat equation, SIAM J. Math. Anal., 18 (1987), 694-702.  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), 456-466. 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), 103-120. 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), DAN USSR, 216 (1974), 733-736). Soviet Math Doklady, 15 (1974), 873-877. Google Scholar

[8]

M. A. Krasnosel'skiĭ and A .V. Pokrovskiĭ, "Systems with Hysteresis," (transl. of "Sistemy s Gisterezisom'', Nauka, Moscow, 1983) Springer-Verlag, Berlin, 1989.  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, vol. 23, Amer. Math. Soc., Providence, 1968.  Google Scholar

[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 Google Scholar

[11]

T. I. Seidman, Switching systems I, Control and Cybernetics, 19 (1990), 63-92.  Google Scholar

[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. Google Scholar

[13]

B. Stoth, "Periodische Lösungen Von Linearen Thermostat Problemen," Diplomthesis: (Report SFB 256), Univ. Bonn, 1987. Google Scholar

[14]

W. Szczechla, "Periodicity for Certain Switching Systems," Ph.D thesis, UMBC, 1993. Google Scholar

show all references

References:
[1]

G. Birkhoff, "Lattice Theory," $2^{nd}$ rev. ed., Amer. Math. Soc. Colloq. Publ., 25 AMS, Providence, 1948. Google Scholar

[2]

A. Friedman and L.-S. Jiang, Periodic solutions for a thermostat control problem, Comm. PDE, 13 (1988), 515-550.  Google Scholar

[3]

K. Glashoff and J. Sprekels, The regulation of temperature by thermostats and set-valued integral equations, J. Int. Eqns., 4 (1982), 95-112.  Google Scholar

[4]

G. Gripenberg, On periodic solutions of a thermostat equation, SIAM J. Math. Anal., 18 (1987), 694-702.  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), 456-466. 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), 103-120. 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), DAN USSR, 216 (1974), 733-736). Soviet Math Doklady, 15 (1974), 873-877. Google Scholar

[8]

M. A. Krasnosel'skiĭ and A .V. Pokrovskiĭ, "Systems with Hysteresis," (transl. of "Sistemy s Gisterezisom'', Nauka, Moscow, 1983) Springer-Verlag, Berlin, 1989.  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, vol. 23, Amer. Math. Soc., Providence, 1968.  Google Scholar

[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 Google Scholar

[11]

T. I. Seidman, Switching systems I, Control and Cybernetics, 19 (1990), 63-92.  Google Scholar

[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. Google Scholar

[13]

B. Stoth, "Periodische Lösungen Von Linearen Thermostat Problemen," Diplomthesis: (Report SFB 256), Univ. Bonn, 1987. Google Scholar

[14]

W. Szczechla, "Periodicity for Certain Switching Systems," Ph.D thesis, UMBC, 1993. Google Scholar

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