Effect of positive feedback on Devil's staircase input-output relationship

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August 2013, 6(4): 1095-1112. doi: 10.3934/dcdss.2013.6.1095

Effect of positive feedback on Devil's staircase input-output relationship

Alexei Pokrovskii ^1,2, and Dmitrii Rachinskii ^1,2,

1.	Department of Applied Mathematics
2.	University College, Cork

Received January 2011 Revised February 2012 Published December 2012

Abstract
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We consider emerging hysteresis behaviour in a closed loop systemthat includes a nonlinear link $f$ of the Devil's staircase (Cantorfunction) type and a positive feedback. This type of closed loopsarises naturally in analysis of networks where local ``negative''coupling of network elements is combined with ``positive'' couplingat the level of the mean-field interaction (in the limit case whenthe impact of each individual vertex is infinitesimal, while thenumber of vertices is growing). For the Cantor function $f$, takenas a model, and for a monotonically increasing input, we present thecorresponding output of the system explicitly, showing that theoutput is piecewise constant and has a finite number of equal jumps.We then discuss hysteresis loops of the system for genericnon-monotone inputs. The results are presented in the context of differential equations describingnonlinear control systems with almost immediate linear feedback, i.e., in the limit where the time of propagation of the signalthrough the feedback loop tends to zero.

Keywords: Transducer, feedback, differential system, delay, Cantor function, hysteresis loop, non-ideal relay, Preisach operator..

Mathematics Subject Classification: Primary: 93C10, 93C15; Secondary: 34H0.

Citation: Alexei Pokrovskii, Dmitrii Rachinskii. Effect of positive feedback on Devil's staircase input-output relationship. Discrete and Continuous Dynamical Systems - S, 2013, 6 (4) : 1095-1112. doi: 10.3934/dcdss.2013.6.1095

References:

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[29]	in "Advances in Condensed Matter and Statistical Physics" (eds. E. Korutcheva and R. Cuerno), Nova Science Publishers, New York, (2004), 263-286.
[30]	Hydrol. Earth Syst. Sci., 11 (2007), 443-459.
[31]	Mathematical Modelling of Natural Phenomena, 7 (2012), 1-30. doi: 10.1051/mmnp/20127313.
[32]	Discrete and Continuous Dynamical Systems B, 11 (2009), 997-1018. doi: 10.3934/dcdsb.2009.11.997.
[33]	Functional Differential Equations, 6 (1999), 411-438.
[34]	Chapter 1 in "Singular Perturbations and Hysteresis" (eds. M. Mortell, R. E. O'Malley, A. Pokrovskii and V. Sobolev), SIAM, (2005), 1-54.
[35]	NoDEA Nonlinear Differential Equations Appl., 6 (1999), 267-288. doi: 10.1007/s000300050076.
[36]	IEEE Trans. Power Deliv., 22 (2007), 919-929.
[37]	Phys. Rev Lett., 70 (1993), 3347.
[38]	Nature, 410 (2001), 242-250.
[39]	in "The Science of Hysteresis, II" (eds. G. Bertotti and I. Mayergoyz), Elsevier, (2005), 107-168.
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[43]	Proceedings of the National academy of Sci. of USA, 99 (2002), 5766-5771. doi: 10.1073/pnas.082090499.

show all references

References:

[1]	IEEE Control Systems Magazine, 29 (2009), 44-69.
[2]	Physica B, 403 (2008), 301-304.
[3]	The American Economic Review, 84 (1994), Papers and Proceedings of the Hundred and Sixth Annual Meeting of the American Economic Association, 406-411.
[4]	J. Phys. C: Solid State Phys., 16 (1983), 2497-2508.
[5]	J. Dynamics & Differential Equations, 24 (2012), 713-759.
[6]	IEEE Control System Magazine, 19 (1999), 111-117.
[7]	Environmental Modeling & Assessment, 16 (2011), 313-333.
[8]	J. Math. Anal. Appl., 319 (2006), 94-109. doi: 10.1016/j.jmaa.2006.02.060.
[9]	Springer, 1996. doi: 10.1007/978-1-4612-4048-8.
[10]	Physica A, 246 (1997), 407-418.
[11]	Oxford University Press, 2005.
[12]	J. Phys.: Conf. Ser., 55 (2006), 55-62.
[13]	Chapter 2 in "Singular Perturbations and Hysteresis" (eds. M. Mortell, R. E. O'Malley, A. Pokrovskii and V. Sobolev), SIAM, (2005), 61-72. doi: 10.1137/1.9780898717860.
[14]	Physica B, 403 (2008), 231-236.
[15]	Nature Physics, 1 (2005), 13-14.
[16]	in "Enciclopedia of Complexity and Systems Science" (eds. C. Marchetti and R. A. Meyers), Springer, (2009), 5021-5037.
[17]	Sensors and Actuators A: Physical, 147 (2008), 127-136.
[18]	Westview Press, 2003.
[19]	Longmans, London, 1964.
[20]	Phys. Rev. B., 54 (1996), 18-21.
[21]	NoDEA Nonlinear Differential Equations Appl., 9 (2002), 93-115. doi: 10.1007/s00030-002-8120-2.
[22]	Springer, 1989. doi: 10.1007/978-3-642-61302-9.
[23]	Gakkotosho, Tokyo, 1996.
[24]	Chapter 3 in "Singular Perturbations and Hysteresi" (eds. M. Mortell, R. E. O'Malley, A. Pokrovskii and V. Sobolev), SIAM, (2005), 73-100. doi: 10.1137/1.9780898717860.ch3.
[25]	Physica D, 241 (2012), 2010-2028. doi: 10.1016/j.physd.2011.05.005.
[26]	Journal of Intelligent Material Systems and Structures, 19 (2008), 1411-1424.
[27]	W. H. Freeman and Company, 1982.
[28]	Springer, 1991. doi: 10.2172/6911694.
[29]	in "Advances in Condensed Matter and Statistical Physics" (eds. E. Korutcheva and R. Cuerno), Nova Science Publishers, New York, (2004), 263-286.
[30]	Hydrol. Earth Syst. Sci., 11 (2007), 443-459.
[31]	Mathematical Modelling of Natural Phenomena, 7 (2012), 1-30. doi: 10.1051/mmnp/20127313.
[32]	Discrete and Continuous Dynamical Systems B, 11 (2009), 997-1018. doi: 10.3934/dcdsb.2009.11.997.
[33]	Functional Differential Equations, 6 (1999), 411-438.
[34]	Chapter 1 in "Singular Perturbations and Hysteresis" (eds. M. Mortell, R. E. O'Malley, A. Pokrovskii and V. Sobolev), SIAM, (2005), 1-54.
[35]	NoDEA Nonlinear Differential Equations Appl., 6 (1999), 267-288. doi: 10.1007/s000300050076.
[36]	IEEE Trans. Power Deliv., 22 (2007), 919-929.
[37]	Phys. Rev Lett., 70 (1993), 3347.
[38]	Nature, 410 (2001), 242-250.
[39]	in "The Science of Hysteresis, II" (eds. G. Bertotti and I. Mayergoyz), Elsevier, (2005), 107-168.
[40]	Springer, 1998.
[41]	Syst. Synth. Biol., 1 (2007), 59-87.
[42]	Springer, 1994.
[43]	Proceedings of the National academy of Sci. of USA, 99 (2002), 5766-5771. doi: 10.1073/pnas.082090499.

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