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April  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

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 & Continuous Dynamical Systems - S, 2013, 6 (4) : 1095-1112. doi: 10.3934/dcdss.2013.6.1095
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show all references

References:
[1]

IEEE Control Systems Magazine, 29 (2009), 44-69. Google Scholar

[2]

Physica B, 403 (2008), 301-304. Google Scholar

[3]

The American Economic Review, 84 (1994), Papers and Proceedings of the Hundred and Sixth Annual Meeting of the American Economic Association, 406-411. Google Scholar

[4]

J. Phys. C: Solid State Phys., 16 (1983), 2497-2508. Google Scholar

[5]

J. Dynamics & Differential Equations, 24 (2012), 713-759. Google Scholar

[6]

IEEE Control System Magazine, 19 (1999), 111-117. Google Scholar

[7]

Environmental Modeling & Assessment, 16 (2011), 313-333. Google Scholar

[8]

J. Math. Anal. Appl., 319 (2006), 94-109. doi: 10.1016/j.jmaa.2006.02.060.  Google Scholar

[9]

Springer, 1996. doi: 10.1007/978-1-4612-4048-8.  Google Scholar

[10]

Physica A, 246 (1997), 407-418. Google Scholar

[11]

Oxford University Press, 2005. Google Scholar

[12]

J. Phys.: Conf. Ser., 55 (2006), 55-62. Google Scholar

[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.  Google Scholar

[14]

Physica B, 403 (2008), 231-236. Google Scholar

[15]

Nature Physics, 1 (2005), 13-14. Google Scholar

[16]

in "Enciclopedia of Complexity and Systems Science" (eds. C. Marchetti and R. A. Meyers), Springer, (2009), 5021-5037. Google Scholar

[17]

Sensors and Actuators A: Physical, 147 (2008), 127-136. Google Scholar

[18]

Westview Press, 2003.  Google Scholar

[19]

Longmans, London, 1964. Google Scholar

[20]

Phys. Rev. B., 54 (1996), 18-21. Google Scholar

[21]

NoDEA Nonlinear Differential Equations Appl., 9 (2002), 93-115. doi: 10.1007/s00030-002-8120-2.  Google Scholar

[22]

Springer, 1989. doi: 10.1007/978-3-642-61302-9.  Google Scholar

[23]

Gakkotosho, Tokyo, 1996.  Google Scholar

[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.  Google Scholar

[25]

Physica D, 241 (2012), 2010-2028. doi: 10.1016/j.physd.2011.05.005.  Google Scholar

[26]

Journal of Intelligent Material Systems and Structures, 19 (2008), 1411-1424. Google Scholar

[27]

W. H. Freeman and Company, 1982.  Google Scholar

[28]

Springer, 1991. doi: 10.2172/6911694.  Google Scholar

[29]

in "Advances in Condensed Matter and Statistical Physics" (eds. E. Korutcheva and R. Cuerno), Nova Science Publishers, New York, (2004), 263-286. Google Scholar

[30]

Hydrol. Earth Syst. Sci., 11 (2007), 443-459. Google Scholar

[31]

Mathematical Modelling of Natural Phenomena, 7 (2012), 1-30. doi: 10.1051/mmnp/20127313.  Google Scholar

[32]

Discrete and Continuous Dynamical Systems B, 11 (2009), 997-1018. doi: 10.3934/dcdsb.2009.11.997.  Google Scholar

[33]

Functional Differential Equations, 6 (1999), 411-438.  Google Scholar

[34]

Chapter 1 in "Singular Perturbations and Hysteresis" (eds. M. Mortell, R. E. O'Malley, A. Pokrovskii and V. Sobolev), SIAM, (2005), 1-54.  Google Scholar

[35]

NoDEA Nonlinear Differential Equations Appl., 6 (1999), 267-288. doi: 10.1007/s000300050076.  Google Scholar

[36]

IEEE Trans. Power Deliv., 22 (2007), 919-929. Google Scholar

[37]

Phys. Rev Lett., 70 (1993), 3347. Google Scholar

[38]

Nature, 410 (2001), 242-250. Google Scholar

[39]

in "The Science of Hysteresis, II" (eds. G. Bertotti and I. Mayergoyz), Elsevier, (2005), 107-168. Google Scholar

[40]

Springer, 1998.  Google Scholar

[41]

Syst. Synth. Biol., 1 (2007), 59-87. Google Scholar

[42]

Springer, 1994.  Google Scholar

[43]

Proceedings of the National academy of Sci. of USA, 99 (2002), 5766-5771. doi: 10.1073/pnas.082090499.  Google Scholar

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