# American Institute of Mathematical Sciences

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

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

 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.
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:
 [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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##### 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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