# American Institute of Mathematical Sciences

April  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 & Continuous Dynamical Systems - S, 2013, 6 (4) : 1095-1112. doi: 10.3934/dcdss.2013.6.1095
##### References:
 [1] B. Applebe, D. Flynn, H. McNamara, J. P. O'Kane, A. Pimenov, A. Pokrovskii, D. Rachinskii and A. Zhezherun, Rate-independent hysteresis in terrestrial hydrology,, IEEE Control Systems Magazine, 29 (2009), 44. [2] B. Appelbe, D. Rachinskii and A. Zhezherun, Hopf bifurcation in a van der Pol type oscillator with magnetic hysteresis,, Physica B, 403 (2008), 301. [3] W. B. Arthur, Inductive reasoning and bounded rationality,, The American Economic Review, 84 (1994), 406. [4] S. Aubry, Exact models with a complete Devil's staircase,, J. Phys. C: Solid State Phys., 16 (1983), 2497. [5] Z. Balanov, W. Krawcewicz and D. Rachinskii, Hopf bifurcation in symmetric systems of coupled oscillators with Preisach memory,, J. Dynamics & Differential Equations, 24 (2012), 713. [6] C. C. Bissell, Control engineering in the former USSR,, IEEE Control System Magazine, 19 (1999), 111. [7] M. Brokate, S. McCarthy, A. Pimenov, A. Pokrovskii and D. Rachinskii, Energy dissipation in hydrological systems due to hysteresis,, Environmental Modeling & Assessment, 16 (2011), 313. [8] M. Brokate, A. Pokrovskii and D. Rachinskii, Asymptotic stability of continual sets of periodic solutions to systems with hysteresis,, J. Math. Anal. Appl., 319 (2006), 94. doi: 10.1016/j.jmaa.2006.02.060. [9] M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions,", Springer, (1996). doi: 10.1007/978-1-4612-4048-8. [10] D. Challet and Y.-C. Zhang, Emergence of cooperation and organization in an evolutionary game,, Physica A, 246 (1997), 407. [11] D. Challet, M. Marsili and Y.-C. Zhang, "Minority Games,", Oxford University Press, (2005). [12] R. Cross, M. Grinfeld and H. Lamba, A mean-field model of investor behaviour,, J. Phys.: Conf. Ser., 55 (2006), 55. [13] R. Cross, M. Grinfeld and H. Lamba, Pittock, Rationality, frustration minimization, hysteresis, and the El Farol Problem,, Chapter 2 in, (2005), 61. doi: 10.1137/1.9780898717860. [14] R. Cross, H. McNamara, A. Pokrovskii and D. Rachinskii, A new paradigm for modelling hysteresis in macroeconomic flows,, Physica B, 403 (2008), 231. [15] K. Dahmen, Nonlinear dynamics: Universal clues in noisy skews,, Nature Physics, 1 (2005), 13. [16] K. Dahmen and Y. Ben-Zion, The physics of jerky motion in slowly driven magnetic and earthquake fault systems,, in, (2009), 5021. [17] D. Davinoa, C. Visonea, C. Ambrosinoa, S. Campopianob, A. Cusanoa and A. Cutoloa, Compensation of hysteresis in magnetic field sensors employing Fiber Bragg Grating and magneto-elastic materials,, Sensors and Actuators A: Physical, 147 (2008), 127. [18] R. L. Devaney, "An Introduction to Chaotic Dynamical Systems,", Westview Press, (2003). [19] A. Draper, "Electrical Circuits, Including Machines,", Longmans, (1964). [20] R. A. Guyer and K. R. McCall, Capillary condensation, invasion percolation, hysteresis, and discrete memory,, Phys. Rev. B., 54 (1996), 18. [21] A. M. Krasnosel'skii and D. Rachinskii, On a bifurcation governed by hysteresis nonlinearity,, NoDEA Nonlinear Differential Equations Appl., 9 (2002), 93. doi: 10.1007/s00030-002-8120-2. [22] M. Krasnosel'skii and A. Pokrovskii, "Systems With Hysteresis,", Springer, (1989). doi: 10.1007/978-3-642-61302-9. [23] P. Krejci, "Hysteresis, Convexity and Dissipation in Hyperbolic Equations,", Gakkotosho, (1996). [24] P. Krejci, Hysteresis in singularly perturbed problems,, Chapter 3 in, (2005), 73. doi: 10.1137/1.9780898717860.ch3. [25] P. Krejci, J. P. O'Kane, A. Pokrovskii and D. Rachinskii, Properties of solutions to a class of differential models incorporating Preisach hysteresis operator,, Physica D, 241 (2012), 2010. doi: 10.1016/j.physd.2011.05.005. [26] K. Kuhnen, Compensation of parameter-dependent complex hysteretic actuator nonlinearities in smart material systems,, Journal of Intelligent Material Systems and Structures, 19 (2008), 1411. [27] B. B. Mandelbrot, "The Fractal Geometry of Nature,", W. H. Freeman and Company, (1982). [28] I. D. Mayergoyz, "Mathematical Models of Hysteresis,", Springer, (1991). doi: 10.2172/6911694. [29] E. Moro, The Minority game: An introductory guide,, in, (2004), 263. [30] J. P. O'Kane and D. Flynn, Thresholds, switches and hysteresis in hydrology from the pedon to the catchment scale: A non-linear systems theory,, Hydrol. Earth Syst. Sci., 11 (2007), 443. [31] A. Pimenov, T. C. Kelly, A. Korobeinikov, M. J. A. O'Callaghan, A. Pokrovskii and D. Rachinskii, Memory effects in population dynamics: Spread of infectious disease as a case study,, Mathematical Modelling of Natural Phenomena, 7 (2012), 1. doi: 10.1051/mmnp/20127313. [32] A. Pimenov and D. Rachinskii, Linear stability analysis of systems with Preisach memory,, Discrete and Continuous Dynamical Systems B, 11 (2009), 997. doi: 10.3934/dcdsb.2009.11.997. [33] A. Pokrovskii, F. Holland, J. McInerney, M. Suzuki and T. Suzuki, Robustness of an analog dynamic memory system to a class of information transmission channels perturbations,, Functional Differential Equations, 6 (1999), 411. [34] A. Pokrovskii and V. Sobolev, Naive view on singular perturbation and hysteresis,, Chapter 1 in, (2005), 1. [35] D. Rachinskii, Asymptotic stability of large-amplitude oscillations in systems with hysteresis,, NoDEA Nonlinear Differential Equations Appl., 6 (1999), 267. doi: 10.1007/s000300050076. [36] A. Rezaei-Zareet, M. Sanaye-Pasand, H. Mohseni, S. Farhangi and R. Iravani, Analysis of ferroresonance modes in power transformers using Preisach-type hysteretic magnetizing inductance,, IEEE Trans. Power Deliv., 22 (2007), 919. [37] J. P. Sethna, K. Dahmen, S. Kartha, J. A. Krumhansl, B. W. Robetrs and J. D. Shore, Hysteresis and hierarchies: Dynamics of disorder-driven first-order phase transitions,, Phys. Rev Lett., 70 (1993). [38] J. P. Sethna, K. Dahmen and C. R. Myers, Crackling noise,, Nature, 410 (2001), 242. [39] J. P. Sethna, K. A. Dahmen and O. Perkovic, Random-Field Ising Models of Hysteresis,, in, II (2005), 107. [40] E. D. Sontag, "Mathematical Control Theory,", Springer, (1998). [41] E. D. Sontag, Monotone and near-monotone biochemical networks,, Syst. Synth. Biol., 1 (2007), 59. [42] A. Visintin, "Differential Models of Hysteresis,", Springer, (1994). [43] D. J. Watts, A simple model of global cascades on random networks,, Proceedings of the National academy of Sci. of USA, 99 (2002), 5766. doi: 10.1073/pnas.082090499.

show all references

##### References:
 [1] B. Applebe, D. Flynn, H. McNamara, J. P. O'Kane, A. Pimenov, A. Pokrovskii, D. Rachinskii and A. Zhezherun, Rate-independent hysteresis in terrestrial hydrology,, IEEE Control Systems Magazine, 29 (2009), 44. [2] B. Appelbe, D. Rachinskii and A. Zhezherun, Hopf bifurcation in a van der Pol type oscillator with magnetic hysteresis,, Physica B, 403 (2008), 301. [3] W. B. Arthur, Inductive reasoning and bounded rationality,, The American Economic Review, 84 (1994), 406. [4] S. Aubry, Exact models with a complete Devil's staircase,, J. Phys. C: Solid State Phys., 16 (1983), 2497. [5] Z. Balanov, W. Krawcewicz and D. Rachinskii, Hopf bifurcation in symmetric systems of coupled oscillators with Preisach memory,, J. Dynamics & Differential Equations, 24 (2012), 713. [6] C. C. Bissell, Control engineering in the former USSR,, IEEE Control System Magazine, 19 (1999), 111. [7] M. Brokate, S. McCarthy, A. Pimenov, A. Pokrovskii and D. Rachinskii, Energy dissipation in hydrological systems due to hysteresis,, Environmental Modeling & Assessment, 16 (2011), 313. [8] M. Brokate, A. Pokrovskii and D. Rachinskii, Asymptotic stability of continual sets of periodic solutions to systems with hysteresis,, J. Math. Anal. Appl., 319 (2006), 94. doi: 10.1016/j.jmaa.2006.02.060. [9] M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions,", Springer, (1996). doi: 10.1007/978-1-4612-4048-8. [10] D. Challet and Y.-C. Zhang, Emergence of cooperation and organization in an evolutionary game,, Physica A, 246 (1997), 407. [11] D. Challet, M. Marsili and Y.-C. Zhang, "Minority Games,", Oxford University Press, (2005). [12] R. Cross, M. Grinfeld and H. Lamba, A mean-field model of investor behaviour,, J. Phys.: Conf. Ser., 55 (2006), 55. [13] R. Cross, M. Grinfeld and H. Lamba, Pittock, Rationality, frustration minimization, hysteresis, and the El Farol Problem,, Chapter 2 in, (2005), 61. doi: 10.1137/1.9780898717860. [14] R. Cross, H. McNamara, A. Pokrovskii and D. Rachinskii, A new paradigm for modelling hysteresis in macroeconomic flows,, Physica B, 403 (2008), 231. [15] K. Dahmen, Nonlinear dynamics: Universal clues in noisy skews,, Nature Physics, 1 (2005), 13. [16] K. Dahmen and Y. Ben-Zion, The physics of jerky motion in slowly driven magnetic and earthquake fault systems,, in, (2009), 5021. [17] D. Davinoa, C. Visonea, C. Ambrosinoa, S. Campopianob, A. Cusanoa and A. Cutoloa, Compensation of hysteresis in magnetic field sensors employing Fiber Bragg Grating and magneto-elastic materials,, Sensors and Actuators A: Physical, 147 (2008), 127. [18] R. L. Devaney, "An Introduction to Chaotic Dynamical Systems,", Westview Press, (2003). [19] A. Draper, "Electrical Circuits, Including Machines,", Longmans, (1964). [20] R. A. Guyer and K. R. McCall, Capillary condensation, invasion percolation, hysteresis, and discrete memory,, Phys. Rev. B., 54 (1996), 18. [21] A. M. Krasnosel'skii and D. Rachinskii, On a bifurcation governed by hysteresis nonlinearity,, NoDEA Nonlinear Differential Equations Appl., 9 (2002), 93. doi: 10.1007/s00030-002-8120-2. [22] M. Krasnosel'skii and A. Pokrovskii, "Systems With Hysteresis,", Springer, (1989). doi: 10.1007/978-3-642-61302-9. [23] P. Krejci, "Hysteresis, Convexity and Dissipation in Hyperbolic Equations,", Gakkotosho, (1996). [24] P. Krejci, Hysteresis in singularly perturbed problems,, Chapter 3 in, (2005), 73. doi: 10.1137/1.9780898717860.ch3. [25] P. Krejci, J. P. O'Kane, A. Pokrovskii and D. Rachinskii, Properties of solutions to a class of differential models incorporating Preisach hysteresis operator,, Physica D, 241 (2012), 2010. doi: 10.1016/j.physd.2011.05.005. [26] K. Kuhnen, Compensation of parameter-dependent complex hysteretic actuator nonlinearities in smart material systems,, Journal of Intelligent Material Systems and Structures, 19 (2008), 1411. [27] B. B. Mandelbrot, "The Fractal Geometry of Nature,", W. H. Freeman and Company, (1982). [28] I. D. Mayergoyz, "Mathematical Models of Hysteresis,", Springer, (1991). doi: 10.2172/6911694. [29] E. Moro, The Minority game: An introductory guide,, in, (2004), 263. [30] J. P. O'Kane and D. Flynn, Thresholds, switches and hysteresis in hydrology from the pedon to the catchment scale: A non-linear systems theory,, Hydrol. Earth Syst. Sci., 11 (2007), 443. [31] A. Pimenov, T. C. Kelly, A. Korobeinikov, M. J. A. O'Callaghan, A. Pokrovskii and D. Rachinskii, Memory effects in population dynamics: Spread of infectious disease as a case study,, Mathematical Modelling of Natural Phenomena, 7 (2012), 1. doi: 10.1051/mmnp/20127313. [32] A. Pimenov and D. Rachinskii, Linear stability analysis of systems with Preisach memory,, Discrete and Continuous Dynamical Systems B, 11 (2009), 997. doi: 10.3934/dcdsb.2009.11.997. [33] A. Pokrovskii, F. Holland, J. McInerney, M. Suzuki and T. Suzuki, Robustness of an analog dynamic memory system to a class of information transmission channels perturbations,, Functional Differential Equations, 6 (1999), 411. [34] A. Pokrovskii and V. Sobolev, Naive view on singular perturbation and hysteresis,, Chapter 1 in, (2005), 1. [35] D. Rachinskii, Asymptotic stability of large-amplitude oscillations in systems with hysteresis,, NoDEA Nonlinear Differential Equations Appl., 6 (1999), 267. doi: 10.1007/s000300050076. [36] A. Rezaei-Zareet, M. Sanaye-Pasand, H. Mohseni, S. Farhangi and R. Iravani, Analysis of ferroresonance modes in power transformers using Preisach-type hysteretic magnetizing inductance,, IEEE Trans. Power Deliv., 22 (2007), 919. [37] J. P. Sethna, K. Dahmen, S. Kartha, J. A. Krumhansl, B. W. Robetrs and J. D. Shore, Hysteresis and hierarchies: Dynamics of disorder-driven first-order phase transitions,, Phys. Rev Lett., 70 (1993). [38] J. P. Sethna, K. Dahmen and C. R. Myers, Crackling noise,, Nature, 410 (2001), 242. [39] J. P. Sethna, K. A. Dahmen and O. Perkovic, Random-Field Ising Models of Hysteresis,, in, II (2005), 107. [40] E. D. Sontag, "Mathematical Control Theory,", Springer, (1998). [41] E. D. Sontag, Monotone and near-monotone biochemical networks,, Syst. Synth. Biol., 1 (2007), 59. [42] A. Visintin, "Differential Models of Hysteresis,", Springer, (1994). [43] D. J. Watts, A simple model of global cascades on random networks,, Proceedings of the National academy of Sci. of USA, 99 (2002), 5766. doi: 10.1073/pnas.082090499.
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