American Institute of Mathematical Sciences

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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
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.   Google Scholar

[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.   Google Scholar

[3]

W. B. Arthur, Inductive reasoning and bounded rationality,, The American Economic Review, 84 (1994), 406.   Google Scholar

[4]

S. Aubry, Exact models with a complete Devil's staircase,, J. Phys. C: Solid State Phys., 16 (1983), 2497.   Google Scholar

[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.   Google Scholar

[6]

C. C. Bissell, Control engineering in the former USSR,, IEEE Control System Magazine, 19 (1999), 111.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[9]

M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions,", Springer, (1996).  doi: 10.1007/978-1-4612-4048-8.  Google Scholar

[10]

D. Challet and Y.-C. Zhang, Emergence of cooperation and organization in an evolutionary game,, Physica A, 246 (1997), 407.   Google Scholar

[11]

D. Challet, M. Marsili and Y.-C. Zhang, "Minority Games,", Oxford University Press, (2005).   Google Scholar

[12]

R. Cross, M. Grinfeld and H. Lamba, A mean-field model of investor behaviour,, J. Phys.: Conf. Ser., 55 (2006), 55.   Google Scholar

[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.  Google Scholar

[14]

R. Cross, H. McNamara, A. Pokrovskii and D. Rachinskii, A new paradigm for modelling hysteresis in macroeconomic flows,, Physica B, 403 (2008), 231.   Google Scholar

[15]

K. Dahmen, Nonlinear dynamics: Universal clues in noisy skews,, Nature Physics, 1 (2005), 13.   Google Scholar

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K. Dahmen and Y. Ben-Zion, The physics of jerky motion in slowly driven magnetic and earthquake fault systems,, in, (2009), 5021.   Google Scholar

[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.   Google Scholar

[18]

R. L. Devaney, "An Introduction to Chaotic Dynamical Systems,", Westview Press, (2003).   Google Scholar

[19]

A. Draper, "Electrical Circuits, Including Machines,", Longmans, (1964).   Google Scholar

[20]

R. A. Guyer and K. R. McCall, Capillary condensation, invasion percolation, hysteresis, and discrete memory,, Phys. Rev. B., 54 (1996), 18.   Google Scholar

[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.  Google Scholar

[22]

M. Krasnosel'skii and A. Pokrovskii, "Systems With Hysteresis,", Springer, (1989).  doi: 10.1007/978-3-642-61302-9.  Google Scholar

[23]

P. Krejci, "Hysteresis, Convexity and Dissipation in Hyperbolic Equations,", Gakkotosho, (1996).   Google Scholar

[24]

P. Krejci, Hysteresis in singularly perturbed problems,, Chapter 3 in, (2005), 73.  doi: 10.1137/1.9780898717860.ch3.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[27]

B. B. Mandelbrot, "The Fractal Geometry of Nature,", W. H. Freeman and Company, (1982).   Google Scholar

[28]

I. D. Mayergoyz, "Mathematical Models of Hysteresis,", Springer, (1991).  doi: 10.2172/6911694.  Google Scholar

[29]

E. Moro, The Minority game: An introductory guide,, in, (2004), 263.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[34]

A. Pokrovskii and V. Sobolev, Naive view on singular perturbation and hysteresis,, Chapter 1 in, (2005), 1.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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).   Google Scholar

[38]

J. P. Sethna, K. Dahmen and C. R. Myers, Crackling noise,, Nature, 410 (2001), 242.   Google Scholar

[39]

J. P. Sethna, K. A. Dahmen and O. Perkovic, Random-Field Ising Models of Hysteresis,, in, II (2005), 107.   Google Scholar

[40]

E. D. Sontag, "Mathematical Control Theory,", Springer, (1998).   Google Scholar

[41]

E. D. Sontag, Monotone and near-monotone biochemical networks,, Syst. Synth. Biol., 1 (2007), 59.   Google Scholar

[42]

A. Visintin, "Differential Models of Hysteresis,", Springer, (1994).   Google Scholar

[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.  Google Scholar

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.   Google Scholar

[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.   Google Scholar

[3]

W. B. Arthur, Inductive reasoning and bounded rationality,, The American Economic Review, 84 (1994), 406.   Google Scholar

[4]

S. Aubry, Exact models with a complete Devil's staircase,, J. Phys. C: Solid State Phys., 16 (1983), 2497.   Google Scholar

[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.   Google Scholar

[6]

C. C. Bissell, Control engineering in the former USSR,, IEEE Control System Magazine, 19 (1999), 111.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[9]

M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions,", Springer, (1996).  doi: 10.1007/978-1-4612-4048-8.  Google Scholar

[10]

D. Challet and Y.-C. Zhang, Emergence of cooperation and organization in an evolutionary game,, Physica A, 246 (1997), 407.   Google Scholar

[11]

D. Challet, M. Marsili and Y.-C. Zhang, "Minority Games,", Oxford University Press, (2005).   Google Scholar

[12]

R. Cross, M. Grinfeld and H. Lamba, A mean-field model of investor behaviour,, J. Phys.: Conf. Ser., 55 (2006), 55.   Google Scholar

[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.  Google Scholar

[14]

R. Cross, H. McNamara, A. Pokrovskii and D. Rachinskii, A new paradigm for modelling hysteresis in macroeconomic flows,, Physica B, 403 (2008), 231.   Google Scholar

[15]

K. Dahmen, Nonlinear dynamics: Universal clues in noisy skews,, Nature Physics, 1 (2005), 13.   Google Scholar

[16]

K. Dahmen and Y. Ben-Zion, The physics of jerky motion in slowly driven magnetic and earthquake fault systems,, in, (2009), 5021.   Google Scholar

[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.   Google Scholar

[18]

R. L. Devaney, "An Introduction to Chaotic Dynamical Systems,", Westview Press, (2003).   Google Scholar

[19]

A. Draper, "Electrical Circuits, Including Machines,", Longmans, (1964).   Google Scholar

[20]

R. A. Guyer and K. R. McCall, Capillary condensation, invasion percolation, hysteresis, and discrete memory,, Phys. Rev. B., 54 (1996), 18.   Google Scholar

[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.  Google Scholar

[22]

M. Krasnosel'skii and A. Pokrovskii, "Systems With Hysteresis,", Springer, (1989).  doi: 10.1007/978-3-642-61302-9.  Google Scholar

[23]

P. Krejci, "Hysteresis, Convexity and Dissipation in Hyperbolic Equations,", Gakkotosho, (1996).   Google Scholar

[24]

P. Krejci, Hysteresis in singularly perturbed problems,, Chapter 3 in, (2005), 73.  doi: 10.1137/1.9780898717860.ch3.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[27]

B. B. Mandelbrot, "The Fractal Geometry of Nature,", W. H. Freeman and Company, (1982).   Google Scholar

[28]

I. D. Mayergoyz, "Mathematical Models of Hysteresis,", Springer, (1991).  doi: 10.2172/6911694.  Google Scholar

[29]

E. Moro, The Minority game: An introductory guide,, in, (2004), 263.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[34]

A. Pokrovskii and V. Sobolev, Naive view on singular perturbation and hysteresis,, Chapter 1 in, (2005), 1.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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).   Google Scholar

[38]

J. P. Sethna, K. Dahmen and C. R. Myers, Crackling noise,, Nature, 410 (2001), 242.   Google Scholar

[39]

J. P. Sethna, K. A. Dahmen and O. Perkovic, Random-Field Ising Models of Hysteresis,, in, II (2005), 107.   Google Scholar

[40]

E. D. Sontag, "Mathematical Control Theory,", Springer, (1998).   Google Scholar

[41]

E. D. Sontag, Monotone and near-monotone biochemical networks,, Syst. Synth. Biol., 1 (2007), 59.   Google Scholar

[42]

A. Visintin, "Differential Models of Hysteresis,", Springer, (1994).   Google Scholar

[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.  Google Scholar

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