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

Previous Article
Topology and homoclinic trajectories of discrete dynamical systems
DCDS-S Home
This Issue
Next Article

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

Abstract
Related Papers

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.
[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.

[1]	Ábel Garab. Unique periodic orbits of a delay differential equation with piecewise linear feedback function. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2369-2387. doi: 10.3934/dcds.2013.33.2369
[2]	Eduardo Liz, Gergely Röst. On the global attractor of delay differential equations with unimodal feedback. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1215-1224. doi: 10.3934/dcds.2009.24.1215
[3]	Pavel Krejčí. The Preisach hysteresis model: Error bounds for numerical identification and inversion. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : 101-119. doi: 10.3934/dcdss.2013.6.101
[4]	Michela Eleuteri, Pavel Krejčí. An asymptotic convergence result for a system of partial differential equations with hysteresis. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1131-1143. doi: 10.3934/cpaa.2007.6.1131
[5]	Monica Lazzo, Paul G. Schmidt. Convergence versus periodicity in a single-loop positive-feedback system 2. Periodic solutions. Conference Publications, 2011, 2011 (Special) : 941-952. doi: 10.3934/proc.2011.2011.941
[6]	Monica Lazzo, Paul G. Schmidt. Convergence versus periodicity in a single-loop positive-feedback system 1. Convergence to equilibrium. Conference Publications, 2011, 2011 (Special) : 931-940. doi: 10.3934/proc.2011.2011.931
[7]	Zhenyu Lu, Junhao Hu, Xuerong Mao. Stabilisation by delay feedback control for highly nonlinear hybrid stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-18. doi: 10.3934/dcdsb.2019052
[8]	Pavel Krejčí, Giselle A. Monteiro. Inverse parameter-dependent Preisach operator in thermo-piezoelectricity modeling. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3051-3066. doi: 10.3934/dcdsb.2018299
[9]	Eugenii Shustin. Exponential decay of oscillations in a multidimensional delay differential system. Conference Publications, 2003, 2003 (Special) : 809-816. doi: 10.3934/proc.2003.2003.809
[10]	Laurent Imbert, Michael J. Jacobson, Jr., Arthur Schmidt. Fast ideal cubing in imaginary quadratic number and function fields. Advances in Mathematics of Communications, 2010, 4 (2) : 237-260. doi: 10.3934/amc.2010.4.237
[11]	Benjamin B. Kennedy. A periodic solution with non-simple oscillation for an equation with state-dependent delay and strictly monotonic negative feedback. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 47-66. doi: 10.3934/dcdss.2020003
[12]	Sun Yi, Patrick W. Nelson, A. Galip Ulsoy. Delay differential equations via the matrix lambert w function and bifurcation analysis: application to machine tool chatter. Mathematical Biosciences & Engineering, 2007, 4 (2) : 355-368. doi: 10.3934/mbe.2007.4.355
[13]	Yulin Zhao, Siming Zhu. Higher order Melnikov function for a quartic hamiltonian with cuspidal loop. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 995-1018. doi: 10.3934/dcds.2002.8.995
[14]	Eugenii Shustin. Dynamics of oscillations in a multi-dimensional delay differential system. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 557-576. doi: 10.3934/dcds.2004.11.557
[15]	István Győri, Ferenc Hartung, Nahed A. Mohamady. Boundedness of positive solutions of a system of nonlinear delay differential equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 809-836. doi: 10.3934/dcdsb.2018044
[16]	Hong Il Cho, Gang Uk Hwang. Optimal design and analysis of a two-hop relay network under Rayleigh fading for packet delay minimization. Journal of Industrial & Management Optimization, 2011, 7 (3) : 607-622. doi: 10.3934/jimo.2011.7.607
[17]	Yunping Jiang, Yuan-Ling Ye. Convergence speed of a Ruelle operator associated with a non-uniformly expanding conformal dynamical system and a Dini potential. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4693-4713. doi: 10.3934/dcds.2018206
[18]	Daniel Grieser. A natural differential operator on conic spaces. Conference Publications, 2011, 2011 (Special) : 568-577. doi: 10.3934/proc.2011.2011.568
[19]	Alain Bensoussan, Shaokuan Chen, Suresh P. Sethi. Linear quadratic differential games with mixed leadership: The open-loop solution. Numerical Algebra, Control & Optimization, 2013, 3 (1) : 95-108. doi: 10.3934/naco.2013.3.95
[20]	Sylvia Novo, Carmen Núñez, Rafael Obaya, Ana M. Sanz. Skew-product semiflows for non-autonomous partial functional differential equations with delay. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4291-4321. doi: 10.3934/dcds.2014.34.4291

2017 Impact Factor: 0.561

American Institute of Mathematical Sciences