American Institute of Mathematical Sciences

Advanced
August  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]

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

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

[1]

Ábel Garab. Unique periodic orbits of a delay differential equation with piecewise linear feedback function. Discrete & Continuous Dynamical Systems, 2013, 33 (6) : 2369-2387. doi: 10.3934/dcds.2013.33.2369
[2]

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
[3]

Eduardo Liz, Gergely Röst. On the global attractor of delay differential equations with unimodal feedback. Discrete & Continuous Dynamical Systems, 2009, 24 (4) : 1215-1224. doi: 10.3934/dcds.2009.24.1215
[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]

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
[6]

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
[7]

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
[8]

Kasthurisamy Jothimani, Kalimuthu Kaliraj, Sumati Kumari Panda, Kotakkaran Sooppy Nisar, Chokkalingam Ravichandran. Results on controllability of non-densely characterized neutral fractional delay differential system. Evolution Equations & Control Theory, 2021, 10 (3) : 619-631. doi: 10.3934/eect.2020083
[9]

Zhenyu Lu, Junhao Hu, Xuerong Mao. Stabilisation by delay feedback control for highly nonlinear hybrid stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4099-4116. doi: 10.3934/dcdsb.2019052
[10]

Izumi Takagi, Conghui Zhang. Existence and stability of patterns in a reaction-diffusion-ODE system with hysteresis in non-uniform media. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3109-3140. doi: 10.3934/dcds.2020400
[11]

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
[12]

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
[13]

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, 2020, 13 (1) : 47-66. doi: 10.3934/dcdss.2020003
[14]

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
[15]

Hasib Khan, Cemil Tunc, Aziz Khan. Green function's properties and existence theorems for nonlinear singular-delay-fractional differential equations. Discrete & Continuous Dynamical Systems - S, 2020, 13 (9) : 2475-2487. doi: 10.3934/dcdss.2020139
[16]

Chun Liu, Jan-Eric Sulzbach. The Brinkman-Fourier system with ideal gas equilibrium. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021123
[17]

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
[18]

Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020107
[19]

Sebastián Buedo-Fernández. Global attraction in a system of delay differential equations via compact and convex sets. Discrete & Continuous Dynamical Systems - B, 2020, 25 (8) : 3171-3181. doi: 10.3934/dcdsb.2020056
[20]

Eugenii Shustin. Dynamics of oscillations in a multi-dimensional delay differential system. Discrete & Continuous Dynamical Systems, 2004, 11 (2&3) : 557-576. doi: 10.3934/dcds.2004.11.557

2020 Impact Factor: 2.425

Tools

Metrics

  • PDF downloads (59)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences