American Institute of Mathematical Sciences

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January  2016, 21(1): 227-243. doi: 10.3934/dcdsb.2016.21.227

Realization of arbitrary hysteresis by a low-dimensional gradient flow

Dmitrii Rachinskii 1,

1. 

Department of Mathematical Sciences, The University of Texas at Dallas, Richardson, TX 75080, United States

Received  May 2015 Revised  July 2015 Published  November 2015

We consider gradient systems with an increasing potential that depends on a scalar parameter. As the parameter is varied, critical points of the potential can be eliminated or created through saddle-node bifurcations causing the system to transit from one stable equilibrium located at a (local) minimum point of the potential to another minimum along the heteroclinic connections. These transitions can be represented by a graph. We show that any admissible graph has a realization in the class of two dimensional gradient flows. The relevance of this result is discussed in the context of genesis of hysteresis phenomena. The Preisach hysteresis model is considered as an example.
Keywords: gradient system, hysteresis, Multi-stability, graph., heteroclinic connection.
Mathematics Subject Classification: Primary: 34C55; Secondary: 37J4.
Citation: Dmitrii Rachinskii. Realization of arbitrary hysteresis by a low-dimensional gradient flow. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 227-243. doi: 10.3934/dcdsb.2016.21.227
References:
[1]

A. Visintin, Differential Models of Hysteresis, Springer, 1994. doi: 10.1007/978-3-662-11557-2.  Google Scholar

[2]

I. D. Mayergoyz, Mathematical Models of Hysteresis and Their Applications, Elsevier, 1993. Google Scholar

[3]

J. C. Mallison, The Foundations of Magnetic Recording, Academic Press, 2012. Google Scholar

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J. Lubliner, Plasticity Theory, Dover, 2008. Google Scholar

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M. Brokate, P. Krejci and D. Rachinskii, Some analytical properties of the multidimensional continuous Mroz model of plasticity, Control Cybernet, 27 (1998), 199-215.  Google Scholar

[6]

V. Lampaert, F. Al-Bender and J. Swevers, A generalized Maxwell-slip friction model appropriate for control purposes, Physics and Control, Proceedings, International Conference, 4 (2013), 1170-1177. doi: 10.1109/PHYCON.2003.1237071.  Google Scholar

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M. Ruderman, F. Hoffmann and T. Bertram, Modeling and identification of elastic robot joints with hysteresis and backlash, IEEE Tran. Ind. Electron, 56 (2009), 3840-3847. doi: 10.1109/TIE.2009.2015752.  Google Scholar

[8]

K. Kuhnen, Compensation of parameter-dependent complex hysteretic actuator nonlinearities in smart material systems, J. Intell. Mater. Syst., 19 (2008), 1411-1424. doi: 10.1177/1045389X08089690.  Google Scholar

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D. Daniele, P. Krejči and C. Visone, Fully coupled modeling of magnetomechanical hysteresis through thermodynamic compatibility, Smart Mater. Struct., 22 (2013), 095009. Google Scholar

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M. Eleuteri, J. Kopfova and P. Krejči, A new phase field model for material fatigue in an oscillating elastoplastic beam, Discret. Cont. Dyn. S. A, 35 (2015), 2465-2495. doi: 10.3934/dcds.2015.35.2465.  Google Scholar

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J.-Y. Parlange, Capillary hysteresis and relationship between drying and wetting curves, Water Resources Res., 12 (1976), 224-228. doi: 10.1029/WR012i002p00224.  Google Scholar

[12]

A. Rahunanthan, F. Furtado, D. Marchesin and M. Piri, Hysteretic enhancemnet of carbon dioxide trapping in deep aquifers, Comput. Geosci., 18 (2014), 899-912. Google Scholar

[13]

M. Brokate, N. D. Botkin and O. A. Pykhteev, Numerical simulation for a two-phase porous medium flow problem with rate independent hysteresis, Physica B, 407 (2012), 1336-1339. doi: 10.1016/j.physb.2011.06.048.  Google Scholar

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E. K. H. Salje and K. A. Dahmen, Crackling noise in disordered materials, Annual Review of Condensed Matter Physics, 5 (2014), 233-254. doi: 10.1146/annurev-conmatphys-031113-133838.  Google Scholar

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M. Dimian and P. Andrei, Noise-Driven Phenomena in Hysteretic Systems, Springer, 2014. doi: 10.1007/978-1-4614-1374-5.  Google Scholar

[16]

P. L. Gurevich, W. Jaeger and A. L. Skubachevskii, On periodicity of solutions for thermocontrol problems with hysteresis-type switches, SIAM J. Math. Anal., 41 (2009), 733-752. doi: 10.1137/080718905.  Google Scholar

[17]

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

[18]

L. Landau and E. Lifshitz, Statistical Physics, Pergamon Press, Oxford, 1969. Google Scholar

[19]

R. E. O'Malley, Singular Perturbation Methods for Ordinary Differential Equations, Springer, 1991. doi: 10.1007/978-1-4612-0977-5.  Google Scholar

[20]

A. Visintin, A Weiss-type model of ferromagnetism, Physica B, 275 (2000), 87-91. doi: 10.1016/S0921-4526(99)00712-7.  Google Scholar

[21]

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), 443-459. doi: 10.5194/hess-11-443-2007.  Google Scholar

[22]

P. Krejči, J. P. O'Kane, A. Pokrovskii and D. Rachinskii, Stability results for a soil model with singular hysteretic hydrology, J. Phys.: Conf. Ser., 268 (2011), 012016. Google Scholar

[23]

B. Athukorallage, E. Aulisa, R. Iyer and L. Zhang, Macroscopic theory for capillary pressure hysteresis, Langmuir, 31 (2015), 2390-2397. doi: 10.1021/la504495c.  Google Scholar

[24]

M. Brokate, S. MacCarthy, A. Pimenov, A. Pokrovskii and D. Rachinskii, Modelling energy dissipation due to soil-moisture hysteresis, Environ. Model. Assess., 16 (2011), 313-333. doi: 10.1007/s10666-011-9258-2.  Google Scholar

[25]

I. Rychlik, A new definition of the rainflow cycle counting method, Internat. J. Fatigue, 9 (1987), 119-121. doi: 10.1016/0142-1123(87)90054-5.  Google Scholar

[26]

P. Krejči, H. Lamba, S. Melnik and D. Rachinskii, Analytical solution for a class of network dynamics with mechanical and financial applications, Phys. Rev. E, 90 (2014), 032822. Google Scholar

[27]

R. Cross, H. McNamara, A. Pokrovskii and D. Rachinskii, A new paradigm for modelling hysteresis in macroeconomic flows, Physica B, 403 (2008), 231-236. doi: 10.1016/j.physb.2007.08.017.  Google Scholar

[28]

R. Cross, H. Hutchinson, H. Lamba and D. Strachan, Reflections on Soros: Mach, Quine, Arthur and far-from-equilibrium dynamics, J. Econ. Methodology, 20 (2014), 357-367. doi: 10.1080/1350178X.2013.859406.  Google Scholar

[29]

S. Melnik, J. A. Ward, J. P. Gleeson and M. A. Porter, Multi-stage complex contagions, Chaos, 23 (2013), 013124, 13pp. doi: 10.1063/1.4790836.  Google Scholar

[30]

C. Chiu, F. C. Hoppensteadt and W. Jäger, Analysis and computer simulation of accretion patterns in bacterial cultures, J. Math. Biol., 32 (1994), 841-855. doi: 10.1007/BF00168801.  Google Scholar

[31]

G. Friedman, S. McCarthy and D. Rachinskii, Hysteresis can grant fitness in stochastically varying environment, PLOS ONE, 9 (2014), e103241. doi: 10.1371/journal.pone.0103241.  Google Scholar

[32]

A. Pimenov, T. C. Kelly, A. Korobeinikov, M. J. A. O'Callaghan, A. V. Pokrovskii and D. Rachinskii, Memory effects in population dynamics: Spread of infectious disease as a case study, Math. Model. Nat. Phenom., 7 (2012), 204-226. doi: 10.1051/mmnp/20127313.  Google Scholar

[33]

, The Science of Hysteresis,, (eds. I. D. Mayergoyz and G. Bertotti), (2005).   Google Scholar

[34]

, Singular Perturbations and Hysteresis,, (eds. M. P. Mortell, (2005).  doi: 10.1137/1.9780898717860.  Google Scholar

[35]

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

[36]

P. Krejči and V. Recupero, BV solutions of rate independent differential inclusions, Math. Bohem., 139 (2014), 607-619.  Google Scholar

[37]

P. Krejči, Hysteresis, Convexity and Dissipation in Hyperbolic Equations, Gakkotosho, 1996.  Google Scholar

[38]

A. Mielke, R. Rossi and G. Savaré, Variational convergence of gradient flows and rate-independent evolutions in metric spaces, Milan J. Math., 80 (2012), 381-410. doi: 10.1007/s00032-012-0190-y.  Google Scholar

[39]

K. J. Aström, Oscillations in systems with relay feedback, in Adaptive Control, Filtering and Signal Processing (eds. K. Aström, G. Goodwin and P. Kumar), Springer, 74 (1995), 1-25. doi: 10.1007/978-1-4419-8568-2_1.  Google Scholar

[40]

M. Brokate, A. Pokrovskii and D. Rachinskii, Asymptotic stability of continuum sets of periodic solutions to systems with hysteresis, J. Math. Anal. Appl., 319 (2006), 94-109. doi: 10.1016/j.jmaa.2006.02.060.  Google Scholar

[41]

A. Pimenov and D. Rachinskii, Linear stability analysis of systems with Preisach memory, Discret. Contin. Dyn. S. B, 11 (2009), 997-1018. doi: 10.3934/dcdsb.2009.11.997.  Google Scholar

[42]

A. Krasnosel'skii and D. Rachinskii, Bifurcation of forced periodic oscillations for equations with Preisach hysteresis, J. Phys.: Conf. Ser., 22 (2005), 93-102. doi: 10.1088/1742-6596/22/1/006.  Google Scholar

[43]

A. Krasnosel'skii and D. Rachinskii, On a bifurcation governed by hysteresis nonlinearity, NoDEA Nonlinear Differential Equations Appl., 9 (2002), 93-115. doi: 10.1007/s00030-002-8120-2.  Google Scholar

[44]

P. Krejči, 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), 2010-2028. doi: 10.1016/j.physd.2011.05.005.  Google Scholar

show all references

References:
[1]

A. Visintin, Differential Models of Hysteresis, Springer, 1994. doi: 10.1007/978-3-662-11557-2.  Google Scholar

[2]

I. D. Mayergoyz, Mathematical Models of Hysteresis and Their Applications, Elsevier, 1993. Google Scholar

[3]

J. C. Mallison, The Foundations of Magnetic Recording, Academic Press, 2012. Google Scholar

[4]

J. Lubliner, Plasticity Theory, Dover, 2008. Google Scholar

[5]

M. Brokate, P. Krejci and D. Rachinskii, Some analytical properties of the multidimensional continuous Mroz model of plasticity, Control Cybernet, 27 (1998), 199-215.  Google Scholar

[6]

V. Lampaert, F. Al-Bender and J. Swevers, A generalized Maxwell-slip friction model appropriate for control purposes, Physics and Control, Proceedings, International Conference, 4 (2013), 1170-1177. doi: 10.1109/PHYCON.2003.1237071.  Google Scholar

[7]

M. Ruderman, F. Hoffmann and T. Bertram, Modeling and identification of elastic robot joints with hysteresis and backlash, IEEE Tran. Ind. Electron, 56 (2009), 3840-3847. doi: 10.1109/TIE.2009.2015752.  Google Scholar

[8]

K. Kuhnen, Compensation of parameter-dependent complex hysteretic actuator nonlinearities in smart material systems, J. Intell. Mater. Syst., 19 (2008), 1411-1424. doi: 10.1177/1045389X08089690.  Google Scholar

[9]

D. Daniele, P. Krejči and C. Visone, Fully coupled modeling of magnetomechanical hysteresis through thermodynamic compatibility, Smart Mater. Struct., 22 (2013), 095009. Google Scholar

[10]

M. Eleuteri, J. Kopfova and P. Krejči, A new phase field model for material fatigue in an oscillating elastoplastic beam, Discret. Cont. Dyn. S. A, 35 (2015), 2465-2495. doi: 10.3934/dcds.2015.35.2465.  Google Scholar

[11]

J.-Y. Parlange, Capillary hysteresis and relationship between drying and wetting curves, Water Resources Res., 12 (1976), 224-228. doi: 10.1029/WR012i002p00224.  Google Scholar

[12]

A. Rahunanthan, F. Furtado, D. Marchesin and M. Piri, Hysteretic enhancemnet of carbon dioxide trapping in deep aquifers, Comput. Geosci., 18 (2014), 899-912. Google Scholar

[13]

M. Brokate, N. D. Botkin and O. A. Pykhteev, Numerical simulation for a two-phase porous medium flow problem with rate independent hysteresis, Physica B, 407 (2012), 1336-1339. doi: 10.1016/j.physb.2011.06.048.  Google Scholar

[14]

E. K. H. Salje and K. A. Dahmen, Crackling noise in disordered materials, Annual Review of Condensed Matter Physics, 5 (2014), 233-254. doi: 10.1146/annurev-conmatphys-031113-133838.  Google Scholar

[15]

M. Dimian and P. Andrei, Noise-Driven Phenomena in Hysteretic Systems, Springer, 2014. doi: 10.1007/978-1-4614-1374-5.  Google Scholar

[16]

P. L. Gurevich, W. Jaeger and A. L. Skubachevskii, On periodicity of solutions for thermocontrol problems with hysteresis-type switches, SIAM J. Math. Anal., 41 (2009), 733-752. doi: 10.1137/080718905.  Google Scholar

[17]

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

[18]

L. Landau and E. Lifshitz, Statistical Physics, Pergamon Press, Oxford, 1969. Google Scholar

[19]

R. E. O'Malley, Singular Perturbation Methods for Ordinary Differential Equations, Springer, 1991. doi: 10.1007/978-1-4612-0977-5.  Google Scholar

[20]

A. Visintin, A Weiss-type model of ferromagnetism, Physica B, 275 (2000), 87-91. doi: 10.1016/S0921-4526(99)00712-7.  Google Scholar

[21]

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), 443-459. doi: 10.5194/hess-11-443-2007.  Google Scholar

[22]

P. Krejči, J. P. O'Kane, A. Pokrovskii and D. Rachinskii, Stability results for a soil model with singular hysteretic hydrology, J. Phys.: Conf. Ser., 268 (2011), 012016. Google Scholar

[23]

B. Athukorallage, E. Aulisa, R. Iyer and L. Zhang, Macroscopic theory for capillary pressure hysteresis, Langmuir, 31 (2015), 2390-2397. doi: 10.1021/la504495c.  Google Scholar

[24]

M. Brokate, S. MacCarthy, A. Pimenov, A. Pokrovskii and D. Rachinskii, Modelling energy dissipation due to soil-moisture hysteresis, Environ. Model. Assess., 16 (2011), 313-333. doi: 10.1007/s10666-011-9258-2.  Google Scholar

[25]

I. Rychlik, A new definition of the rainflow cycle counting method, Internat. J. Fatigue, 9 (1987), 119-121. doi: 10.1016/0142-1123(87)90054-5.  Google Scholar

[26]

P. Krejči, H. Lamba, S. Melnik and D. Rachinskii, Analytical solution for a class of network dynamics with mechanical and financial applications, Phys. Rev. E, 90 (2014), 032822. Google Scholar

[27]

R. Cross, H. McNamara, A. Pokrovskii and D. Rachinskii, A new paradigm for modelling hysteresis in macroeconomic flows, Physica B, 403 (2008), 231-236. doi: 10.1016/j.physb.2007.08.017.  Google Scholar

[28]

R. Cross, H. Hutchinson, H. Lamba and D. Strachan, Reflections on Soros: Mach, Quine, Arthur and far-from-equilibrium dynamics, J. Econ. Methodology, 20 (2014), 357-367. doi: 10.1080/1350178X.2013.859406.  Google Scholar

[29]

S. Melnik, J. A. Ward, J. P. Gleeson and M. A. Porter, Multi-stage complex contagions, Chaos, 23 (2013), 013124, 13pp. doi: 10.1063/1.4790836.  Google Scholar

[30]

C. Chiu, F. C. Hoppensteadt and W. Jäger, Analysis and computer simulation of accretion patterns in bacterial cultures, J. Math. Biol., 32 (1994), 841-855. doi: 10.1007/BF00168801.  Google Scholar

[31]

G. Friedman, S. McCarthy and D. Rachinskii, Hysteresis can grant fitness in stochastically varying environment, PLOS ONE, 9 (2014), e103241. doi: 10.1371/journal.pone.0103241.  Google Scholar

[32]

A. Pimenov, T. C. Kelly, A. Korobeinikov, M. J. A. O'Callaghan, A. V. Pokrovskii and D. Rachinskii, Memory effects in population dynamics: Spread of infectious disease as a case study, Math. Model. Nat. Phenom., 7 (2012), 204-226. doi: 10.1051/mmnp/20127313.  Google Scholar

[33]

, The Science of Hysteresis,, (eds. I. D. Mayergoyz and G. Bertotti), (2005).   Google Scholar

[34]

, Singular Perturbations and Hysteresis,, (eds. M. P. Mortell, (2005).  doi: 10.1137/1.9780898717860.  Google Scholar

[35]

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

[36]

P. Krejči and V. Recupero, BV solutions of rate independent differential inclusions, Math. Bohem., 139 (2014), 607-619.  Google Scholar

[37]

P. Krejči, Hysteresis, Convexity and Dissipation in Hyperbolic Equations, Gakkotosho, 1996.  Google Scholar

[38]

A. Mielke, R. Rossi and G. Savaré, Variational convergence of gradient flows and rate-independent evolutions in metric spaces, Milan J. Math., 80 (2012), 381-410. doi: 10.1007/s00032-012-0190-y.  Google Scholar

[39]

K. J. Aström, Oscillations in systems with relay feedback, in Adaptive Control, Filtering and Signal Processing (eds. K. Aström, G. Goodwin and P. Kumar), Springer, 74 (1995), 1-25. doi: 10.1007/978-1-4419-8568-2_1.  Google Scholar

[40]

M. Brokate, A. Pokrovskii and D. Rachinskii, Asymptotic stability of continuum sets of periodic solutions to systems with hysteresis, J. Math. Anal. Appl., 319 (2006), 94-109. doi: 10.1016/j.jmaa.2006.02.060.  Google Scholar

[41]

A. Pimenov and D. Rachinskii, Linear stability analysis of systems with Preisach memory, Discret. Contin. Dyn. S. B, 11 (2009), 997-1018. doi: 10.3934/dcdsb.2009.11.997.  Google Scholar

[42]

A. Krasnosel'skii and D. Rachinskii, Bifurcation of forced periodic oscillations for equations with Preisach hysteresis, J. Phys.: Conf. Ser., 22 (2005), 93-102. doi: 10.1088/1742-6596/22/1/006.  Google Scholar

[43]

A. Krasnosel'skii and D. Rachinskii, On a bifurcation governed by hysteresis nonlinearity, NoDEA Nonlinear Differential Equations Appl., 9 (2002), 93-115. doi: 10.1007/s00030-002-8120-2.  Google Scholar

[44]

P. Krejči, 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), 2010-2028. doi: 10.1016/j.physd.2011.05.005.  Google Scholar

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