American Institute of Mathematical Sciences

Advanced
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 and 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.

[2]

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

[3]

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

[4]

J. Lubliner, Plasticity Theory, Dover, 2008.

[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.

[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.

[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.

[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.

[9]

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

[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.

[11]

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

[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.

[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.

[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.

[15]

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

[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.

[17]

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

[18]

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

[19]

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

[20]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[33]

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

[34]

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

[35]

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

[36]

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

[37]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

show all references

References:
[1]

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

[2]

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

[3]

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

[4]

J. Lubliner, Plasticity Theory, Dover, 2008.

[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.

[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.

[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.

[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.

[9]

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

[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.

[11]

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

[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.

[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.

[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.

[15]

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

[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.

[17]

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

[18]

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

[19]

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

[20]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[33]

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

[34]

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

[35]

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

[36]

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

[37]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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