Nonlinear Dynamics

, Volume 88, Issue 4, pp 2435–2455 | Cite as

Dual input–output pairs for modeling hysteresis inspired by mem-models

  • Jin-Song Pei
  • François Gay-Balmaz
  • Joseph P. Wright
  • Michael D. Todd
  • Sami F. Masri
Original Paper


We call attention to a dual-pair concept for modeling hysteresis involving instantaneous switching: Specifically, there are two input–output pairs for each hysteresis model under one specific input, namely a differential pair and an integral pair. Currently in engineering mechanics, only one pair is being recognized and utilized, not the other. Whereas this dual-pair concept is inherent in the differential and algebraic forms of memristors and memcapacitors, the concept has not been carried over to memristive system theory, nor to memcapacitive system theory. We show that the “zero-crossing” feature in memristors, memcapacitors, and memristive/memcapacitive models (i.e., the “mem-models”) is also a feature of the differential pairs of well-known non-mem-models, examples of which are Ramberg–Osgood, Bouc–Wen, bilinear hysteresis, and classical Preisach. The dual-pair concept thus connects mem-models and non-mem-models, thereby facilitating the modeling of hysteresis, and raising a set of scientific questions for further studies that might not otherwise come to awareness.


Hysteresis Memcapacitive system Piecewise smooth system Hybrid dynamical system Classical Preisach model Mullins effect 

List of symbols






Absement, the first time integral of displacement, x


The first time derivative of r


Resisting force or characteristic force of an element


General momentum, the first time integral of r

\(\rho \)

The first time integral of p



\(\mathbf {y}\)

State variables, see Tables 1, 2, 3 and 4


Internal state variable


Driving force


See Tables 1 and 3


See Tables 2 and 4

\(\mathbf {g}\)

See Tables 1 and 3

\(\mathbf {f}\)

See Tables 2 and 4



The first author would like to acknowledge the Vice President for Research of the University of Oklahoma for the support of the Faculty Investment Program.


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Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  • Jin-Song Pei
    • 1
  • François Gay-Balmaz
    • 2
  • Joseph P. Wright
    • 3
  • Michael D. Todd
    • 4
  • Sami F. Masri
    • 5
  1. 1.School of Civil Engineering and Environmental ScienceUniversity of OklahomaNormanUSA
  2. 2.CNRS LMDEcole Normale Supérieure de ParisParisFrance
  3. 3.Division of Applied ScienceWeidlinger Associates Inc.New YorkUSA
  4. 4.Department of Structural EngineeringUniversity of California, San DiegoLa JollaUSA
  5. 5.Sonny Astani Department of Civil and Environmental EngineeringUniversity of Southern CaliforniaLos AngelesUSA

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