Journal ArticleDOI

# A Survey of the Hysteretic Duhem Model

01 Nov 2018-Archives of Computational Methods in Engineering (Springer Netherlands)-Vol. 25, Iss: 4, pp 965-1002
TL;DR: The relationship between the Duhem model as a mathematical representation and hysteresis as the object of that representation is surveyed in this paper, where the authors present a survey of the relationships between the two models.
Abstract: The Duhem model is a simulacrum of a complex and hazy reality: hysteresis. Introduced by Pierre Duhem to provide a mathematical representation of thermodynamical irreversibility, it is used to describe hysteresis in other areas of science and engineering. Our aim is to survey the relationship between the Duhem model as a mathematical representation, and hysteresis as the object of that representation.

### 2 Introduction and literature review

• Give P infinitely small variations, the length l and temperature T will also experience infinitely small variations, and a new equilibrium may be achieved.
• There are several generalizations of the original Duhem model (5).
• Multistability means that “the system must have multiple attracting equilibria for a constant input value” [6].

### 3 Terminology and notations

• This section presents those results obtained in Ref. [43] that are relevant to the present paper.
• Instead, the authors of Ref. [54] propose a definition that decides whether a given generalized Duhem model is a hysteresis or not (this is Definition 4).
• Finally the relationship between Definition 4 and strong consistency is analyzed in Section 12.1. 6 A summary of the results obtained in Ref. [35].
• Lemma 1 shows that the input u has been “normalized” so that the resulting function ψu is such that ψ̇u has norm 1 with respect to the new time variable %.

### 10 A note on minor loops

• The minor loops of the Duhem model have not been studied formally in the available literature.
• For this reason, the authors provide in this section the formal definition of a minor loop and analyze the behavior of the minor loops of the scalar semilinear Duhem model in Section 11.9.
• Figure 3 illustrates what has been exposed up till now.
• Depending on the particular field in which hysteresis is observed, minor loops may have some additional properties that may be formalized mathematically.
• Property (i) says that the major loop and the minor loop intersect at only one point when umax,1 < umax,2. Property (ii) says that the minor loop is located inside the major loop.

### 12 Relationships between concepts

• In this section the authors explore the connections that exist between the concepts presented in this paper.
• The authors use the case study of the semilinear Duhem model to illustrate these connections and motivate the open problems proposed in Section 13.
• 1 Relationship between Definition 4 and strong consistency.

### 12.1.1 Comments on Definition 4

• The authors have seen in Section 5.2 that Ref. [54] proposes a definition that aims to decide whether a given generalized Duhem model is a hysteresis or not.
• (iii) Check whether Condition (i) of Definition 4 holds.
• Indeed, the concept of Cauchy sequence can be used to prove the existence of x0,γ without actually having to find the explicit expression of x0,γ .
• This equality is obtained thanks to the explicit expression (103) of the initial condition x0,γ .
• To sum up, the linearity with respect to the state in the differential equation that describes the scalar semilinear Duhem model, is crucial to prove that Condition (i) of Definition 4 holds.

### 12.1.2 Comments on strong consistency

• The analysis of the consistency of the semilinear Duhem model is provided in Section 11.2, and it uses both the linearity with respect to the state, and the fact that the initial condition in Equation (70) does not change with γ.
• For the generalized Duhem model (17) that may not be linear with respect to the state, Lemma 6 provides sufficient conditions that provide the expression of the corresponding rate independent Duhem model.
• Ensuring these sufficient conditions may not be easy if the model is nonlinear with respect to the state.
• Instead, the concept of Cauchy sequence is used in Ref. [35] to prove the desired convergence property.
• Again, the linearity of the model is used to derive a Lyapunov function which allows mathematical analysis.

### 12.1.3 Relationship between the hysteresis loop derived from Definition 4 and the one derived from strong consistency

• The hysteresis loop derived from strong consistency is the set Gu of Equation (31).
• It can be checked that, for the scalar semilinear Duhem model, the authors have Cu = Gu. However, for the generalized Duhem model, at the time of the submission of the present paper they do not know whether the sets Cu and Gu are equal or not.
• This means that, in order to evaluate the effect of perturbations on the hysteresis loop of the model (43)–(45), Proposition 2 needs to be enhanced to take into account different initial conditions.
• This observation leads to formulating Open Prob- lem 2 in Section 13.2.

### 13 Open problems

• The motivation for Open Problem 1 is provided in Section 12.1.3.
• Consider that the generalized Duhem model (17)– (19) satisfies Assumption 1 so that the authors can define the operators Ho and Hs of Section 5.1.
• The motivation for Open Problem 2 is provided in Section 12.2.
• Consider the scalar rate-independent Duhem model (34)–(36) where the output is the state x.

### A On the existence and uniqueness of solutions of differential equations

• In this section the authors present some existence and uniqueness theorems for the solutions of ordinary differential equations.
• The differential equation (129)–(130) has a solution on some nonempty open interval I 3 t0, in the sense that there exists an absolutely continuous function x : I → Rn such that the following properties (i)–(iii) are satisfied.

### D Proof of Theorem 9

• This is the aim of the following analysis.
• The uniform convergence of z̄γ (restricted to the interval [0, T ]) to z̄ has thus been demonstrated, which completes the proof.

### F Proof of Lemma 10

• It can be checked from Equation (204) that this last equality cannot hold.
• A similar argument can be used for Equation (205).

### G Proof of Lemma 11

• Observe that, for Theorem 5 to hold, it is needed that A1 and −A2 are both stable.
• Using Theorem 7 it follows that the function ϕ◦u that characterizes the hysteresis loop satisfies the differential state equation (79) and the output equation (80).
• This study was funded by the Spanish Ministry of Economy, Industry and Competitiveness (grant number DPI2016-77407-P (AEI/FEDER, UE)).

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A Survey of the Hysteretic Duhem Model
Facal Ikhouane
Abstract The Duhem model is a simulacrum of a com-
plex and hazy reality: hysteresis. Introduced by Pierre
Duhem to provide a mathematical representation of
thermodynamical irreversibility, it is used to describe
hysteresis in other areas of science and engineering. Our
aim is to survey the relationship between the Duhem
model as a mathematical representation, and hysteresis
as the object of that representation.
Keywords Duhem model · Diﬀerential equations ·
Hysteresis
1 Prolegomenon
Citing a reference allows the author of a scientiﬁc ar-
ticle to attribute work and ideas to the correct source.
Nonetheless, the process of describing that work and
these ideas assumes some interpretation, at least of
their relative importance. In order to ensure that the
interpretation is reliable, we use, whenever adequate,
a quotation from the reference so that the reader has
is even more important when the source is not easily
available like Ref. [67] or is not written in English like
Refs. [16]–[22] among others, in which case we provide
a translation. This is our approach to the literature re-
view of Section 2.
Supported by grant DPI2016-77407-P (AEI/FEDER, UE) of
the Spanish Ministry of Economy, Industry and Competitive-
ness.
F. Ikhouane
Universitat Politecnica de Catalunya, Department of Mathe-
matics. Barcelona East School of Engineering, carrer Eduard
Maristany, 16, 08019, Barcelona, Spain.
E-mail: faycal.ikhouane@upc.edu
In Sections 4–9 we proceed diﬀerently since our aim
in these sections is to provide an accurate description of
the results presented in the references under study. Be-
cause of the diversity of notations and nomenclature in
these references, quotations may not be the best way to
transmit that accurate description. Instead, we summa-
rize the references using a unifying framework provided
in Ref. [35]. The references we have chosen in Sections
4–9 are, in our opinion, those that are relevant to the
subject of this study.
Our aim in this work is also to shed light on the
relationships between the concepts introduced in this
paper. To this end, we use a special form of the Duhem
model, the scalar semilinear one, as a case study.
2 Introduction and literature review
A brief history of the Duhem model. The term hys-
teresis was coined by J. A. Ewing in 1881 to describe
a speciﬁc relationship between the torsion of a mag-
netized wire and its polarization (although the phe-
nomenon of hysteresis has been known and described
by several authors before that date as shown in the lit-
erature review provided in Ref. [65]).
Quoting from Ewing’s paper[28]: These curves ex-
hibit, in a striking manner, a persistence of previous
state, such as might be caused by molecular friction.
The curves for the back and forth twists are irreversible,
and include a wide area between them. The change of
polarization lags behind the change of torsion. To this
action . . . the author now gives the name Hyster¯esis (. . .
to be behind)”.
In 1887 Lord Rayleigh models the relationship be-
tween a magnetizing force F [F
max
, F
max
] and the
corresponding magnetization M using two polynomials

2 Fay¸cal Ikhouane
[59, p. 240]:
M = αF + βF
2
max
1
1
2
1
F
F
max
2
!
when F is decreasing,
M = αF + βF
2
max
1 +
1
2
1 +
F
F
max
2
!
when F is increasing,
where α, β and F
max
are constants.
However, the ﬁrst in-depth study of hysteresis is due
to Pierre Duhem
1
in the period 1896–1902. A detailed
review of Duhem’s work on hysteresis may be found
in [67, Chapter IV] so that we provide here only those
elements of that extensive work that are directly related
to the present paper.
To understand the motivation for Duhem’s work
we quote from [67, p. 306]: “take a metallic wire un-
der strain by means of a load. We can take the length
of the wire and its temperature as variables that deﬁne
its state. The gravity weight P will represent the ex-
ternal action. At temperature T and under the load P
the wire may be at equilibrium with length l. Give P
inﬁnitely small variations, the length l and temperature
T will also experience inﬁnitely small variations, and
a new equilibrium may be achieved. In this last state,
give the gravity weight and temperature variations equal
in absolute value, but of opposite signs to the previous
ones. The length l should experience a variation equal
to the previous one with opposite sign. However, exper-
imentation shows that this is not the case. In general,
to the expansion of the wire corresponds a smaller con-
traction, and the diﬀerence lasts with time.”
This permanent deformation is the subject of a seven-
memoirs research by Duhem, see Refs. [16]–[22]. In his
ﬁrst memoir submitted to the section of sciences of the
Acad´emie de Belgique on October 13, 1894, and re-
viewed by the mathematician Charles Lagrange in Ref.
[44],
2
Duhem writes: “The attempts to make the diﬀer-
ent kinds of permanent deformations compatible with
the principles of thermodynamics have been few up till
now. Only one of these attempts, due to M. Marcel
Brillouin, appears to us worthy of interest.” [16, p. 3].
Duhem analyzes the work of Brillouin and concludes
that it is not compatible with the principles of thermo-
1
For a detailed study of the life and work of Pierre-
Maurice-Marie Duhem (9 June 1861 14 September 1916)
see Refs. [39] or [67].
2
We are indebted to Jean Fran¸cois Stoﬀel for this informa-
tion.
As an alternative, Duhem starts a theory of per-
manent deformations by considering the simplest case:
that of a system deﬁned by one normal variable x and
its absolute temperature T . Denoting F(x, T ) the in-
ternal thermodynamic potential of the system, Duhem
writes [16, p. 8]: “Let X be the external action to which
this system is subject. The condition of equilibrium of
the system will be
X =
F(x, T )
x
. (1)
Let (x, T, X) and (x + dx, T + dT, X + dX) be two equi-
libria of the system, inﬁnitely close to each other; owing
to equality [(1)] we get
dX =
2
F(x, T )
x
2
dx +
2
F(x, T )
x∂T
dT. (2)
Equation (2) does not take into account the fact that
the modiﬁcations of equilibria are not reversible. So
Duhem introduces a term f(x, T, X)|dx| to be added
to the right-hand side of Equation (2), where f is a
continuous function of the three variables x, T , and X.
For an isothermal modiﬁcation (that is when T is main-
tained constant) we get [16, pp. 9–10]:
dX
dx
=
(
f
1
(x, T, X) for an increasing x,
f
2
(x, T, X) for a decreasing x,
(3)
where
f
1
(x, T, X) =
2
F(x, T )
x
2
+ f(x, T, X),
f
2
(x, T, X) =
2
F(x, T )
x
2
f(x, T, X).
(4)
Observe that, when the input is piecewise monotone,
the model (3) is equivalent to the model (5) proposed
in Refs. [3] and [43, p. 282]:
˙x(t) =
(
φ

x(t), u(t)
˙u(t) for ˙u(t) 0,
φ
r
x(t), u(t)
˙u(t) for ˙u(t) 0,
(5)
where φ

and φ
r
are functions that satisfy some con-
ditions, the function u is the input (which is x using
Duhem’s notation), the function x the state (which is
X using Duhem’s notation), and t is time.
To the best of our knowledge, the ﬁrst reference that
called the form (5) “Duhem model” is Ref. [48] in 1993.
3
Indeed, the authors of Ref. [43] attributed erroneously
Duhem’s model to Madelung [63, p. 797].
4
3
Ref. [48] cites a translation into German of the original
memoir Ref. [16] which is written in French.
4
Quoting from Ref. [48, p. 96]: “the Madelung paper does
not use a diﬀerential equation or integral operator. In fact,
Madelung allows nonuniqueness of trajectories through a
point . . . which would make a diﬀerential equation model dif-
ﬁcult.”

A Survey of the Hysteretic Duhem Model 3
Between 1916 when P. Duhem dies and 1993 when
his model of hysteresis is ﬁnally attributed to him, Duhem’s
work on hysteresis does not have a relevant impact. Ma-
jor references on hysteresis like Refs. [8], [12] or [56] do
not cite his memoirs. Several authors propose diﬀer-
ent forms of the Duhem model without a direct refer-
ence to Duhem’s memoirs. This is the case of the Cole-
man and Hodgdon model of magnetic hysteresis [12],
the Dahl model of friction [13], the model (5) in Ref.
[3], and a generalized form of the model (5) in Ref. [43,
p. 95]. In 1952, Everett cites brieﬂy Duhem’s work as
follows [24, p. 751]: “From a thermodynamic standpoint
the introduction of an additional variable whose value
depends on the history of the system is suﬃcient for a
formal discussion (cf. Duhem
[ref]
derstanding of the phenomenon [of hysteresis], however,
a molecular interpretation is desirable.”
A general theory of physics based on a molecular
interpretation was precisely what Duhem rejected. In a
review of his work presented in 1913 for his application
to the Acad´emie des Sciences, Duhem writes that his
“doctrine should note imitate the numerous mechanical
theories proposed by physicists hitherto; to the observ-
able properties that apparatus measure, it will not sub-
stitute hidden movements of hypothetical bodies” [67,
p. 74].
In recent times, Duhem’s phenomenological approach
is becoming more accepted [5,9,46,52,57]. Indeed, “hys-
teretic phenomena arising in structural and mechanical
systems are so complicated that there has been no well-
accepted mathematical model which can describe all ob-
served hysteretic characteristics.” [52, p. 1408]. More-
over, the Preisach model which was believed to describe
the constitutive behavior of magnetic hysteresis, has
shown to be a phenomenological model [49, p. 2].
Several reasons are invoked for the use of Duhem’s
model to describe hysteresis. On the one hand, “diﬀer-
ential equation-based models lead to a particularly sim-
ple phenomenological description [46, p. C8-545]. On
the other hand, the “Duhem models [sic] . . . have the
advantage that [they] require a small amount of mem-
ory so they are suitable in practical and low cost ap-
plications.” [9, p. 628]. Finally, many phenomenological
models of friction or hysteresis can be seen as particular
cases of a more general form of the Duhem model: this is
the case for example of the Dahl [13], the LuGre [2,11],
or the Maxwell-slip models [30]. Thus “recast[ing] each
model in the form of a generalized . . . Duhem model . . .
provide[s] a uniﬁed framework for comparing the hys-
teretic nature of these models.” [57, p. 91].
There are several generalizations of the original Duhem
model (5). The following generalization is proposed in
[43, p. 95]: ˙x(t) = f
t, x(t), u(t), ˙u(t)
. In [64, p. 141]
the terms φ

x(t), u(t)
and φ
r
x(t), u(t)
in (5) are re-
placed by
φ

x, u

(t) and
φ
r
x, u

(t) respectively,
where φ

and φ
r
are causal operators. In Ref. [54] Duhem’s
model is generalized as ˙x(t) = f
x(t), u(t)
g
˙u(t)
whilst
[64, p. 145] proposes the following form for vector hys-
teresis: ˙x(t) = f
x(t), u(t), π( ˙u)
| ˙u(t)| where π( ˙u 6=
0) = ˙u/| ˙u|.
Why are there different generalized forms of
the Duhem model ? To answer this question we have
to recall the concept of rate independence.
5
To the best of our knowledge, the earliest author
to state clearly rate independence is R. Bouc in Ref.
[8], although that property was known before Bouc’s
work. Due to the importance of rate independence in
the study of hysteresis, and the fact that Ref. [8] is not
available in English, we quote from [8, p. 17]:“Consider
the graph with hysteresis of Fig. 1 where F is not a
function of x. To the value x = x
0
correspond four
values of F.
Fig. 1: Graph “Force–Displacement” with hysteresis.
. . . If we consider now x as a function of time, the
value of the force at instant t will depend not only on
the value x(t), but also on all past values of function
x since the origin instant where it is deﬁned. If β is
that instant (x(β) = F(β) = 0, β −∞), then we
denote F(t) = A
x(·), t
the value of the force at instant
t”, where x(·) “represents” the whole function on the
interval [β, t]
[footnote]
. Our aim is to explicit functional
A
x(·), t
.
To this end, we make the following assumption: the
graph of Fig. 1 remains the same for all increasing func-
5
The term “rate independence” is attributed to Truesdell
and Noll (Section 99, Encyclopedia of Phyics, volume III/3,
1965) by Visintin [64, p. 13]. We read Section 99 of the 2004
edition [62] of the original treatise by Truesdell and Noll but
found no clear evidence of the correctness of the attribution.

4 Fay¸cal Ikhouane
tion x(·) between 0 and x
1
, decreasing between the val-
ues x
1
and x
2
, etc. The functional will no longer depend
explicitly on time and we will write F(t) = A
x(·)
(t).
We can say: If x(t
j
) and x(t
j+1
) are two extremal val-
ues, consecutive in time, we have for all t [t
j
, t
j+1
]
A
x(·)
(t) = f
j
x(t)
,
where f
j
is a function of only the variable x(t).
We can also say: If φ : R R is a class C
1
func-
tion whose derivative is strictly positive for t β with
φ(β) = β, and if we consider the function y(t) = x
φ(t)
which is a “compression or an “expansion of x by in-
tervals, then the graphs
A
y(·)
, y
and
A
x(·)
, x
are identical and we have
A
x(·)
(t) = A
y(·)

φ
1
(t)
.
The exact deﬁnition of rate independence varies from
author to author. For example, Visintin requires the
time-scale-change φ to be a strictly increasing time home-
omorphism [64, p. 13] whilst Oh and Bernstein con-
sider that φ is continuous, piecewise C
1
, nondecreasing,
φ(0) = 0, and lim
t→∞
φ(t) = [54]. Loosely speaking,
rate independence means that the graph of hysteresis
(output versus input) is invariant with respect to any
change in time scale.
Rate independence is used by Visintin to deﬁne hys-
teresis :“Deﬁnition. Hysteresis = Rate Independent Mem-
ory Eﬀect.”[64, p. 13]. However, “this deﬁnition excludes
any viscous-type memory” [64, p. 13] because it leads
to rate-dependent eﬀects that increase with velocity.
A deﬁnition based on rate independence assumes that
“the presence of hysteresis loops is not . . . an essential
feature of hysteresis.” [64, p. 14].
This point of view is challenged by Oh and Bern-
stein who consider hysteresis as a “nontrivial quasi-dc
input-output closed curve” [54, p. 631] and propose a
modiﬁed version of the Duhem model which can repre-
sent rate-dependent or rate-independent eﬀects. A char-
acterization of hysteresis systems using hysteresis loops
is also addressed by Ikhouane in Ref. [35] through the
concepts of consistency and strong consistency.
In light of what has been said it becomes clear that,
in Ref. [64], the generalizations of Duhem’s model are
done in such a way that rate independence is preserved,
whilst a deﬁnition of hysteresis based on hysteresis loops
in Ref. [54] is compatible with a generalized form of the
Duhem model that may be rate dependent or rate in-
dependent.
Why are there different models of hysteresis?
In Ref. [16] Duhem proposes his model to account for
the irreversibility in the modiﬁcations of equilibria ob-
served experimentally in magnetic hysteresis [16, Chap-
ter IV], sulfur [17], red phosphorus [19, Chapter III],
and in diﬀerent processes of metallurgy [19].
Preisach [56] uses “plausible hypotheses concerning
the physical mechanisms of magnetization [49, p. 1] to
elaborate a model of magnetic hysteresis. This model
is also proposed and studied by Everett and co-workers
[24]–[27] who postulate “that hysteresis is to be attributed
in general to the existence in a system of a very large
number of independent domains, at least some of which
can exhibit metastability.” [24, p. 753].
Krasnosel’skiˇı and Pokrovskiˇı point out to the is-
sue of admissible inputs, as “it is by no means clear a
priori for any concrete transducer with hysteresis, how
to choose the relevant classes of admissible inputs” [43,
p. 5]. This is why they introduce the concept of vibro-
correctness which allows the determination of the out-
put of a hysteresis transducer that corresponds to any
continuous input, once we know the outputs that cor-
respond to piecewise monotone continuous inputs [43,
p. 6]. The models that Krasnosel’skiˇı and Pokrovskiˇı
propose (ordinary play, generalized play, hysteron) are
vibro-correct, although the authors acknowledge the
existence of hysteresis models that may not be vibro-
correct like the Duhem model.
6
Hysteresis models based on a feedback interconnec-
tion between a linear system and a static nonlinearity
are proposed in Ref. [55]. The authors study “hysteresis
arising from a continuum of equilibria . . . and hysteresis
arising from isolated equilibria” [55, p 101].
A review of hysteresis models is provided in Ref. [48]
and a detailed study of these (and other) models may
be found in Refs. [7], [10], [14], [37], [49], [64].
In light of what has been said, the diversity of hys-
teresis models is due to the wide range of areas to which
hysteresis is concomitant, and the diversity of methods
and assumptions underlying the elaboration of these
models.
Note that all mathematical models of hysteresis share
a common property: they model hysteresis. This fact
leads us to our next question.
What is hysteresis? A description found in many
papers is that hysteresis “refers to the systems that have
memory, where the eﬀects of input to the system are
experienced with a certain delay in time. [33, p. 210].
This description is misleading as it applies also to dy-
namic linear systems. Indeed, when the output y is re-
lated to the input u by ˙x = Ax + Bu and y = Cx
which is a possible description of a linear system, the
output is given by y(t) = C
exp(tA)x
0
+
R
t
0
exp
(t
τ)A
Bu(τ )dτ
where x
0
is the initial state and t 0 is
time. We can see that y(t) depends on the integral of a
function that incorporates u(τ) for all τ [0, t], which
means that the linear system does have memory. How-
6
called the Madelung model in Ref. [43].

A Survey of the Hysteretic Duhem Model 5
ever, “hysteresis is a genuinely nonlinear phenomenon
[10, p. vii].
Mayergoyz considers hysteresis as a rate-independent
phenomenon which is “consistent with existing experi-
mental facts.” [49, p. xvi]. However, “for very fast in-
put variations, time eﬀects become important and the
given deﬁnition of rate-independent hysteresis fails.”
[49, p. xvi]. Also, “in the existing literature, hysteresis
phenomenon is by and large linked with the formation of
hysteresis loops (looping). This may be misleading and
create the impression that looping is the essence of hys-
teresis. In this respect, the given deﬁnition of hysteresis
emphasizes the fact that history dependent branching
constitutes the essence of hysteresis, while looping is a
particular case of branching.” [49, pp. xvi–xvii].
Following Mayergoyz, “All rate-independent hystere-
sis nonlinearities fall into two general classiﬁcations:
(a) hysteresis nonlinearities with local memories, and
(b) hysteresis nonlinearities with nonlocal memories.”
[49, pp. xvii]. In a hysteresis with a local memory, the
state or output at time t t
0
is completely deﬁned by
the state or output at instant t
0
, and the input on [t
0
, t].
This is the case for example of a hysteresis given by a
diﬀerential equation. Hysteresis with a nonlocal mem-
ory is a hysteresis which is not with local memory. This
is the case for example of the Preisach model. “How-
ever, the notion of hysteresis nonlinearities with local
memories is not consistent with experimental facts.”
[49, pp. xix–xx]. Hodgdon, on the other hand, writes
in relation with the use of a special case of the Duhem
model to represent ferromagnetic hysteresis: “These re-
sults are in good agreement with the manufacturer’s dc
hysteresis data and with experiments” [34, p. 220].
In Ref. [54], Oh and Bernstein consider the gener-
alized Duhem model ˙x = f(x, u)g( ˙u) and y = h(x, u)
with u the input, y the output and x the state. The au-
thors assume the existence of a unique solution of the
diﬀerential equation on the time interval [0, [. They
also assume the existence of a T –periodic solution x
T
for any T –periodic input u
T
with one increasing part
and one decreasing part, which means that the graph
{(u
T
, x
T
)} is a closed curve. Finally they assume that
when T the graph {(u
T
, x
T
)} converges with re-
spect to the Hausdorﬀ metric to a closed curve C. If we
can ﬁnd (a, b
1
) C and (a, b
2
) C with b
1
6= b
2
, the
curve C is not trivial and the generalized Duhem model
is a hysteresis.
In a PhD thesis advised by Bernstein [15], Drinˇci´c
considers systems of the form ˙x = f(x, u) and y =
h(x, u) for which hysteresis is deﬁned as in Ref. [54].
The system is supposed to be step convergent, that
is lim
t→∞
x(t) exists for all initial conditions and for
all constant inputs. It is noted that there exists “a
close relationship” [15, p. 6] between the curve C and
the input-output equilibria map, that is the set E =

u, h(lim
t→∞
x(t), u)

where u is constant and
f
lim
t→∞
x(t), u
= 0. In particular, the “system . . .
is hysteretic if the multivalued map E has either a con-
tinuum of equilibria or a bifurcation [15, p. 7].
In Ref. [6] Bernstein states that “a hysteretic sys-
tem must be multistable; conversely, a multistable sys-
tem is hysteretic if increasing and decreasing input sig-
nals cause the state to be attracted to diﬀerent equilib-
ria that give rise to diﬀerent outputs.” Multistability
means that “the system must have multiple attracting
equilibria for a constant input value” [6].
In Ref. [50], Morris presents six examples of hys-
teresis systems taken from the areas of electronics, bi-
ology, mechanics, and magnetics; hysteresis being un-
derstood as a “characteristic looping behavior of the
input-output graph” [50, p. 1]. The author explains the
qualitative behavior of these systems from the point
of view of multistability. For “the diﬀerential equations
used to model the Schmitt trigger, cellular signaling and
a beam in a magnetic ﬁeld” it is observed that “these
systems, all possess, for a range of constant inputs, sev-
eral stable equilibrium points.” [50, p. 13]. The author
observes that the systems are rate dependent for high
input rates.
For the play operator, the Preisach model and the
Bouc-Wen model which are rate independent, “these
models present a continuum of equilibrium points.” [50,
p. 13]. These observations lead the author to conclude
that “hysteresis is a phenomenon displayed by forced
dynamical systems that have several equilibrium points;
along with a time scale for the dynamics that is consid-
erably faster than the time scale on which inputs vary.”
[50, p. 13]. Morris proposes the following deﬁnition.
“A hysteretic system is one which has (1) multiple
stable equilibrium points and (2) dynamics that are con-
siderably faster than the time scale at which inputs are
varied.” [50, p. 13].
In Ref. [35], Ikhouane considers a hysteresis opera-
tor H that associates to an input u and initial condi-
tion ξ
0
an output y = H(u, ξ
0
), all belonging to some
appropriate spaces.” [35, p. 293]. It is assumed that the
operator H is causal and satisﬁes the property that con-
stant inputs lead to constant outputs. Examples include
all rate-independent models [47, Proposition 2.1], some
rate-dependent models, models with local memory like
the various generalizations of the Duhem model, and
models with nonlocal memory like the Preisach model.
The author introduces two changes in time scale: (1)
a linear one which is applied to a given input, and (2) a
-possibly- nonlinear one which is the total variation of
the original input. When the input is composed with the

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### "A Survey of the Hysteretic Duhem Mo..." refers methods in this paper

• ...Finally, many phenomenological models of friction or hysteresis can be seen as particular cases of a more general form of the Duhem model: this is the case for example of the Dahl [13], the LuGre [2,11], or the Maxwell-slip models [30]....

[...]

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• ...Preisach [56] uses “plausible hypotheses concerning the physical mechanisms of magnetization” [49, p....

[...]

• ...[8], [12] or [56] do not cite his memoirs....

[...]