Noname manuscript No.
(will be inserted by the editor)
A Survey of the Hysteretic Duhem Model
Fay¸cal Ikhouane
Received: date / Accepted: date
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
a direct access to the cited source. This direct access
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 Polit`ecnica 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-
dynamics [16, p. 6] (see also [19, pp. 5–7]).
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]
). To advance our un-
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
![Fig. 16: ϕ◦u(%) versus ψu(%) for % ∈ [0, %4]. The marker ◦ corresponds to the point ( ψu(%1), ϕ◦u(%1) ) whilst the marker ? corresponds to the point ( ψu(%3 = %5), ϕ◦u(%3 = %5) ) .](/figures/fig-16-ph-u-versus-psu-for-0-4-the-marker-corresponds-to-the-2xq6bf7h.webp)
![Fig. 15: ψu(%) versus % for % ∈ [0, 3]. We have %1 = 1, %2 = 1.5, %3 = %5 = 2, %4 = 3.](/figures/fig-15-psu-versus-for-0-3-we-have-1-1-2-1-5-3-5-2-4-3-2a5j59ny.webp)
![Fig. 8: ϕ◦u (%) versus ψu(%) for % ∈ [0, 2].](/figures/fig-8-ph-u-versus-psu-for-0-2-3en35qg0.webp)
