AVERAGING PRINCIPLE FOR DIFFERENTIAL EQUATIONS WITH … · AVERAGING PRINCIPLE FOR DIFFERENTIAL EQUATIONS WITH HYSTERESIS 3 operator W, that is, to call the operators W and W0 diﬀerent

AVERAGING PRINCIPLE FOR DIFFERENTIAL EQUATIONSWITH HYSTERESIS

A. POKROVSKII O.RASSKAZOV A. VLADIMIROV

Abstract. The goal of this paper is to extend the averaging technique to newclasses of hysteresis operators and oscillating functions as well as to bring moreconsistency into the exposition. In the first part of the paper, making accenton polyhedral vector sweeping processes, we keep in mind possible applicationsto the queueing theory where these processes arise naturally. In the secondpart we concentrate on the systems with the classical Preisach nonlinearity.

Key words: Averaging technique, Hysteresis, Sweeping processes, Queueingtheory, Preisach model.

A part of this paper was written during the last author’s visit to Boole Centre for Researchin Informatics and Department of Applied Mathematics, University College Cork, financial sup-port from which is gratefully acknowledged. Vadimirov was also partially supported by grant No.03-01-00258 of the Russian Foundation for Basic Research, by grant for Scientific Schools No.1532.2003.1 of the President of Russian Federation and by the project ”Development of mathe-matical methods of analysis of distributed asynchronous computational networks” of the programof fundamental investigations of OITVS RAN (Branch of Information Technologies and ComputerSystems of Russian Academy of Sciences) ”New physical and structural methods in infocommu-nications”. A. Pokrovskii and O. Rasskazov were partially supported by the Enterprise Ireland,Grants SC/2000/138 and SC/2003/376.

A. Pokrovskii and O. Passkazov: Department of Applied Mathematics, University College,Cork, IRELAND. E-mails: [email protected],[email protected]. Vladimirov: Institute for Informa-tion Transmission, 19, Bol’shoi Karetnyi ln., Moscow, Russia. E-mail: [email protected].

1

2 A. POKROVSKII O.RASSKAZOV A. VLADIMIROV

Contents

1. Introduction 22. Sweeping processes 32.1. Hysteresis operators 32.2. Sweeping processes 42.3. Averaging method and the limit hysteresis operator 52.4. Oscillating sweeping processes 62.5. Limit operators for oscillating sweeping processes 73. Averaging in systems with Preisach nonlinearity 73.1. Preisach model 73.2. Averaging of Preisach nonlinearity 103.3. Numerical example 113.4. Application to qualitative analysis of systems with hysteresis 13References 15

1. Introduction

For the classical Bogolyubov-Krylov-Mitropol’skii averaging principle, see, forinstance, [2]. Briefly, this principle asserts that the evolution of slowly changingvariables in an oscillating differential system is well approximated by solutions of the“averaged” differential equation where oscillations in the right-hand are smoothedout. The length of time interval where this approximation holds is inverse propor-tional to the rate of change of slow variables.

The system is described by an ODE with a small right-hand side (a functionf multiplied by a small parameter ε); this guarantees the slowness of change ofthe state variable x if f is uniformly bounded. The function f , on the contrary,oscillates rapidly, that is, for a fixed x it is an oscillating function of “quick” timet. There might be other “quick” variables y that depend on x and t in a ratherstraightforward manner. Thus, we study the equation

(1.1) x = εf(x, y, t).

If y is just a function of (x, t), there is no need to consider y at all: the dependenceof f on y can be handled by using a modified function f(x, t) = f(x, y(x, t), t) inthe right-hand side of (1.1). In the literature, the case where y is a solution ofanother ODE with “large” right-hand side is often addressed.

In this paper we consider the case of y being the output of a hysteresis operatorW with a rapidly oscillating input u(t) = g(x(t), t), that is,

(1.2) y(t) = Wg(x(t), t).

In general, the operator W itself may depend on the “slow” parameter x but herewe do not consider this case.

In standard hysteresis references [3, 9, 10, 19], the output z(t) = Wu(t) for agiven input u(t), t ≥ 0, is, usually, not uniquely determined unless an initial stateω0 is given. Wherever possible, we will modify the hysteresis operator in order tomake the map u(·) → z(·) single-valued and, thus, to get rid of the state variableω. A straightforward method to do this is to set some continuous dependenceω0 = Ω(u(·)). For instance, one may assume ω0 to be fixed for a particular hysteresis

AVERAGING PRINCIPLE FOR DIFFERENTIAL EQUATIONS WITH HYSTERESIS 3

operator W , that is, to call the operators W and W ′ different if they have differentinitial states ω0 6= ω′0.

We use the papers [8, 20] as a basis for further development of the theory. Aprofound research of forced oscillations with a simple hysteresis operator was madein [1].

Our goal is to extend the averaging technique to new classes of hysteresis opera-tors and oscillating functions as well as to bring more consistency into the exposi-tion. In the first part of the paper (Sections 2), making accent on polyhedral vectorsweeping processes, we keep in mind possible applications to the queueing theorywhere these processes arise naturally if certain finite-dimensional service disciplinesare used, [6, 17]. The averaging technique in information networks can be useful insituations where large amounts of information are transmitted and their inflow is“irregular” as, say, in the Internet, and where the control of the network is basedonly on some rather slowly changing parameters such as averaged delays of mes-sages during a time interval of a given length, etc. In the second part (Section 3)we concentrate on the systems with the classical Preisach nonlinearity.

2. Sweeping processes

2.1. Hysteresis operators. In this section we recall basic notions and definitionsof mathematical hysteresis theory, see [3, 9, 10, 19]. Let W be a finite-dimensionalinput-output operator with inputs u(t) : I → Rn and outputs z(t) : I → Rm, whereI = R+ or I = [0, T ] for some T ≥ 0. The operator Wu(·) = z(·) is a hysteresis oneif it possesses the following two properties:

(1) Causality. The action of W does not depend on the future, that is, ifu(t) = u′(t), 0 ≤ t ≤ T , for a pair of inputs u(·) and u′(·), then z(t) = z′(t),0 ≤ t ≤ T , where z = Wu and z′ = Wu′.

(2) Rate-independence. The operator W is invariant to monotone continuouschanges of time. Namely, if ϕ(t) is a continuous nondecreasing function from R+

to R+ such that ϕ(0) = 0, and z = Wu, then z′ = Wu′ where u′(t) = u(ϕ(t)) andz′(t) = z(ϕ(t)), t ∈ R+.

Particular cases of hysteresis operators are plays and stops, both scalar and vec-tor (they find applications in the theory of elastic-plastic media), and the Ishlinskiimodel (with a fixed initial state), a particular (continuous) case of the Preisachmodel [9, 13] which is used in the magnetization theory. We have m = n for playsand stops and m = n = 1 for the Ishlinskii model.

As a simple example of play let us consider the operator

Wu(t) = z(t) = min0, inf0≤s≤t

u(s).

The corresponding stop operator V u(t) = u(t) −Wu(t) is a one-dimensional Sko-rokhod problem [18]. This is one of rare cases where the output z(·) can be foundexplicitly by the input u(·). The properties of causality and rate-independence areimmediate.

Plays and stops are short-memory hysteresis operators which means that theoutput z(t) for t ≥ T depends only on the value z(T ) and on the restriction of theinput u(·) to the interval [T, t]. Whenever possible we will assume that the initialstate z(0) is fixed; this assumption leads to the unique dependence of z(·) on u(·).

The Ishlinskii operator is not a short-memory one; its evolution is determinedby the input u(·) and the initial state ω(0) (infinite-dimensional, in general). In


fact, one may consider the Ishlinskii operator as a short-memory one by taking thevariable state ω(t) for the output. For a general hysteresis operator it is alwayspossible to consider formally the restriction of the input u(·) to the interval [0, t] asthe state ω(t) or even as the output. It is clear that this “canonical” operator is,indeed, causal, rate-independent and short-memory.

The Ishlinskii model is a uniformly continuous operator, that is, continuous withrespect to the sup-norm ‖u(·)‖ = supt |u(t)|; the same is true for certain plays andstops, but not for all of them in RN , N > 2.

The play operator is determined by its characteristic set Z (a closed convex setin Rn) and by the rule of choice of the initial state z(0) = G(u(0)) ∈ Z + u(0).The set Z is called regular if the corresponding play is uniformly continuous forsome uniformly continuous map G : u(0) → z(0), say, for G(u) = z ∈ Z + u :‖z‖ = miny∈Z+u ‖y‖. This definition does not depend on the choice of G(·). Notethat G(u) can be, itself, considered as a hysteresis operator on the singleton timeinterval I = 0. In this paper we restrict consideration to the class of polyhedralsets Z.

2.2. Sweeping processes. The sweeping process introduced by J. J. Moreau [15,16], see also [14], is a formalization of a “lazy” point motion constrained to achanging domain. It is a generalization of the vector play operator, that is, aplay with time-dependent characteristic set Z(t). As we will see below, sweepingprocesses may arise, in particular, as the result of averaging of plays.

Let us define a polyhedral sweeping process in Rn for the time interval t ≥ 0.Let p1, . . . , pk be a finite set of unit vectors in Rn. For a vector b ∈ Rk, we definethe polyhedral set

Z(b) = z ∈ Rn : 〈z, pi〉 ≤ bi, i = 1, . . . , k,and use notation B = b ∈ Rk : Z(b) 6= ∅, Z = Z(b) : b ∈ B.

The input to the sweeping process is a continuous vector-function

b(t) = b1(t), . . . , bk(t), t ≥ 0,

such that the set Z(b(t)) is not empty for all t ≥ 0. Note that Z(t) = Z(b(t)) is aHausdorff-continuous set-valued map because the map Z(b) is, obviously, Lipschitzcontinuous from B to 2R

n

, where 2Rn

is endowed with the Hausdorff metric.Let us define the output z(t) of the sweeping process Z(t) as follows. We set

z(0) = PZ(0)(0), that is, z(0) is the orthogonal projection of the origin onto theclosed convex set Z(0). As is known, this projection is uniquely determined andthe map z → PZ(0)(z) is a contraction in Rn.

An absolutely continuous function z(t) : R+ → Rn is a solution of the sweepingprocess Z(t) if z(0) = PZ(0)(0), if z(t) ∈ Z(t), t ≥ 0, and if

z(t) ∈ NZ(t)(z(t)) for almost all t ≥ 0,

whereNZ(z) = y ∈ Rn : 〈y, x− z〉 ≥ 0 for all x ∈ Z

is the inner normal cone to Z at z ∈ Z.If the interior of Z(t) is not empty for all t ≥ 0 then, for any z0 ∈ Z(0), there

exists a unique solution z(t) satisfying z(0) = z0, [14]. Moreover, this solutiondepends in a Lipschitz continuous way both on the process characteristic set Z(t)


(with respect to the Hausdorff metric) and on the initial point z0 ∈ Z(0), see[21, 9, 5, 10, 11]. Namely, there exists an L > 0 such that

(2.1) ‖z(t)− z′(t)‖ ≤ L(‖z0 − z′0‖+ sup0≤s≤t

‖b′(s)− b(s)‖), t ≥ 0,

where z′(t) is the solution of the process Z ′(t) = Z(b′(t)) from z′0 ∈ Z ′(0) and b′(t)is a continuous vector-function from R+ to Rk such that Z(b′(t)) has nonemptyinterior for all t ∈ R+.

Hence, the definition of a solution can be extended by continuity to the set ofall nonempty inputs Z(t) = Z(b(t)) for continuous b(t) ∈ B, and the Lipschitzbound (2.1) still holds for the extended operator. Note that the output of theextended sweeping process is no longer absolutely continuous. For instance, if Z(t)is a singleton z(t) for each t, then Wu(t) = z(t) for any input u(t), but z(t) is ageneral continuous vector-function.

2.3. Averaging method and the limit hysteresis operator. Here we recallsome results of [8, 20]. The ODE system in consideration is

(2.2) x(t) = εf(t, x, w), x ∈ Rn, w ∈ Rm,

where

(2.3) w = Wg(t, x(t))

is the output of the hysteresis operator W and ε > 0 is a small parameter. Thesolutions are traced on a finite interval of the “slow” time τ = εt. The functionsf(t, x, w) and g(t, x) are assumed periodic in t and uniformly continuous in all theirvariables. In [8], both f and g were assumed to be almost periodic in t. Herewe only consider the periodic case for simplicity. All the results, however, can beextended to the almost periodic case.

In order to write the averaged equations, we must first define the limit hysteresisoperator W ∗ as follows. We should demonstrate first that W transmits some classesof “limit oscillatory” inputs to the same classes. In the next section we will studythis property in detail. For simplicity, let us speak about the class V of limit1-periodic inputs (and outputs). A function f(t), t ≥ 0, is limit 1-periodic iflimt→∞ ‖f(t)− g(t)‖ = 0 for some 1-periodic function g(t). Since g(·) is, obviously,unique, we introduce the operator P by g(·) = Pf(·).

Let us assume that W transfers V into V (this will be proved below for polyhedralsweeping processes). We define the class of inputs to the limit operator W ∗ as theclass of vector-functions v(t, σ) ∈ Rn, periodic in t and uniformly continuous in thescalar variable σ ≥ 0. Informally, the parameter σ is increasing infinitely slowly ast → +∞.

For the formal definition, we consider a sequence of δ-slow approximations ϕδ(t, σ) =δt for 0 ≤ t ≤ δ−1σ and ϕδ(t, σ) = σ for t ≥ δ−1σ. Then we define the family ofoutputs

ξδ(t, s) = PWv(t, ϕδ(t, σ)).

Suppose that, for each s there exists a limit lim ξδ(t, s) uniformly by t. This limitdefines the operator W ∗v(t, s) = u(t, s).

The operator W ∗, if it exists, can be regarded as a hysteresis input-outputoperator in the (infinite-dimensional) space Q of 1-periodic functions. Namely,


Q = C(S1), where S1 is the unit circle and C(S1) is the space of continuous func-tions on S1 endowed with the max-norm ‖f(·)‖ = maxt∈S1 |f(t)|. The propertiesof causality and rate-independence of W ∗ follow immediately from its definition.

Let us denote

Φx(s) = limT→+∞

∫ T

0

f(t, x(s),W ∗g(t, x(s))dt.

The averaging method replaces (2.2) by the equation

(2.4) x(t) = εΦx(t)

The following theorem adapted from [8] will be used here for polyhedral sweepingprocesses.

Theorem 2.1. Solutions of (2.4) converge to the corresponding solutions of (2.2)as ε → +0 uniformly on any ε-dependent interval [0, aε−1], a > 0, if the limitoperator W ∗ exists.

2.4. Oscillating sweeping processes. In order to clarify the form of limit hys-teresis operators in specific cases, we will study the behavior of sweeping processeson oscillatory inputs of different kind. Let us recall the following well-known result(see, for instance, [14]).

Lemma 2.2. If z(t) and z′(t) are solutions of the same sweeping process Z(t), then‖z′(t1)− z(t1)‖ ≤ ‖z′(t0)− z(t0)‖ for any pair 0 ≤ t0 ≤ t1. Moreover, there existsa D = D(p1, . . . , pk) > 0 such that the total variation of z′(t) − z(t) on R+ doesnot exceed D‖z′(0)− z(0)‖.

We will formulate and prove all the results for the case of periodic inputs b(t),t ≥ 0, of the sweeping process keeping in mind that most of the results can beextended to almost periodic and other “recurrent” inputs. Since the process israte-independent, we may assume without loss of generality that b(t) is a 1-periodicfunction. Let us first assume that Z(t) is bounded for all t (and, hence, uniformlybounded for t ∈ R+). Because of the continuous dependence of the solution on theinitial point z0, the Brouwer fixed point principle gives us existence of a (unique)1-periodic solution z(t), z(0) = z0.

The assumption of boundedness of Z(t) can be lifted:

Lemma 2.3. Let Z(t) be continuous and periodic. Then all the solutions z(t) ofZ(t) are bounded and their amplitudes supt,t′ ‖z(t)− z(t′)‖ are uniformly bounded.

Proof. It suffices to consider the case of nonempty interior of the recession cones ofall Z(t). In this case the assertion follows immediately from (2.1) and Lemma 2.2.All the solutions in this case converge to constant solutions. ¤

Let us denote by Y = Y (Z) = Y (b) the set of all initial points z(0) of 1-periodicsolutions z(t) of the process Z(t). Obviously, Y is closed. Lemma 2.2 implies theconvexity of Y .

Lemma 2.4. The set Y belongs to Z.

Proof. Let z(t) be a 1-periodic solution of Z(t). Let us show that

Y = y ∈ Z(0) : y + z(t)− z(0) ∈ Z(t), 0 ≤ t < 1.This is true for a ”decomposed” (discontinuous) process, where no more than oneconstraint 〈z, pi〉 ≤ βi(t) is present at each t ≥ 0. This happens because, for a


decomposed process, two solutions are parallel if and only if their difference belongsto the intersection of all hyperplanes that are active somewhere for any one of them.Then we get the general result by limit considerations. ¤

The following result is well known.

Lemma 2.5. Any solution of a periodic process Z(·) converges to some periodicsolution of Z(·) as t → +∞.

The assertion follows from the existence of a periodic solution and from the factthat the variation of difference of two arbitrary solutions of Z(·) is bounded.

Lemma 2.6. The set Y = Y (b(t)) depends in a Lipschitz way on the periodicinput b(t), where the metrics di and do on the input and output spaces are definedas follows: do(Y, Y ′) is the Hausdorff distance and di(b, b′) = supt≥0 ‖b(t)− b′(t)‖.Proof. By Lemma 2.5, if an initial point y0 is distant from Y , the solution fromy(0) is distant from periodic. Since this solution depends in a Lipschitz way on b(·),we get the statement of the theorem. ¤

2.5. Limit operators for oscillating sweeping processes. Let b(t, σ) be de-fined on E = R×R+, continuous in both variables, 1-periodic in t, and such thatZ(b(t, σ)) is nonempty for all (t, σ) ∈ E. We denote Y (σ) = Y (b(t, σ)) andz(t, σ) = z(t)−z(0), where z(t) is an arbitrary periodic solution of Z(t) = Z(b(t, σ)).The averaged sweeping process will be defined as the process on t ∈ [0, 1] with thecharacteristic set Y (σ).

Here we will distinguish the following three main cases.(C1) The oscillatory component of a solution is constant. This happens when,

for each σ ∈ R+, the intersection⋂

t∈[0,1] Z(t, σ) is not empty.(C2) The set Y (σ) is a singleton for each σ ∈ R+. In this case the limit behavior

of solutions of Z(t) does not depend on the initial state.(C3) In the intermediate case, Y (σ) is not a singleton for some values of σ and

z(t, σ) depends essentially on t.

Theorem 2.7. The limit hysteresis operator W ∗ for the sweeping process W = Z(·)transfers the input u(t, σ) to the output v(t, σ) = y(σ) + z(t, σ), where y(·) is theoutput of the averaged process Y (·) and z(t, σ) = z(t) − z(0) for some periodicsolution z(·) of Z(t, σ).

Hence, the existence of limit hysteresis operator is proved for the class of polyhe-dral sweeping processes with periodic inputs. In the most general case its output isthe sum of the slow component y(σ) which is the output of the averaged sweepingprocess Y (σ) and the fast component z(t, σ). All the results can be extended to theclass of almost periodic processes and some of them to more general “recurrent”classes of oscillating inputs and outputs.

Finally, let us note that the averaging results of this paper are also true for a wideclass of sweeping processes with oblique reflection (see, for instance, [4, 5, 11, 12]for definitions).

3. Averaging in systems with Preisach nonlinearity

3.1. Preisach model. The nonideal relay (with the threshold values α < β) isthe simplest hysteretic transducer. This transducer is denoted by Rα,β ; its output


y(t) can take one of two values 0 or 1: at any moment the relay is either ‘switchedoff’ or ‘switched on’. The dynamics of the relay is usually described by the picturebelow, Figure 1.

β

y

x0

1

α

Figure 1: Nonideal relay Rα,β

The variable output y(t)

(3.1) y(t) = Rα,β [t0, η0]x(t), t ≥ t0,

depends on the variable input x(t) (t ≥ t0) and on the initial state η0. Herethe input is an arbitrary continuous scalar function; η0 is either 0 or 1. The scalarfunction y(t) has at most a finite number of jumps on any finite interval t0 ≤ t ≤ t1.

The output behaves rather ‘lazily’: it prefers to be unchanged, as long as thephase pair (x(t), y(t)) belongs to the union of the bold lines in the picture above.The values of function (3.1) at a moment t are defined by the following explicitformula:

y(t) = Rα,β [t0, η0]x(t) =

η0, if α < x(τ) < β for all τ ∈ [t0, t];

1, if there exists t1 ∈ [t0, t] such thatx(t1) ≥ β, x(τ) > α for all τ ∈ [t1, t];

0, if there exists t1 ∈ [t0, t] such thatx(t1) ≤ α, x(τ) < β for all τ ∈ [t1, t].

The equalities y(t) = 1 for x(t) ≥ β and y(t) = 0 for x(t) ≤ α always hold fort ≥ t0.

Figure 2 gives another geometrical illustration of the definition of a nonidealrelay. Here we combine on the same graph a typical input and the correspondingoutput of the relay.

y(t)

β

α

1

0 t

x,yx(t)

Figure 2: Input and output or nonideal relay Rα,β


Consider a family R of relays Rø = Rαø,βø with threshold values αw, βw, w ∈ Ω.The index set Ω may be finite or infinite, and we call such family a bundle of relays.Suppose that the set Ω is endowed with a probability measure µ. We furthersuppose that both functions αø, βø are measurable with respect to µ.

We call any measurable function η(ø) : Ω → 0, 1 the initial state of the bundleR.

For any initial state η0(w) and any continuous input x(t), t ≥ t0, we define thefunction

(3.2) y(t) = y[t0, η0](t) =∫

Ω

Rø[t0, η0(ø)]x(t) dµ, t ≥ t0.

We refer to this model as Preisach nonlinearity (or when convenient, Preisachmodel or Preisach operator). Here y(t) is the output of the Preisach model fromthe initial state η0 and the input x(t).

If the measure µ has a finite support w1, . . . , wN , then the definition (3.2) maybe rewritten as a parallel connection of the relays Røj = Rαøj

,βøjwith the weights

µj = µ(øj) > 0:

y(t) = y[t0, η0](t) =N∑

j=1

µjRøj [t0, η0(øj)]x(t), t ≥ t0.

Fig. 3 illustrates this discrete situation.

........µ

µ

µµ

2

1

Σ

Rα β 1 1

R

R

R

α β

α β

α β

2 2

3 3

N N

x(t) y(t)3

N

........

........

Figure 3: Weighted parallel connection of a finite number of nonideal relays

Thus, from the point of view of system theory, the Preisach nonlinearity is aparallel connection of a continuous bundle of nonideal relays.

For t > t0, it is also convenient to interpret the function

η(ω) : ω → Rø[t0, η0(ω)]x(t)

as the state of the Preisach nonlinearity at the moment t.


3.2. Averaging of Preisach nonlinearity. Let us give a simple example for anexplicit description of the limit nonlinearity. Let P be a Preisach nonlinearity andlet function u(t, s) be T (s)-periodic. We suppose also that the set Ω coincides withthe half plane α < β.

It is sufficient to consider the case when the initial state η0 is the characteristicfunction of a set S, which look like the coloured set in Figure 4. That is, S islocated to the left of the line α = β and below a continuous linear curve L = LS ,with a non-positive slope. We denote the set of states η describing the sets S likeabove by E∗. Additionally, we assume that the coordinates of the intersection of Lwith the diagonal coincide with the current value of the input x.

S=S(t) α

β

S=S(t)

−1

−1 1

β

α

(a) The set S (b) The set S(0)Figure 4: The sets S, S(0)

Now we will correspond to a given function u(t, s) a varying set S(s) ∈ S∗. Foreach s ≥ 0 introduce

(3.3) u−(s) = mint

u(t; s) and u+(s) = maxt

u(t; s).

For simplicity we consider only the case when both these functions are piece-wisemonotone.

We define S(0) as the part of S which is located to the left from the vertical lineα = u−(0), that is,

S(0) =

(α, )¯∈ S : α < u−(0)

.

see Figure 4,(b).Suppose that S(σ) is defined for 0 ≤ σ ≤ s0 and that the functions (3.3) are

monotone at the interval s0 ≤ s. Let us consider four cases:

Case 1: u−(s) ≥ u−(s0), and u+(s) ≤ u−(s0).Case 2: u−(s) ≥ u−(s0), and u+(s) ≥ u−(s0).Case 3: u−(s) ≤ u−(s0), and (u−(s), u+(s)) 6∈ S(s0).Case 4: u−(s) ≤ u−(s0), and (u−(s), u+(s)) ∈ S(s0).

Figure 5 provide the ‘picture definitions’ for the corresponding set S(s) which isthe union of the blue and yellow regions. The bold red line is here given by theparametric equation α = u−(σ), β = u+(σ).

For instance in Case 1 S(s) is the union of the set S(s0) and the set(α, β) : u−(s0) ≤ α < u−(s), and α < ¡

¯u+(α)

.


u (s )0+

−u (s)

u (s)+

0−u (s )

α

β

S=S(t)

u (s )0+

−u (s)

0−u (s )

u (s)+

α

β

S=S(t)

(a) Case 1 (b) Case 2

u (s )0+

u (s)+

−u (s)

0−u (s )

α

β

S=S(t)

u (s )0+

0−u (s )

u (s)+

−u (s)

α

β

S=S(t)

(c) Case 3 (d) Case 4Figure 5: Set S(s)

The limit Preisach nonlinearity P ∗[η0] can be described as

(P ∗[η0]xs)(t) = (P [η(s)]xs)(t),

where xs(t) = u(s, t) and η(s) is the characteristic function of the set S(s).

3.3. Numerical example. Here we demonstrate how the techniques describedabove works at a particular example. Consider our standard scalar differential-operator equation of the form

(3.4)x′′ + x = εy(t),

y(t) = Φ(η(t))2,η(t) = (Γ[η0])x(t).

where Φ(η(t)) is a Preisach nonlinearity with a measure µ. We suppose that µ isa Lebesgue measure on the triangle ∆(−1, 1) = −1 ≤ α ≤ β ≤ +1 and zeroanywhere else.

Using the substitution

x = r cos(θ + t), x′ = −r sin(θ + t)

the equation (3.4) can be rewritten in the form of the system

(3.5)

dr/dt = −ε sin(t + θ)y(t),dθ/dt = ε cos(t + θ)y(t)/r,

y(t) = Φ(η(t))2,η(t) = (Γ[η0])(r(t) cos(θ(t) + t)).


Consider the initial conditions

r(0) = r0 ∈ (0, 1], θ(0) = 0, η(0) = η0.

For those initial conditions, the system (3.5) has a unique solution. The initialstate of the Preisach nonlinearity η0 is chosen to be union of the colored areas onFigure 6.

r

−r

−1

1

cPreisach input range[−r,r]

Figure 6: Set S(s) for (3.5) is in cyan; all possible sets described bycharacteristic function η belong to the union of the colored areas

Using averaging method from the previous subsection and the properties of theLebesgue measure, we can write differential equation for the average r of r

dr/dt = − 12π

∫ 2π+θ

θ

ε sin(s + θ)Φ((Γ[η0])(r cos(θ + s)))2ds

= − 12π

∫ π

0

ε sin(s)(c +r2

2(cos(s)− 1)2)2ds

− 12π

∫ 2π

π

ε sin(s)(c + 2r2 − r2

2(1− cos(s))2)2ds

= −4εr2(c + r2)3π

.

Here c > 0 and r is decreasing in time, that is the case of the evolution of a Preisachnonlinearity shown on Figure 4 (a). The solution x = r sin(t + θ) has symmetricminima and maxima which means that c is the area the cyan colored polygon inFigure 6. Thus c = 1− r2 and the equation above can be written as

dr

dt= −4εr2

3π, r(0) = r0

and has the solution

r(t) =3πr0

3π + 4εr0t.


Similary, one can write down and solve the equation for the average θ of θ. Fromthe averaging principle

dθ

dt=

12πr

∫ 2θ

θ

ε cos(s + θ)Φ((Γ[η0])(r cos(θ + s)))2ds

=1

2πr

∫ π

0

ε cos(s)(c +r2

2(cos(s)− 1)2)2ds

12πr

∫ 2π

π

ε cos(s)(c + 2r2 − r2

2(1− cos(s))2)2ds

= εr(c + r2).

Now the equation for θ

dθ

dt= ε

3πr0

3π + 4r0εt, θ(0) = 0

has the solution

θ(t) = −3/4(π log(3π)− π log(3π + 4εr0t)).

Below we compare the results of the averaging method with the straightforwardnumerical integrations. We consider two values of parameter ε: ε = 0.01 andε = 0.001. Initial conditions are the same in both cases, r0 = 1 and we comparethe solutions on the time interval [0, 1000].

0 20 40 60 80 100-1

-0.5

0

0.5

1

900 920 940 960 980 1000

-0.2

-0.1

0

0.1

0.2

(a) ε = 0.01, t ∈ [0, 100] (b) ε = 0.01, t ∈ [900, 1000]

0 20 40 60 80 100

-1

-0.5

0

0.5

1

900 920 940 960 980 1000

-1

-0.5

0

0.5

1

(c) ε = 0.001, t ∈ [0, 100] (d) ε = 0.001, t ∈ [900, 1000]Figure 7: Black dashed – numerical solution; Red solid – solution of averaging

equations; Brown solid – graphs r(t) and −r(t)

3.4. Application to qualitative analysis of systems with hysteresis. Follow-ing [7], we will demonstrate how the averaging method could work in qualitative


analysis of the scalar differential-operator equation of the form

x′′ + x = φ(t) + εy(t),y(t) = Φ(η(t)),(3.6)η(t) = (Γ[η0])x(t), ,

(3.7)

with the Preisach nonlinearity. We will suppose that the corresponding measure µis finite, positive, absolutely continuous with bounded density, defined in the halfplane α < β. Suppose that the external force φ(t) is a sum of periodic functionsφi(t), i = 1, . . . , n, with periods Ti, for which

(3.8)n∑

i=1

ki2π

Ti+ kn+1 6= 0

for any integers km, m = 1, . . . , n + 1. As we know, this equation for any initialconditions x(0) = x0, x′(0) = x′0 has a unique solution x(t, x(0), x′(0), ε)

We denote by ui(t), i = 1, . . . , n, the periodic solution (it is unique) of the equa-tion

u′′ + u = φi(t),

and we set

(3.9) u(t) =n∑

i=1

ui(t).

By means of the usual substitution

x = u(t) + r cos(θ + t), x′ = u′(t)− r sin(θ + t)

the equation (3.6) can be rewritten in the form of the system

dr

dt= −ε sin(t + θ)y(t),

dθ

dt= ε cos(t + θ)y(t)/r,

y(t) = Φ(η(t)),η(t) = (Γ[η0])x(t),x(t) = u(t) + r(t) cos(θ(t) + t).

For any initial conditions

r(0) = r0 > 0, θ(0) = θ0, η(0) = η0

this system has a unique solution

(r(t; r0, θ0, η0, ε), θ(t; r0, θ0, η0, ε), η(t; r0, θ0, η0, ε)) .

We denote by Ω(r), r > 0 the following domain in the half plane α < β:

Ω(r) =

β <

n∑

i=1

max ui(t) + r or α >

n∑

i=1

min ui(t)− r

.


For r > 0, (α, β) ∈ Ω(r) and 0 < θi < 2π, i = 1, . . . , n + 1, we set

F (r; α, β, θ1, . . . θn+1) = limt0→−∞

(R[χ(α, β); α, β)x) (0)

x(t) =n∑

i=1

ui

(t− t0 +

θi

2πTi

)+ r cos(t− t0 + θn+1).

The limit of the right-hand side always exists. For fixed values of r > 0 and(α, β) ∈ Ω(r) it is convenient to consider this function to be defined on the (n+1)-dimensional torus Hn+1. For r > 0 we set

P (r) =∫

Ω(r)

dµ(α, β)∫

Hn+1F (r; α, β, Θ) sin θn+1 dΘ.

Since in any domain 0 < r < R, P (r) satisfies a Lipschitz condition, each Cauchyproblem

dr

dt= −εP (r), ˜r(0) = r0

has a unique solution r(t; r0, ε).

Theorem 3.1. ([7]) The function P(r) is strictly positive for r > 0.

Corollary 3.2. Let R > 0. Then to each δ > 0 there correspond numbers ε > 0and L > 0 such that for 0 < ε < ε0, and |x0|+ |x′0|

|x(t; x0, x′0)− u(t)|+ |x′(t; x0, x

′0)− u′(t)| < δ, t > L/ε.

References

[1] P.-A. Bliman, A. M. Krasnoselskii, and M. Sorine. Dither in systems with hysteresis. Rapportde recherche n 2690, INRIA, 1995.

[2] N. N. Bogoliubov and Yu. A. Mitropolsky. Asymptotic Methods in the Theory of Non-LinearOscillations. Hindustan, Delhi, 1961.

[3] M. Brokate and J. Sprekels. Hysteresis and phase transitions. Springer-Verlag, Berlin, 1996.[4] P. Drabek, P. Krejci, and P. Takac. Nonlinear differential equations, volume 404 of Research

Notes in Mathematics. Chapman & Hall/CRC, London, 1999.[5] P. Dupuis and H. Ishii. On Lipschitz continuity of the solution mapping to the Skorokhod

problem with applications. Stoch. and Stoch. Rep., 35:31–62, 1991.[6] P. Dupuis and K. Ramanan. Convex duality and the Skorokhod problem, i, ii. Probab. Theory

Relat. Fields, 115:153–195, 197–236, 1999.[7] T.S. Gilman and A.V. Pokrovskii. Forced vibrations of an oscillator with hysteresis taken

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1988.[10] P. Krejci. Hysteresis, convexity and dissipation in hyperbolic equations. Gakkotosho, Tokyo,

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space of regulated functions. Set-Valued Anal., 11:91–110, 2003.[13] I. D. Mayergoyz. Mathematical models of hysteresis. Springer-Verlag, New York, 1991.[14] M. D. P. Monteiro Marques. Differential inclusions in nonsmooth mechanical problems—

shocks and dry friction. Birkhauser, Basel, 1993.[15] J. J. Moreau. Rafle par un convexe variable: premiere partie. SAC, 1, expose no 15, Mont-

pellier, 1971.[16] J. J. Moreau. Evolution problem associated with a moving convex set in a Hilbert space.

Journ. of Dif. Eq., 20:347–374, 1977.


[17] I. Nedaiborshch, K. Nikolaev, and A. Vladimirov. Lipschitz continuity and unique solvabilityof fluid models of queueing networks. Information Processes, Electronic Scientific Journal,3:138–150, 2003.

[18] A. V. Skorokhod. Stochastic equations for diffusion processes in a bounded region. Theor. ofProb. and its Appl., 6:264–274, 1961.

[19] A. Visintin. Differential models of hysteresis. Applied mathematical sciences. Springer, NewYork, 1994.

[20] A. A. Vladimirov. Averaging properties of multidimensional hysteresis operators. SovietMath. Dokl., 40(3):623–626, 1990.

[21] A. A. Vladimirov and A. F. Kleptsyn. On some hysteresis elements. Avtomat. i Telemekh.,(7):165–169, 1982. Russian.

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AVERAGING PRINCIPLE FOR DIFFERENTIAL EQUATIONS WITH … · AVERAGING PRINCIPLE FOR DIFFERENTIAL EQUATIONS WITH HYSTERESIS 3 operator W, that is, to call the operators W and W0 diﬀerent