EFFECT of POSITIVE FEEDBACK on DEVIL's STAIRCASE INPUT-OUTPUT RELATIONSHIP Alexei Pokrovskii Dmitrii Rachinskii 1. Introductio

DISCRETE AND CONTINUOUS doi:10.3934/dcdss.2013.6.1095 DYNAMICAL SYSTEMS SERIES S Volume 6, Number 4, August 2013 pp. 1095–1112

EFFECT OF POSITIVE FEEDBACK ON DEVIL’S STAIRCASE INPUT-OUTPUT RELATIONSHIP

Alexei Pokrovskii Department of Applied Mathematics University College Cork, Ireland

Dmitrii Rachinskii Department of Applied Mathematics University College Cork, Ireland and Department of Mathematical Sciences University of Texas at Dallas, USA

Abstract. We consider emerging hysteresis behaviour in a closed loop system that includes a nonlinear link f of the Devil’s staircase (Cantor function) type and a positive feedback. This type of closed loops arises naturally in analysis of networks where local “negative” coupling of network elements is combined with “positive” coupling at the level of the mean-field interaction (in the limit case when the impact of each individual vertex is infinitesimal, while the number of vertices is growing). For the Cantor function f, taken as a model, and for a monotonically increasing input, we present the corresponding output of the system explicitly, showing that the output is piecewise constant and has a finite number of equal jumps. We then discuss hysteresis loops of the system for generic non-monotone inputs. The results are presented in the context of differential equations describing nonlinear control systems with almost imme- diate linear feedback, i.e., in the limit where the time of propagation of the signal through the feedback loop tends to zero.

1. Introduction. 1.1. Motivation. For the last two decades, the interest to emerging hysteresis behaviour in systems with simple functional elements is growing rapidly. The ﬁrst example we refer to is the recently discovered scaling laws universal to diverse systems which exhibit avalanches and crackling (Birkhausen) noise; examples include ferromagnetic materials, sand piles, earthquake fault systems, partially saturated porous media, phase transitions in solids, and social systems [15,16,20,38,43]. The roots of this universality have been revealed in terms of the Ising model of ferromagnetic hysteresis, see the review [39] and references therein. A general character- isation of emergent memory eﬀects in systems of interacting spins with “positive” interactions, such as the Ising model, has been obtained in [37]. This is the so-called return point rate-independent memory.

2010 Mathematics Subject Classification. Primary: 93C10, 93C15; Secondary: 34H05. Key words and phrases. Transducer, feedback, differential system, delay, Cantor function, hysteresis loop, non-ideal relay, Preisach operator. This publication has emanated from research conducted with the financial support of Russian Foundation for Basic Research, grant 10-01-93112.

1095 1096 ALEXEI POKROVSKII AND DMITRII RACHINSKII

As the second example of emergent strongly nonlinear behaviour of spin interaction systems, we mention the article [4] where Devil’s staircases are rigorously proved to be the description of the relationship between the external magnetic field and the mean-field magnetic induction of antiferrimagnetic materials. Devil’s staircases are in the heart of fractal type structures as they have the property of self-similarity, see comprehensive discussion in [18,27]. Another characteristic feature of a Devil’s staircase is that its derivative is equal to zero almost everywhere. We will focus on the classical Cantor function as a prominent example of the Devil’s staircase. The model of antiferromagnetic material used in [4] is of rather general nature. Loosely speaking, this is a network where each vertex can be in one of the two states, either ‘on’ or ‘off’. The greater is the external field, the more pressure is exercised upon all vertices to be ‘on’. On the other hand, each one of the vertices is in conflict with its neighbours tending to have a different state than them. The Devil’s staircase in the Aubry theorem [4] describes the limit case when the number N of vertices is growing, while the impact of each individual vertex becomes infinitesimally small. This type of networks is relevant to important phenomena such as the so-called minority games [3,10,11,29] in different subject areas including economics, biology, psychology and social sciences. Informally speaking, in this paper we are interested in the question what happens when local “negative” coupling of network elements is combined with the “positive” coupling at the level of the mean-field interaction. That is, in the first approximation, we are interested in dynamics of the systems with the block-diagram shown in Fig.1, where f is the Devil’s staircase.

u u+kx x f

Figure 1. System with positive feedback (k > 0); u = u(t) is input and x = x(t) is output.

This combination of “negative” and “positive” interactions is characteristic of various systems across science and technology. We mention modern computer memory technologies that include both ferro- and antiferromagnetic components (per- sonal communication by Prof. Gary Friedman) and models of economics combining ideas from [13] and [12]. Conditions ensuring monotonicity of networks involving both positive and negative feedbacks have been obtained in [41] and applied in the context of modelling control and signalling biochemical networks with spin interaction models and diﬀerential equations. As a prototype example, consider the following simple model. Let the input u = u(t) be dynamics of temperature in a certain town (measured, say, daily at noon) and x = x(t) be overall amount of ice cream sold during the lunch time by street retailers. We suppose that retailers can occupy various positions. If a particular position P is occupied, then the neighbouring positions are less likely to be occupied, to avoid competition. Thus, the distribution of the occupied positions is of “antiferromgnetic” type, and following [4] we expect that for given prices DEVIL’S STAIRCASE AND POSITIVE FEEDBACK 1097

Į 0 ȕ u

Figure 2. Nonideal relay.

the relationship between u and x is of the Devil’s staircase type (for a very large number of retailers). On the other hand, the larger is the overall number of retailers, the lower are prices, thus adding a “mean-field” positive feedback into our model; therefore, the relationship between u and x should be qualitatively similar to that depicted in Fig.1. The main observation of this paper is that the principal result of introducing a positive feedback in a system with the Devil’s staircase input-output relationship is a specific quantization of this relationship. To describe what kind of quantization it is, we need one more notion. The nonideal relay (with thresholds α, β, α < β) is the simplest hysteretic transducer; this transducer plays an important role below, and we include here an informal definition. The transducer’s output x(t), t ≥ t0, can take one of the two values, either 0 or 1, at any moment; the relay is said to be switched off or switched on, respectively. The dynamics of the output are usually illustrated by Fig.2. The variable output

x(t) = (Rα,β[t0, r0]u)(t), t ≥ t0, (1) depends on the variable continuous input u(t), t ≥ t0, and on the initial state r0 ∈ {0, 1}. The output behaves ‘lazily’–it prefers to be unchanged as long as the phase pair (u(t), x(t)) belongs to either of the two bold lines in Fig.2. The relay switches on when the input reaches the threshold value β and switches of when the input reaches the threshold value α. From the physical point of view, the deﬁnition of the output is veriﬁable only for those inputs that do not have local minima equal to α and local maxima equal to β. In the context of this paper, a useful remark aside is that the nonideal relay itself may be interpreted as the manifestation of emerging hysteretic behaviour in the block-diagram shown on Fig.1 when f is an ideal switch with the input-output relationship

0 if u(t) ≤ β, x(t) = (W u)(t) := (2) 1 if u(t) > β, and k = β − α (see further discussions in the next subsection). 1098 ALEXEI POKROVSKII AND DMITRII RACHINSKII

Below, we show that the input-output relationships for the block-diagram shown in Fig.1 with the Cantor function f are given by the formula M 1 X x(t) = (R [t , r0 ]u)(t). (3) M αm,βm 0 m m=1 The number M of relays increases rapidly and each individual relay becomes “nar- rower” as k → 0, so that the accumulated distributions of the thresholds αm and of the thresholds βm both approach the Cantor Middle Third set. An explicit characterization of the thresholds αm, βm is presented in Section2. Note that the meaning of the input-output relationships for the block-diagram shown in Fig.1 is not straightforward; it is discussed in the next subsection. The sum of relays (3) is a speciﬁc case of the discrete Preisach hysteresis transducer [9], which is used for modelling magnetic materials, plasticity, capillary hysteresis, design of sensors and actuators and other applications [1,2,5,7,8,14,17,21, 23, 25, 26, 30–32, 35, 36, 42]. According to the Mayergoyz theorem [28], the Preisach model is a rate-independent transducer with return point memory, which is charac- terized by one additional property, the congruency of hysteresis loops corresponding to any ﬁxed simple periodic input and varying initial state. 1.2. Mathematical models of systems with functional elements and feedbacks. The method of block-diagrams, which is common in engineering and control theory, provides a convenient toolbox to describe dynamical systems. In particular, systems with proportional feedbacks are of special interest [40]. For example, con-

y x u y=u+kx x W W

x (a) (b)

u y=u+kz x W

L z İ x (c)

Figure 3. (a) Graphical representation of an operator W . (b) System with proportional feedback. (c) System with delay in an information transmission channel. sider a transducer W (an operator, in the mathematical language) that transforms an input y(t), t ≥ t0, into an output x(t), t ≥ t0, see Fig.3(a). Figure3(b) depicts the block-diagram of the feedback system which includes the transducer W and the proportional feedback with a coefficient k > 0. An informal explanation of the system in Fig.3(b) is that for a given input signal u(t) one measures the corresponding output x(t) and adds it to the current value of the input after multiplying by the coefficient k. We will use the notation W ~ k for the corresponding input-output DEVIL’S STAIRCASE AND POSITIVE FEEDBACK 1099 relationship. In the first approximation, dynamics of this closed loop system with the input u and the output x is described by the simultaneous equations x(t) = (W y)(t), (4) y(t) = u(t) + kx(t). (5) This system has a formal solution −1 x(t) = W −1 − kI u(t) (6) where I is the identity operator. However, formula (6) does not make sense unless both the operators W and W −1−kI are invertible. Indeed, as an example, if W = I, i.e. the output x(t) of the transducer W coincides with its input, and k = 1, system (4), (5) reduces to the equality u(t) = 0. That is, the loop ‘forces’ u to be identically zero. This contradicts, of course, the informal interpretation of u(t) as an input. To some extent, this loop is in the heart of the so-called Shchipanov ideal regulator [6]. Moreover, even if the operator in the right hand side of (6) is well defined, we are not always home and dry. For instance, if (W y)(t) = ay(t) with some constant a satisfying ak > 1, the equality (6) is well defined and leads to the linear input-output relationship au(t) x(t) = . 1 − ak However, this formula implies that the output increases when the input decreases, which is opposite to the effect of a positive feedback on real systems, thus indicating that equations (4), (5) do not provide an adequate model in this case. As the next example, more relevant to the main subject of this paper, consider the ideal switch operator (2). Here system (4), (5) has the unique solution x(t) = 0 for u(t) ≤ β−k and the unique solution x = 1 for u(t) > β. But for β−k < u(t) ≤ β the system has two solutions x(t) = 0, 1 and hence the output is not uniquely defined. Question arises, which one of those two solutions is relevant at a particular time moment. One possible way to deal with the aforementioned failures in modelling the effect of feedback on a transducer W is to take into account small delays in the channels of information transmission. The block-diagram in Fig.3(c) illustrates a model where a delay between measuring the output x(t) and feeding it into the the magnifier k is introduced. In the simplest interpretation, this approach leads to replacing system (4), (5) either with the system x(t) = (W y)(t), (7) y(t) = u(t) + kz(t), (8) z(t) = x(t − ε) (9) or with the system x(t) = (W y)(t), (10) y(t) = u(t) + kz(t), (11) dz ε = x(t) − z(t) (12) dt depending on whether we use the pure delay or the inductance type delay; the delayed signal is denoted by z(t) = (Lεx)(t). If system (7)-(9) (or system (10)- (12), or both of them) has, for a positive ε, a unique solution (xε(t), yε(t), zε(t)) for any given input u(t) from a reasonable class of inputs, then the output x(t) = 1100 ALEXEI POKROVSKII AND DMITRII RACHINSKII

((W ~ k)u)(t) of the block-diagram shown in Fig.3(b) can be interpreted as the limit of xε(t) as ε → 0, i.e.,

((W k)u)(t) = lim xε(t). (13) ~ ε→0 Note that the limit operator W ~ k may depend on additional parameters such as initial data of system (7)-(9) (or (10)-(12), respectively). For example, if W is the ideal switch operator (2), then the input-output relationship x(t) = ((W ~ k)u)(t) obtained in this limit is the nonideal relay transducer (1) shown in Fig.2. A possible rigorous formulation is as follows. Proposition 1. Let W be the ideal switch (2). Suppose that a continuous piecewise monotone input u(t), t ≥ t0, does not have local maxima equal to β and local minima equal to α = β − k for k > 0. Let (xε(t), yε(t), zε(t)) be a solution of system (7)-(9) with the initial data x(t) ≡ x0 for t0 − ε ≤ t < t0 where x0 is either 0 or 1; and let (˜xε(t), y˜ε(t), z˜ε(t)) be a solution of system (10)-(12) with any initial data satisfying u(t0) + kz(t0) 6= β. Then for each t > t0 such that u(t) 6= α and u(t) 6= β,

lim xε(t) = (Rα,β[t0, r0]u)(t), lim x˜ε(t) = (Rα,β[t0, r˜0]u)(t) ε→0 ε→0 where r0 =r ˜0 = 0 if u(t0) < α, r0 =r ˜0 = 1 if u(t0) > β, and r0 = x0, r˜0 = W (u(t0) + kz(t0)) if α < u(t0) < β. For more general classes of limit operators, see [24, 34]. 1.3. Superposition operator with positive feedback. The effects similar to those described in the previous subsection are observed if W is the superposition operator (W y)(t) = f(y(t)) generated by a continuous monotone function f which has intervals of rapid growth. In this case, introduction of the proportional feedback y = u + kx with a coefficient k > 0 leads to the input-output relation u(t) 7→ x(t) formally described by x = f(u + kx). (14) This relation has a simple graphical interpretation, see Fig.4. Indeed, a solution x = x0 of equation (14) for a given u = u0 is the ordinate of an intersection point 1 of the graph of f with the straight line x = k (u − u0). Consequently, if f has an infinite derivative at a point u, f 0(u) = ∞, then equation (14) has a non-unique solution at the point u0 = u − kf(u) for any k > 0. For instance, if the graph of f has the shape shown in Fig.5(a), then equation (14) has three solutions in some interval u− < u < u+. For such a function f, the limit operator (13) generated by system (10)-(12) with (W y)(t) = f(y(t)) is similar to the nonideal relay transducer. In particular, for any input u = u(t) which increases from a value α < u− to a value β > u+, the phase pair (u, x) = (u, (W ~ k)u) of the limit input-output relationship follows the graph of the least solution g−(u) = min{x : x = f(u + kx)} (15) of equation (14) (the lower curve in Fig.5(b)). While for an input u decreasing from the value β to the value α the phase pair (u, x) of the limit transducer follows the graph of the largest solution g+(u) = max{x : x = f(u + kx)} (16) of equation (14) (the upper curve in Fig.5(b)). For a periodic input u = u(t) sweeping back and forth between values α < u− and β > u+, the phase pair makes the hysteresis loop with two vertical jumps on a period at the points of discontinuity DEVIL’S STAIRCASE AND POSITIVE FEEDBACK 1101 of functions (15) and (16). This loop can be used to describe the limit operator (13) for system (10)-(12) for arbitrary continuous inputs in the same way as the rectangular loop in Fig.2 defines the nonideal relay operator. The loop in Fig.5(b) is a canonical example of the limit of relaxation oscillations in slow-fast systems, see [24, 34] and references therein.

x=f(u) x g(v)

x=(u-v)/k

v u u

Figure 4. Graphical interpretation of the implicit function x = g(u) deﬁned by equation (14). For a given u = v, the value x = g(v) of this implicit function is the ordinate of the intersection point (if unique) of the straight line x = k−1(u−v) with the graph x = f(u) of the function f on the (u, x)-plane. In other words, (v, g(v)) are the coordinates of the intersection point in the skew coordinate system (˜u, x˜) = (u + kx, x).

Fig.4 implies that if equation (14) with an arbitrary continuous function f has a solution for u = u1 and for u = u2 > u1, then functions (15) and (16) are well defined on the whole segment u1 ≤ u ≤ u2. This functions are generally different and discontinuous if there are points with f 0(u) = ∞ where equation (14) has multiple solutions, as illustrated by Fig.5. Furthermore, the latter figure suggests that for systems with complex continuous functions f, such as Devil’s staircases, an introduction of a small positive feedback to the input-output relationship x = f(u) generates functions (15) and (16) with many jumps. Taking the Cantor function as a model of f, in Subsection 2.1 we address the questions (a) whether the number of jumps of functions (15), (16) is finite; (b) whether these functions have a nontrivial continuous (singular or regular) compo- nent; and (c) what is the number, size and distribution of their jumps. Functions (15), (16) determine the response of delayed differential systems (7)-(9) and (10)- (12) with the functional element (W y)(t) = f(y(t)) to monotone inputs in the limit of small ε, defining the limit operator W ~ k for this class of inputs. In Subsection 2.2 we describe the hysteresis loop composed of the graphs of functions (15), (16) in terms of the sum (3) of nonidelal relays. This is the so-called outer hysteresis 1102 ALEXEI POKROVSKII AND DMITRII RACHINSKII loop of the limit operator W ~ k corresponding to the inputs which sweep back and forth between a sufficiently large and a sufficiently small value. We conclude Section 2 by discussing the representation (3) of the limit operator W ~ k for differential equations with arbitrary piecewise monotone continuous inputs which start in the region where functions (15) and (16) coincide (that is, u(t0) is either sufficiently large or sufficiently small). Section3 contains the proofs.

x x=(u-v)/k x=f(u) x

x + g (v) x

g - (v)

v u

u u

(a) (b)

Figure 5. (a) Multiple solutions of equation (14) at a point u = v; g−(v) is the least solution, g+(v) is the largest solution. (b) The least solution (15) and the largest solution (16) of equation (14) with the function f from the left panel. The graphs of the least and the largest solutions form a hysteresis loop with two vertical jumps; the curves coincide to the left and right of the jumps.

2. Main result.

2.1. Quantization of input-output operator for monotone inputs. In order to study system (14) with the Cantor function f and the corresponding response functions (15), (16), we first introduce some notation related to the definition of the Cantor Middle Third set and the Cantor function, which will help us to formu- late and prove the results, and which is equivalent to the definition using ternary notation. Consider the sequence of increasing sets, Uj ⊂ Uj+1, defined by

X −n` Uj = {u = 2 3 : 0 < n1 < ··· < nL ≤ j} (17) ` with integer n`, j. That is, u ∈ Uj if there is a positive integer L and positive PL −n` integers n`, ` = 1,...,L such that n1 < ··· < nL ≤ j and u = 2 `=1 3 . Denote S j by U the union U = j Uj of the sets Uj. The set Uj consists of 2 − 1 points DEVIL’S STAIRCASE AND POSITIVE FEEDBACK 1103

j ui ∈ (0, 1); the set U is inﬁnite. For every point L X −n` u = 2 3 ∈ U, 0 < n1 < ··· < nL, `=1 deﬁne also the point v(u) = u − 3−nL . If j j j 0 < u1 < u2 ··· < u2j −1 < 1 (18) j is the partition of the segment 0 ≤ u ≤ 1 by the points ui ∈ Uj from the set (17), then j j j j j j 0 < v(u1) < u1 < v(u2) < u2 ··· < v(u2j −1) < u2j −1 < 1, j hence the intervals Iu = [v(u), u] with u = ui ∈ U do not intersect. Denote 0 Iu = (v(u), u). In this notation, [ 0 C = [0, 1] \ Iu u∈U 0 is the Cantor Middle Third set and the union of the disjoint open intervals Iu is the j j complement to C. The points ui and v(ui ) are the the right and left ends of these open intervals, respectively. Let f be the Cantor function, i.e., f is a unique continuous monotone extension of the function X X f(w) = 2−n` for v(u) ≤ w ≤ u with u = 2 3−n` ∈ U (19) ` ` S from the set u∈U Iu to the segment 0 ≤ u ≤ 1. For convenience, we further extend the function f to the whole real axis by setting f(0) = 0 for u ≤ 0; f(u) = 1 for u ≥ 1, (20) see Fig.6. We will mean this continuous monotone extension when referring to the Cantor function below.

Figure 6. The graph of the Cantor function x = f(u) extended to the whole real axis by relations (19). 1104 ALEXEI POKROVSKII AND DMITRII RACHINSKII

Consider system (14), with positive feedback, with the input-output relationship x = g−(u) defined by (15). Since f increases and satisfies (20), relation (15) implies that the function g− also increases and g−(u) = 0 for u ≤ 0; g−(u) = 1 for u ≥ 1. (21) Theorem 1. For any k > 0 the range of the function g− is a finite subset of m the set of all the fractions 0 ≤ 2j ≤ 1 with integer m, j. More precisely, for 2 j+1 2 j 2 3 ≤ k < 2 3 with j ≥ 1, the increasing piecewise constant left-continuous function g− is defined by

 0, u ≤ u0 = 0,   −  m k(m−1) km j g (u) = 2j , um−1 − 2j < u ≤ um − 2j , m = 1,..., 2 − 1, (22)    k(2j −1) 1, u > u2j −1 − 2j , where 0 < u1 < ··· < u2j −1 < 1 is the partition (18) of the segment 0 ≤ u ≤ 1 by the points un ∈ Uj from the set (17). The proof of this theorem is based on the self-similarity of the graph of the Cantor function, see Section 3. If k ≥ 4/3, then the straight line x = k−1u passes through, or below, the right end of the middle step {(u, x) : 1/3 ≤ u ≤ 2/3, x = 1/2} of the graph of the Cantor function (see, Fig. 6). Hence, the ordinate of the lowest point of intersection between the graph of the Cantor function and the straight line x = k−1(u − v) equals x = 1 for positive values of the parameter v, and x = 0 for v ≤ 0. According to Fig. 4, this ordinate equals g−(v). Therefore, if k ≥ 4/3, then g−(u) = 0 for u ≤ 0 and g−(u) = 1 for u > 0. 2.2. Hysteresis loop. Positive feedback is a well-known mechanism creating hysteresis in the input-output relationship of transducers with time-dependent input u = u(t) and output x = x(t), see [40]. As mentioned in the Introduction, the simplest example is a non-ideal relay. Indeed, if we denote 0 if u ≤ β, s (u) = β 1 if u > β, then (14) becomes the equation

x = sβ(u + kx) (23) which has two solutions g−(u) = 0 and g+(u) = 1 in the interval β − k < u ≤ β, one solution g−(u) = g+(u) = 0 for u ≤ β − k, and one solution g−(u) = g+(u) = 1 for u > β, see Fig.2. In this notation, the superposition operator u(t) 7→ x(t) = sβ(u(t)) generated by the function sβ is the ideal switch (2). Hence, according to Proposition 1, the limit input-output operator (13) of the feedback loop system shown on Fig.3(c) with W = sβ(·) is the non-ideal relay transducer x(t) = (Rα,β[t0, r0]u)(t) with α = β − k, which exhibits a rectangular hysteresis loop. For instance, if u = u(t) increases on the time interval t0 ≤ t ≤ t1 from a 1 value u(t0) = a < α to a value u(t ) = b > β and then decreases on the time interval t1 ≤ t ≤ t2 back from the value b to the value a = u(t2), then − g (u(t)), t0 ≤ t ≤ t1, x(t) = + (24) g (u(t)), t1 < t ≤ t2, DEVIL’S STAIRCASE AND POSITIVE FEEDBACK 1105 and the point (u(t), x(t)) makes a counterclockwise hysteresis loop consisting of a rectangle and two horizontal segments on the input-output diagram, see Fig.7(b); the area of this loop is β − α = k. More generally, the output of the non-ideal relay for any continuous input u(t), t ≥ t0, and an initial state r0 which is either 0 or 1 (this state is determined by the initial data of system (7)-(9), or system (10)-(12)) is deﬁned explicitly by the formula 0, if u(t) ≤ α;   1, if u(t) ≥ β;  r , if u(τ) ∈ (α, β) for all τ ∈ [t , t];  0 0 x(t) = (Rα,β[t0, r0]u)(t) := 0, if there exists t1 ∈ [t0, t) such that u(t1) = α   and u(τ) ∈ (α, β) for τ ∈ (t1, t];  1, if there exists t ∈ [t , t) such that u(t ) = β  1 0 1  and u(τ) ∈ (α, β) for τ ∈ (t1, t]; see, for example, [22]. Here the initial pair (u(t0), r0) is assumed to belong to the join of the two horizontal lines in Fig.7(b), i.e., r0 = 0 if u(t0) ≤ α, r0 = 1 if u(t0) ≥ β, and r0 ∈ {0, 1} for α < u(t0) < β.

(a) x (b) x

x=(u-v)/k

v Į ȕ u a Į ȕ b u

Figure 7. (a) Intersections of the graph of the function x = sβ(u) with the straight lines x = (u−v)/k deﬁne two branches of solutions of equation (23): x = 0 for u ≤ β and x = 1 for u > β − k. (b) The lower step-function solution x = g−(u) and the upper step-function solution x = g+(u) of equation (23). For a periodic input u = u(t) which increases from a value a < α = β − k to a value b > β and then decreases from b to a, the point (u(t), x(t)) with the output (24) makes a square hysteresis loop. It ﬁrst follows the line x = 0 from left to right till it jumps upwards at the point u = β and continues along the line x = 1 as long as u increases from the value a to the value b. Then, while u decreases, the point (u(t), x(t)) moves along the line x = 1 from right to left, jumps downwards at u = α and continues moving left along the line x = 0.

In a similar manner, equation (24) with function f shown in Fig.5(a) creates the hysteresis loop (24) shown in Fig.5(b) for inputs u = u(t) oscillating between two proper values a and b. We now consider hysteresis loops (24) of system (14) with the Cantor function f. 1106 ALEXEI POKROVSKII AND DMITRII RACHINSKII

A slight modiﬁcation of the argument presented in the proof of Theorem1 below leads to the following formula for the function (16). Theorem 2. For the Cantor function f, g+ is the increasing piecewise constant right-continuous function deﬁned by  0, u < v − k ,  1 2j   g+(u) = m km k(m+1) j (25) 2j , vm − 2j ≤ u < vm+1 − 2j , m = 1,..., 2 − 1,    1, u ≥ v2j − k = 1 − k

2 j+1 2 j for 2 3 ≤ k < 2 3 , where 0 < v1 < ··· < v2j −1 < 1 is the partition of the segment 0 ≤ u ≤ 1 by the points v` = v(u`) with u` ∈ Uj. Combining Theorems 1 and 2 results in the next theorem. Theorem 3. Suppose that a continuous input u = u(t) of system (14) with the Cantor function f increases on the time interval t0 ≤ t ≤ t1 from a value u(t0) = a ≤ −k to a value u(t1) = b ≥ 1 and then decreases on the time interval t1 ≤ t ≤ t2 2 j+1 2 j back from the value b to the value a = u(t2). Then, for 2 3 ≤ k < 2 3 , the output (24) of system (14) coincides with the output of the Preisach transducer (3) with the parameters M = 2j, km k(m − 1) α = v − , β = u − , m = 1,..., 2j, (26) m m 2j m m−1 2j 0 0 and r1 = ··· = rM = 0. The statement of Theorem3 follows from explicit formulas (22), (25). Due to the −j relation um−1 −vm = 2 k, which follows from the deﬁnition of um, vm = v(um) for j all 1 ≤ m ≤ 2 , the thresholds (26) of the relays Rαm,βm composing the Preisach transducer (3) satisfy

α1 < β1 < ··· < α2j < β2j (27) and k 1 1 β − α = ··· = β j − α j = − ≥ > 0. (28) 1 1 2 2 2j+1 2 · 3j 2 · 3j+1 Fig. 8(a) presents the counterclockwise hysteresis loop of system (14) and the equivalent Preisach transducer (3) for the input considered in Theorem3. According to relations (27), (28), the loop consists of the 2j congruent rectangles of height 2−j −j −j and width βm − αm = 2 k − 3 and horizontal segments connecting them. Since 2 j+1 2 j 2 3 ≤ k < 2 3 , the area σk of the loop satisfies −j−1 −j −j −j 3 ≤ σk = 2 k − 3 < 3 . We see that the area of the hysteresis loop tends to zero as k → 0, i.e. when the log2 3 feedback decreases. More precisely, σk k log2 3−1 . The distribution of segments [βm−1, αm] tends to the Cantor Middle Third set in this limit. The hysteresis loop shown in Fig. 8(a) is the so-called outer loop of the Preisach operator, connecting the output saturation values 0 and 1 which are achieved for sufficiently small and sufficiently large values of the input, respectively, as in The- orem3. Fig. 8(b) presents one of the minor loops of this operator for a different input. DEVIL’S STAIRCASE AND POSITIVE FEEDBACK 1107

(a) 1

a 0 b u

(b) 1

0 c b u

Figure 8. (a) The outer hysteresis loop of system (14) with the Cantor function f and the equivalent Preisach transducer (3). When the input u = u(t) increases from the value a to the value b, the point (u(t), x(t)) follows the arrows pointing right and upwards (the lower staircase path); when u decreases from b to a, the point (u, x) follows the arrows pointing left and downwards (the upper staircase path). Each path consists of horizontal steps, the point (u, x) makes a vertical jump at the end of a step. (b) A minor loop of the transducer (3) corresponding to an input u which varies between the values c and b. The initial values for this input are 0 0 0 0 u(t0) = c, r1 = r2 = 1, r3 = r4 = 0.

2.3. Input-output operator as a limit of delayed systems. Finally, we discuss, in more detail and rigour, the operator (3) introduced in Theorem 3 as the limit input-output relationship of delayed feedback systems consisting of the operator (W u)(t) = f(u(t)), where f is the Cantor function, and a linear positive feedback with vanishing delay. Let us first introduce a class of models of information transmission channels ex- emplified by systems (7)-(9) and (10)-(12). The effect of the information transmission channel is intrinsically distributed along the channel, but using the method of lumping [19], p. 6, we represent it as a finite dimensional system L = Lε in the feedback path of the block-diagram shown in Fig.3(c). Concentration and separation of resistance, inductance and capacitance features of the information transmission 1108 ALEXEI POKROVSKII AND DMITRII RACHINSKII channel (which is the essence of the lumping method) introduce some errors in the system description. However, the lumping method is shown to be quite accurate in typical cases. Here we consider the case when the element L = Lε is modelled by a general finite dimensional linear system dξ ε = Aξ + bx(t − ε ), z(t) = hc, ξ(t)i, (29) 1 dt 2 involving a delayed term, where ε1, ε2 are small positive parameters which charac- d terise the overall quality of the channel; ε = (ε1, ε2); A is a d × d-matrix; b, c ∈ R ; and h·, ·i denotes the Euclidean scalar product in Rd. The vector variable ξ ∈ Rd represents the internal dynamics of the lumped inductance link, whereas the scalar variable z = hc, ξi is the observable output of the link. We suppose that A is a Hur- witz matrix, i.e, the eigenvalues of A lie in the open left half plane of the complex plane. Moreover, it is assumed (see, [33]) that

det A = −1, hc, A−1bi = −1. (30)

The first of these relations can always be satisfied by an appropriate scaling, since det A < 0 for any Hurwitz matrix A. The second relation means that the channel does not affect constant signals in the sense that for a constant input x(t) = x∗ = const the function z(t) = x∗ represents a possible output of the linear system L (the corresponding internal dynamics is actually given by the equilibrium ξ(t) = −1 −x∗A b). This second relation holds in most situations, for instance it holds for −1 the simplest inductance link with the transfer function (ε1p + 1) . It means that the element L works as a low pass filter, which is a typical feature of inductance elements. Now, the limit input-output operator (13) for system (7), (8), (29) can be speci- fied as follows.

Proposition 2. Suppose that a continuous piecewise monotone input u(t), t ≥ t0, does not have local minima equal to αm and local maxima equal to βm where αm, j βm are deﬁned by relations (26) for m = 1,...,M, M = 2 . For simplicity, suppose additionally that either u(t0) < α1 or u(t0) > βM . Let (xε(t), yε(t), zε(t), ξε(t)) be a solution of system (7), (8), (29) with any continuous initial data x = x0(t), t0 −ε2 ≤ d t ≤ t0 and ξ(t0) = ξ0 ∈ R satisfying the compatibility condition x(t0) = hc, ξ(t0)i. Then the output xε(t) of system (7), (8), (29) satisﬁes pointwise the limit relation

M 1 X x (t) → (R [t , r0 ]u)(t) as ε , ε → 0 ε M αm,βm 0 m 1 2 m=1 at every point t > t0 such that u(t) 6= αm and u(t) 6= βm, m = 1,...,M, where 0 0 rm = 0 if u(t0) < α1 and rm = 1 if u(t0) > βM for all m. This statement can be obtained from Theorem3 using the general argument presented in [34]; we omit further details here.

3. Proof of Theorem1. The proof is based on self-similarity of the Cantor function f, which can be expressed by the relations u f(u) f = for 0 ≤ u ≤ 1, (31) 3 2 DEVIL’S STAIRCASE AND POSITIVE FEEDBACK 1109

L ! L 2 X −n` X −n` f u + 2 3 = f(u) + 2 for 0 ≤ u ≤ , 0 < n1 < ··· < nL, 3nL `=1 `=1 (32) following from its deﬁnition. Suppose 2j+1 2j 2 ≤ k < 2 (33) 3 3 with j ≥ 1. Lemma 1. Relations (33) imply f(u) ≥ k−1u (34) for 0 ≤ u ≤ 3−j.

−j PL −n` Proof. Consider an arbitrary point u ∈ U, 0 < u < 3 , i.e., u = 2 `=1 3 with nL > ··· > n1 ≥ j + 1. The inequalities 2j+1 2n` k ≥ 2 ≥ 2 3 3 imply 2 2−n` ≥ . 3n` k Summing these relations over 1 ≤ ` ≤ L, we obtain L L X 2 X u 2−n` ≥ 3−n` = k k `=1 `=1

PL −n` where `=1 2 = f(u) according to the definition (19) of f. Hence, (34) holds for u ∈ U, 0 < u < 3−j. Moreover, by definition, f is constant on each segment −j Iu = [v(u), u], therefore relation (34) is valid on every segment Iu ⊂ (0, 3 ). As the union of these segments in a dense set in the interval 0 ≤ u ≤ 3−j, we conclude that (34) holds on this interval. We prove formula (22) by induction in m. Relations (21) show that this formula holds for u ≤ u0. To implement the induction step, we assume that formula (22) j j holds for u ≤ um − km/2 with some 0 ≤ m ≤ 2 − 1 and show that it is valid for j j u ≤ um+1 − k(m + 1)/2 . Here we use the notation u2j = 1 for m = 2 − 1. By definition, L L X −n` X −n` −j um = 2 3 with nL ≤ j; f(um) = 2 = 2 m. `=1 `=1 Hence, the property (32) of the function f implies

−j 2 2 f(w + um) = f(w) + 2 m for 0 ≤ w ≤ ≤ 3j 3−nL and from Lemma1 it follows that −j −j −j f(w + um) = f(w) + 2 m ≥ w/3 + 2 m for 0 ≤ w ≤ 3 . Equivalently, u − (u − 2−jmk) f(u) ≥ m for u ≤ u ≤ u + 3−j. (35) k m m 1110 ALEXEI POKROVSKII AND DMITRII RACHINSKII

The deﬁnition of g− can be rewritten as

g−(v) = min{f(u): f(u) = k−1(u − v)}, hence g−(v) is the inﬁmum of the closed set of ordinates of the intersection points of the straight line x = k−1(u − v) with the graph x = f(u) of the function f. By the − −j −j induction assumption, g (um − 2 mk) = 2 m. Hence, the graph x = f(u) does −1 −j −j not intersect the straight line x = k (u − um + 2 mk) for x < 2 m. Therefore

−1 −j −1 −j −j f(u) > k (u − um + 2 mk) for k (u − um + 2 mk) < 2 m. Combining this relation with (35), we obtain

u − (u − 2−jmk) f(u) ≥ m for u ≤ u + 3−j. k m Consequently, u − v f(u) > for u ≤ u + 3−j, v > u − 2−jmk. k m m Since f increases, this relation implies

− −j −j −j g (v) ≥ f(um + 3 ) = 2 (m + 1) for v > um − 2 mk, (36) where the equality is due to the relation

−j −j f(u) = 2 (m + 1) for um + 3 ≤ u ≤ um+1 (37) which follows from the deﬁnition of f. On the other hand, the straight line x = k−1(u − v) intersects the horizontal line x = 2−j(m + 1) at the pointu ˜ = v + 2−j(m + 1)k, hence (37) implies u˜ − v f(˜u) = k at this point whenever

−j −j um + 3 ≤ u˜ = v + 2 (m + 1)k ≤ um+1. Therefore (m + 1)k (m + 1)k g−(v) ≤ f(˜u) = 2−j(m + 1) for u + 3−j − ≤ v ≤ u − . m 2j m+1 2j This relation, inequality (36) and the estimate

−j −j −j um − 2 mk > um + 3 − 2 (m + 1)k, which follows from (33), imply that

− −j −j −j g (v) = 2 (m + 1) for um − 2 mk < v ≤ um+1 − 2 (m + 1)k. (38)

This formula completes the induction step and proves (22) for u ≤ u2j − k = 1 − k. Finally, because g− increases, g−(u) ≤ 1 for all u, and g−(1 − k) = 1 according to (38), we conclude that g−(u) = 1 for u > 1 − k and hence (22) holds on the whole axis. DEVIL’S STAIRCASE AND POSITIVE FEEDBACK 1111

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