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 num- ber 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 be- haviour in systems with simple functional elements is growing rapidly. The first example we refer to is the recently discovered scaling laws universal to diverse sys- tems 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 ferro- magnetic 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, hys- teresis 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 interac- tion 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 approxima- tion, 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 x 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 mem- ory 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 interac- tion 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 x 1 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 2 f0; 1g. 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 trans- ducer [9], which is used for modelling magnetic materials, plasticity, capillary hys- teresis, 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 feed- backs. 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.