Technical Physics, Vol. 49, No. 7, 2004, pp. 805–809. Translated from Zhurnal TekhnicheskoÏ Fiziki, Vol. 74, No. 7, 2004, pp. 1–5. Original Russian Text Copyright © 2004 by Vladimirov, Shtraukh. THEORETICAL AND MATHEMATICAL PHYSICS Entropy Control in Discrete- and Continuous-Time Dynamic Systems S. N. Vladimirov* and A. A. Shtraukh Tomsk State University, Tomsk, 634050 Russia *e-mail: [email protected], [email protected] Received August 14, 2003; in final form, December 22, 2003 Abstract—The dynamics of a modified logistic mapping are considered for a system with the order parameter modulated by an external signal. It is shown that the Kolmogorov–Sinay entropy changes with changing mod- ulation depth, while the harmonic signals and white noise can be used as a modulating signal. The conditions for the excitation of regular and strange nonchaotic attractors in the phase space are established. © 2004 MAIK “Nauka/Interperiodica”. INTRODUCTION modulation of the order parameter, this mapping takes At present, random oscillations have been observed the form in a great variety of objects, starting with crude x = Φ()x , mod 1, mechanical systems and ending with highly organized n + 1 n biological systems. In the theoretical and applied stud- 2π (1) Φ()x = 1 – α 1 + msin ------ n + ϕ x . ies of determinate chaos, one can distinguish a number n 0 T 0 n of priorities such as the elaboration of new scenarios for α ≥ ≤ ≤ the transition from regular to chaotic motion, methods Here, 0 1 is the order parameter; 0 m 1 is the of generating dynamic chaos, the study of the interac- order-parameter modulation depth (control parameter); ϕ tions between chaotic systems and the possible types of T and 0 are, respectively, the period and initial phase their collective behavior, unconventional dynamics and of the external force; and n = 0, 1, 2, …, N is the discrete informational processes, and entropy control in contin- time. Mapping (1) models a nonautonomous dynamic uous- and discrete-time dynamic systems. The latter system subjected to the parametric action of an external area is caused by the needs for both the excitation and periodic force. In the autonomous case m = 0, the order α suppression of chaotic oscillations in the same dynamic parameter 0 determines the degree of randomness of system [1–6]. the motion {xn}, when the Kolmogorov– Sinay entropy Apart from the purely theoretical interest, the con- can be calculated exactly to give K = log α [10]. trol of the degree of motion ordering is also of great 20 applied importance. Dynamic systems with chaotic Piecewise linear mapping (1) represents two rays oscillations can be used in the design of systems for that emanate from the point (x = 0, Φ(x) = 1) at the radio camouflage and electronic countermeasures, angle θ and go to –∞. As the discrete time n varies at α θ θ π noise radiolocation systems, confidential communica- fixed 0 and m, the angle varies from min = – α θ π α tion systems, and systems for nonstandard action on 2 arctan[ 0(1 + m)] to max = – 2 arctan [ 0(1 – m)] biological objects [7–9]. However, in some cases, the with the period T, resulting in a periodic variation of the appearance and development of the determinate chaos Lyapunov characteristic index. regime are highly undesirable. λ Indeed, the local Lyapunov index n is calculated The purpose of this work is to examine one of the analytically according to the relation possible methods of controlling the degree of motion Φ() ordering, namely, through the modulation of the order λ = log d xn parameter by an external signal in the form of a har- n 2 ------------------ dxn monic oscillation or white noise. (2) 2π = log α 1 +,msin ------ n + ϕ 20 T 0 MATHEMATICAL MODEL AND ITS ANALYTIC INVESTIGATION whence it follows that it is a 2π-periodic function of the τ π ϕ As the object for investigation, we chose a discrete- discrete phase n = 2 n/T + 0. During the action of the λ time system representing a modified logistic mapping external force, n takes both positive and negative val- [10]. After the addition of a multiplier describing the ues. Consequently, a strong dependence on the initial 1063-7842/04/4907-0805 $26.00 © 2004 MAIK “Nauka/Interperiodica” 806 VLADIMIROV, SHTRAUKH τ conditions for different n values may either take place related to a certain flow system, and this applies, in full or be absent; i.e., the phase trajectories may either mix measure, to the mapping of interest. Model (1) can or not. identically be transformed to a set of two autonomous λ mappings Due to the periodicity of n, the Lyapunov charac- teristic index Λ defined as its mean along the time real- x = 1 – α ()1 + msin2πψ x , mod 1, ization of {xn}, n + 1 0 n n N 1 (6) ψ = ψ + --- , mod 1, Λ 1 λ n + 1 n = ---- ∑ n, (3) T N n = 1 which now describe the dynamics of a nonlinear con- may be calculated only on the interval [0–2π]. Passing τ tinuous-time system subjected to an external bihar- from the discrete to a continuous n phase, one gets monic excitation with periods T1 and T. 2π 2π 1 1 Indeed, the first of the equations in (6) is obtained Λ = ------ λτ()τd = log α + ------ log ()1 + msinτ dτ from the stroboscopic sections (with the period T ) of 2π ∫ 20 2π ∫ 2 1 0 0 (4) the motion of the original flow system, while the first α action by itself is excluded. One iteration in the discrete 0 2 = log ----- ()11+ – m . time n corresponds to the period T1 of the first harmonic 2 2 action. If Λ < 0, the motion in the attractor is, on average, Since the relation ordered and, conversely, it is chaotic if Λ > 0. For this ψ ψ 1 reason, by setting the right-hand side of expression (4) n = 0 + --- n, mod 1 equal to zero, one can determine the critical value mˆ T that separates the chaotic and nonchaotic regimes, holds for the phase of eternal action, the rotation num- ber Θ is defined as 2 mˆ = ----- α –.1 (5) α 0 ψ – ψ 1 0 Θ ==lim ------------------n 0 --- . (7) → ∞ π It is significant that neither the period of the external n 2 n T force nor its initial phase appears in this relation. If T is an irrational number, the external action is Because of this, the motion type in the system is deter- quasi-periodic; otherwise it is periodic. Thus, the α mined only by the parameters o and m. This implies results obtained for model (1) can be extended in full that only the force and energetic relations play the dom- measure to continuous-time systems of the indicated inant part. The external action does not affect the crude- type. ness property that is inherent in an autonomous system This completes the analytic study of models (1) and [10], and, hence, the main properties of model (1) do (6). In the subsequent sections, the results of numerical not change if the variations of its parameters are small. experiments are presented. According to the familiar theorem in [11], the Kol- mogorov–Sinay entropy (hereinafter entropy) for map- ping (1) is defined by the relation HARMONIC EXTERNAL ACTION ∞ In this section, we analyze the influence of the con- trol parameter m on the dynamics of models (1) and (6) K = ρ()Λx ()x dx, α ∫ with various values of the order parameter 0. Calcula- 0 tions were carried out by formula (4) and numerically where ρ(x) is the invariant probability measure of a using algorithm [13] with quite a long realization of the dynamic system. Since the dynamics of the mapping time series {xn}. Both methods gave identical results under consideration proceed in a restricted phase-space (Fig. 1a). For any value of the order parameter, there domain, the probability measure always exists for the exists the mˆ value which separates the chaotic and non- Λ α time series generated by this mapping [12]. In our case, chaotic oscillation types. Since the function (m, 0) Λ is independent of x, so that, with allowance for the decreases monotonically, one can assert that the control normalization condition, K ≡ Λ. In what follows, we parameter “smoothly” changes the entropy of the sys- will not distinguish between these two notions and tem and, hence, the degree of motion ordering in it. The identify the Lyapunov characteristic index of model (1) character of the motion is fundamentally different for with its entropy. rational and irrational, even if close, rotation numbers. So far, we dealt with the properties of a discrete- If the modulation period is rational, the time series is time system, and nothing was said about the way of periodic (Fig. 1b), otherwise the periodicity is absent constructing model (1). At the same time, it is well (Fig. 1c). At the same time, one can see that the motions known that many of the discrete-time systems can be share some common traits. At the instants the local TECHNICAL PHYSICS Vol. 49 No. 7 2004 ENTROPY CONTROL IN DISCRETE- AND CONTINUOUS-TIME DYNAMIC SYSTEMS 807 0.6 (a) 1.0 (b) 1.0 (c) α 0 = 1.5 0.3 1.2 0.5 0.5 1.05 0 n n Λ x x 0 0 Ð0.3 Ð0.5 Ð0.6 Ð0.5 0 0.2 0.4 0.6 0.8 1.0 0 1 2 3 0 1 2 3 mn/Tn/T Fig. 1. (a) The Lyapunov characteristic index Λ as a function of the controlling parameter m for different values of order parameters α o of system (1).