International Journal of Bifurcation and Chaos, Vol. 17, No. 10 (2007) 3455–3460 c World Scientific Publishing Company
TWO-SCROLL ATTRACTOR IN A DELAY DYNAMICAL SYSTEM ∗ , TATJANA PYRAGIENE ¯ ˇ ˇ ˙ ARUNAS TAMASEVI CIUS ˇ and MANTAS MESKAUSKAS Semiconductor Physics Institute, A. Goˇstauto 11, Vilnius LT-01108, Lithuania ∗
[email protected]
Received February 5, 2006; Revised March 22, 2006 Nonvanishing -shaped nonlinear function has been introduced in delay dynamical system instead of commonly used Mackey–Glass type function. Depending on time delay the system exhibits not only mono-scroll, but also more complex two-scroll hyperchaotic attractors. Delay system with the novel nonlinear function can be implemented as an analogue electronic oscillator. Keywords: Delay dynamical systems; chaos; two-scroll attractor.
1. Introduction Delayed-feedback dynamical systems given by delay differential equation (DDE) dx = −x(t) + F [x(t − τ )], (1) dt where x(t) is a dynamical variable, F (·) is a nonlinear function and τ is the time delay, exhibit very rich dynamics including periodic, chaotic and even hyperchaotic behavior characterized with multiple positive Lyapunov exponents. There are many practical examples in electronics, optics, laser physics, physiology, population biology, economics described by Eq. (1) or similar DDE. Specifically the Mackey–Glass (MG) system [Mackey & Glass, 1977], originally introduced as a mathematical model for hematological disorders, has become the most popular scalar DDE in nonlinear theory and its applications providing an example of an infinite-dimensional system. The nonlinear function in the MG system is the following: ax . (2) FMG (x) = 1 + xm
Here x ≡ x(t − τ ) when inserted in a DDE. The parameters are commonly set to a = 2.0 and m = 10. This type of nonlinearity has been exploited in a number of theoretical and experimental investigations for various purposes: (i) to estimate the Lyapunov exponents [Farmer, 1982] and correlation dimension [Grassberger & Procaccia, 1983] of an infinite-dimensional system; (ii) to develop an electronic circuit [Namaj¯ unas et al., 1995a, 1997; Kittel et al., 1998] imitating dynamical behavior of the MG system; (iii) to test the techniques for controlling [Namaj¯ unas et al., 1995b, 1997] and synchronizing [Tamaˇseviˇcius et al., 1997, 1998] high dimensional hyperchaotic systems; (iv) to demonstrate the possibility of secure communication [Pyragas, 1998]. The nonlinearity given by Eq. (2) with even values of power m is an odd-symmetry function, FMG (−x) = −FMG (x). It asymptotically vanishes at |x| 1 (practically at |x| > 2 because of high value of power m). Though FMG (x) lies in the first and the third quadrants of the F –x plane, the corresponding attractors of the MG system occupy the phase space with either positive or negative values of x only depending on the
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initial conditions. Thus only mono-scroll attractors appear in Eq. (1) with the FMG function. Other types of the nonlinear function F (·) have been also considered for Eq. (1). Among them is a piecewise constant rectangular-shaped function proposed by Haiden and Mackey and considered in [Losson et al., 1993; Schwarz & M¨ ogel, 1994; M¨ ogel et al., 1995], an odd-symmetry piecewise linear five-segment function [Lu & He, 1996; Lu et al., 1998; Thangavel et al., 1998], a piecewise linear tent-shaped function [Schwarz & M¨ ogel, 1994; M¨ ogel et al., 1995; Mykolaitis et al., 2002, 2003], a bell-shaped shifted Gaussian function [Voss, 2001], the third-order polynomial function [Voss, 2002]. We note, however, that the above-mentioned nonlinear functions in Eq. (1) all provide mono-scroll attractors. In the present paper a nonvanishing -shaped function is introduced in the DDE instead of the commonly used vanishing FMG function. Depending on the delay parameter τ the novel system exhibits not only mono-scroll, but also two-scroll chaotic attractors. We demonstrate that chaotic oscillations associated with two-scroll attractor exhibit better spectral characteristics. In addition, we suggest hardware implementation of the system.
2. Nonlinear Function We introduce the following piecewise linear threesegment function: 8 > > > :b(x − 1) + a,
x < −1, −1 ≤ x ≤ 1,
(a > 0, b < 0)
x > 1.
(3) It is -shaped function (Fig. 1). Like the FMG (x) the F3 (x) is an odd-symmetry function. However, in F3(x)
(1,a) a
b x
(−1, −a) Fig. 1. Odd-symmetry nonvanishing three-segment nonlinear function F3 (x).
contrast to FMG (x) it does not vanish at large x but has constant negative slopes b at |x| > 1 and lies in all the four quadrants of the F –x plane.
3. Mono- and Two-Scroll Attractors Equation (1) has been integrated with the novel nonlinear function F3 [x(t − τ )]. The results are presented in Fig. 2 with the projections of the phase trajectories. For small values of the delay parameter τ the system undergoes period-doubling bifurcations [Figs. 2(a) and 2(b)] resulting in chaotic attractors [Fig. 2(c)]. However the phase trajectories do not switch between the mirror attractors in the top right and bottom left corners with two different basins of attraction. Thus only mono-scroll attractors are formed similarly to the MG and other delayed-feedback systems. The phase trajectories stay on the mono-scroll isolated attractors up to τ = 2.2. In the τ interval between 2.2 and 4.0 two-scroll attractors do appear. Phase trajectories chaotically wander in all-plane visiting the previously two isolated mirror attractors, thus merging them in a new “two-in-one” attractor. This twoscroll attractor has a single basin of attraction, but spirals alternately around two unstable steady points x0 = ±(a−b)/(1−b). There are some narrow periodic windows near τ = 3.5 and between the τ values of 4.0 and 5.0. For larger delays, τ > 5 again two-scroll attractors are observed [Fig. 2(d)].
4. Lyapunov Exponents, Kolmogor Entropy and Power Spectra In order to characterize the attractors quantitatively the Lyapunov exponents (LE) and the Kolmogor entropy (KE), i.e. the sum of all positive LE: K = Σλ+ i
(4)
have been calculated from Eq. (1) and are shown in Fig. 3. One can see from Fig. 3, that for the given set of parameters (a = 2 and b = 6) simple chaotic oscillations appear at about τ = 1.23 (single positive LE), while at τ = 1.9 the second positive LE does emerge, indicating hyperchaotic behavior. In the τ interval between 4.0 and 5.0 there are several periodic windows with zero LE. Also there are weakly hyperchaotic domains with relatively low values of LE and KE, though formally characterized with two and three positive LE. Eventually at τ > 5 the number of the positive LE starts to grow rapidly and becomes 5 for τ > 7. Power spectra for
Two-Scroll Attractor in a Delay Dynamical System
(a)
(b)
(c)
(d)
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Fig. 2. Phase portraits from Eqs. (1) and (3) for different delays: (a) τ = 1.00, (b) τ = 1.15, (c) τ = 2.00, (d) τ = 8.00. Parameters of the function F3 (x) are fixed at a = 2.0 and b = −4.0. The plots presented as dashed lines at the bottom left corners in (a)–(c) are isolated mirror attractors obtained for negative initial conditions.
Fig. 3. Non-negative Lyapunov exponents λi and Kolmogorov entropy K versus time delay τ from Eqs. (1) and (3) with a = 2.0, b = −4.0.
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Fig. 4. Power spectra from Eq. (1) at τ = 8. Dotted line is for the MG system with FMG (x); a = 2.0, m = 10. Solid thin line is for the delay system with the -shaped function F3 (x); a = 2.0, b = −4.0. Solid thick line is for the delay system with the same function F3 (x); a = 2.0, b = −4.0, but with the addition of a constant term c = 0.5 to Eq. (1), similarly as introduced in [Thangavel et al., 1998]. Note, that the fundamental frequency f ∗ for the MG system is f ∗ ≈ 1/3τ ≈ 0.04, while for the -shaped function f ∗ ≈ 1/τ ≈ 0.12.
both the mono-scroll MG system and the two-scroll system with F3 (x) are presented in Fig. 4.
5. Circuit Implementation The system described by Eq. (1) with the nonlinear function F3 (x) can be implemented using analogue electronics devices. A possible solution is shown in Fig. 5. The nonlinear unit is the OA1 based stage with two diodes at the input. The OA2 with R5 and R6 compose an amplifying stage. The transfer function of the nonlinear unit with a specific set of parameters is presented in Fig. 6. The integrating unit is simply an RC circuit. Resistor R6 plays also the role of the relaxation term −x(t) in Eq. (1). The feedback loop comprises a delay line DEL and the buffers at the input and the output. Resistor R7 is a matching element. The dimensionless delay parameter τ in Eq. (1) is simply τ = Tdel /RC . Software simulation of the oscillator with a fixed value of Tdel = 3 ms and RC tuned from 4.5 ms to 0.45 ms was performed using the “Electronics Workbench” simulator. The results are shown in Fig. 7. PSpice results are in qualitative agreement with the phase portraits presented in Fig. 2. We note, that in contrast to the piecewise linear function (Fig. 1) the PSpice model uses smooth transfer function (Fig. 6). To equalize these two models, namely to equalize the effective slopes of the transfer functions we have increased in the PSpice model the parameter values by 25%: a = 2.0 → 2.5 and b = −4.0 → −5.0 (Figs. 6 and 7).
Fig. 5. Circuit implementation of the delay system. R1 = R2 = R3 = R5 = 1 kΩ, R4 = R = 3 kΩ, R6 = 4 kΩ, R7 = 200 Ω.
Fig. 6. Transfer function of the nonlinear unit and the amplifying stage. a = 2.5, b = −5.0. Parameters a and b are related to the resistor values as follows: “ i“ ” h “ ” ” R6 R4 R4 R6 2 + 1 , b = a + 1 − + 1 . a = R1R+R R5 R3 R3 R5 2
Two-Scroll Attractor in a Delay Dynamical System
(a)
(b)
(c)
(d)
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Fig. 7. PSpice simulation of the circuit in Fig. 5 for different values of dimensionless delay τ = Tdel /RC : (a) τ = 0.7, (b) τ = 1.0, (c) τ = 1.6, (d) τ = 6.7. a = 2.5, b = −5.0. The plots presented as dashed lines at the bottom left corners in (a)–(c) are isolated mirror attractors obtained for negative initial conditions.
6. Conclusions A delayed-feedback system with a nonvanishing -shaped nonlinear function in contrast to the MG system exhibits not only mono-scroll attractors, but also topologically more complex two-scroll hyperchaotic attractors. From an engineering point of view the two-scroll mode has an advantage since the fundamental frequency f ∗ of chaotic oscillations is three times higher in comparison with the mono-scroll MG type system (Fig. 4). The emergent peaks close to zero frequency also at f ∗ and its higher harmonics can be easily removed by adding an external constant force c (Fig. 4). The system with the -shaped nonlinear function can be implemented as an electronic oscillator (Figs. 5 and 6).
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