Generating Multi-Scroll Chaotic Attractors via Threshold ... - CiteSeerX

Generating Multi-Scroll Chaotic Attractors via Threshold Control Jinhu L¨u

K. Murali

S. Sinha

Henry Leung

Key Laboratory of Systems Department of Physics Institute of Mathematical Department of Electrical and Control, Institute of Anna University Sciences, Taramani and Computer Engineering Systems Science, Academy of Chennai 600 025, India Chennai 600 113, India University of Calgary Mathematics and Systems Science Email: [email protected] Calgary, Canada T2N 1N4 Chinese Academy of Sciences Beijing 100080, China Email: [email protected]

Abstract— This paper proposes a novel threshold control approach for creating multi-scroll chaotic attractors. The general jerk circuit is used as an example to show the working principle of this method. The controlled jerk circuit can emerge various limit cycles and n−scroll chaotic attractors by adjusting the upper threshold, lower threshold, and the width of inner saturated plateau. The dynamical mechanism of the threshold control is then further explored by analyzing the system dynamical behaviors. In particular, this method is effective and simple to implement since we only need monitor a single state variable and reset it if it exceeds the thresholds. It indicates the potential engineering applications for various chaos-based information systems.

I. I NTRODUCTION Chaos control is guiding a chaotic system to reach a desired goal dynamics via various controllers. Over the past two decades, many different approaches or techniques have been proposed to achieve chaos control, such as OGY approach, linear feedback control, inverse optimal control, among many others [1-5]. It is well known that the theoretical basis of most methods is stabilizing unstable periodic orbits via parameter perturbation [3]. In this paper, we introduce a threshold control approach, which clips desired state variable instead of adjusting system parameters. There have been a large number of publications devoted to this research topic on circuits design for generating multiscroll chaotic attractors [6-15]. Suykens and Vandewalle [6] proposed a family of n−double scroll chaotic attractors. L¨u et al. [7-8] introduced a switching manifold approach for generating chaotic attractors with multiple-merged basins of attraction. Yalcin et al. [9] presented a family of scroll grid attractors by using a step function approach, including one-dimensional (1-D) n−scroll, two-dimensional (2-D) n × m−grid scroll, and three-dimensional (3-D) n × m × l−grid scroll chaotic attractors. L¨u et al. [10-11] proposed the hysteresis and saturated series method for generating 1-D n−scroll, 2-D n × m−grid scroll, and 3-D n × m × l−grid scroll chaotic attractors. Last but not least, L¨u and Chen [14] reviewed the main advances in theories, methods, and applications of multiscroll chaos generation.

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In the following, a novel threshold control method is introduced for generating n−scroll chaotic attractors based on a general jerk circuit. The controlled jerk circuit can create various limit cycles and n−scroll chaotic attractors by adjusting its thresholds. The dynamic mechanism of the threshold control is then further investigated by analyzing the system dynamical behaviors. The rest of this paper is organized as follows. In Section II, a novel threshold control method is presented for generating n−scroll chaotic attractors from a general jerk circuit. The emerging various bifurcation and limit cycles are then discussed in Section III. In Section IV, the dynamical mechanism of threshold control is further investigated. Conclusions are drawn in Section VI. II. G ENERATING MULTI - SCROLL CHAOTIC ATTRACTORS VIA THRESHOLD CONTROL

In the following, we use the general jerk circuit as an example to show the working principle of the threshold controller. The general jerk circuit is given by ... x + c x¨ + b x˙ + a x = G(x) , (1) where a, b, c are real parameters and G(x) is a nonlinear threshold function. Linz and Sprott [15] studied the dynamical behaviors of some simple jerk circuits. Since soft nonlinearity is easy to be implemented. Hereafter, assume that G(x) is a soft nonlinearity. In fact, system (1) can be implemented by using the op-amps and diodes. It should be pointed out that the threshold function G(x) has many different algebraic forms, such as step function, hysteresis function, saturated function, and even any functions by cutting the tails that exceed the given threshold values. For simplification, assume that G0 (x ; k1 , k2 ) is described by   k2 , if x > k2 x , if k1 ≤ x ≤ k2 (2) G0 (x ; k1 , k2 ) =  k1 , if x < k1 , where k2 > k1 and k1 , k2 are the lower and upper thresholds, respectively. It is noticed that k1 , k2 are the threshold control

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parameters. When the variable x exceeds the critical threshold k1 (or k2 ), that is, x < k1 (or x > k2 ), the controller is triggered and reset to k1 (or k2 ). For a general case, G1 (x ; k1 , k2 , k3 , k4 ) is given by  k4 − k3 , if x > k4      x − k3 if k3 ≤ x ≤ k4 0 if k2 < x < k3 G1 (x ; k1 , k2 , k3 , k4 ) =   x − k if k1 ≤ x ≤ k2  2   k1 − k2 if x < k1 , (3) where k1 < k2 < k3 < k4 and k1 , k2 , k3 , k4 are the lower, middle-left, middle-right, and upper thresholds, respectively. When the variable x exceeds the critical threshold k1 (or k2 , k3 , k4 ), that is, x < k1 (or k2 < x < k3 , k2 < x < k3 , x > k4 ), the controller is triggered and reset to k1 − k2 (or 0, 0, k4 − k3 ). It should be pointed out that threshold control is different from other chaos control methods since there is no parameter perturbation in the original system. It only need monitor a single state variable and clip the desired time series by resetting the initial conditions. The essence is slightly limiting the dynamic range of original system by snipping the state variable. Since a chaotic attractor exists many temporary patterns, such as unstable periodic orbit, we can clip them to the desired dynamical behaviors by selecting suitable threshold controller. III. B IFURCATION AND DYNAMICAL BEHAVIORS OF THE CONTROLLED SYSTEMS

Fig. 2. (a) Double scroll, where k1 = −1 and k2 = 1; (b) 3−scroll, where k1 = −2, k2 = −1, k3 = 1 and k4 = 2.

Similarly, our numerical simulations also show that system (1) with (3) can create various period-doubled limit cycles, and 1−, 2−, 3−scroll chaotic attractors. Fig. 2 (b) shows a three-scroll chaotic attractor. Fig. 3 shows the bifurcation diagram of x vs k2 with k1 = −10 and its maximal Lyapunov exponents (MLE); Fig. 4 displays the bifurcation diagram of x vs k1 with k2 = 1 and its MLE; Fig. 5 shows the bifurcation diagram of x vs (k3 − k2 ) with k1 = −2, k4 = 2 and its MLE. When k1 = −1, system (1) with threshold controller (2) has different dynamical behaviors: 1) period 2 cycle: 0.15 ≤ k2 < 0.22; 2) period 4 cycle: 0.09 < k2 < 0.15; 3) period 8 cycle: k2 ≈ 0.08; 4) period 3 cycle: 0.297 ≤ k2 < 0.31; 5) period 6 cycle: k2 ≈ 0.32; 6) chaos: 0.254 < k2 < 0.296; 7) chaos: 0.4 < k2 < 0.58; 8) period 2 cycle: 0.59 < k2 < 0.64. When k1 = −2 and k4 = 2, system (1) with threshold controller (3) has different dynamical behaviors: 1) period 1 cycle: 0.1 ≤ k2 − k3 ≤ 0.4; 2) period 2 cycle: 0.44 ≤ k2 − k3 ≤ 0.52; 3) period 4 cycle: 0.54 ≤ k2 − k3 ≤ 0.55; 4) 3−scroll chaotic attractor: 1.2 ≤ k2 − k3 ≤ 2.0. IV. DYNAMICAL MECHANISMS OF THRESHOLD CONTROLLERS G0 (x) AND G1 (x) A. System (1) with threshold controller (2) Jerk system (1) with threshold controller (2) is given by ... x + cx ¨ + b x˙ + a x = G0 (x) . (4)

Fig. 1. (a) Period 2 cycle, where k1 = −2 and k2 = 10; (b) period 4 cycle, where k1 = −1 and k2 = 10; (c) period 3 cycle, where k1 = −0.3 and k2 = 1; (d) single scroll, where k1 = −0.4 and k2 = 1.

Obviously, system (4) has three different subspaces: V1 = { X | x ≥ k2 }, V2 = { X | k1 ≤ x ≤ k2 }, V3 = { X | x ≤ k1 }, where X = (x, x, ˙ x ¨)T ∈ R3 . Under the coordinate transform (x, x, ˙ x ¨) → (−x, −x, ˙ −¨ x), system (4) has a natural symmetry for k1 = −k2 . Moreover,... system (4) is dissipative in each subspace for ∂∂ xx˙ + ∂∂ xx¨˙ + ∂∂ xx¨ = −c < 0 . If X ∈ V1 , V3 , the jerk system is described by ... ¨˜ + b x x˜ + c x ˜˙ + a x ˜ = 0. (5) where

Our numerical simulations show that system (1) with (2) can generate various period-doubled limit cycles, one-scroll and two-scroll attractors. Fig. 1 shows various limit cycles and single scroll chaotic attractor. Fig. 2 (a) shows a double-scroll chaotic attractor.

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(˜ x, x˜˙ , x¨˜)T = (x − (˜ x, x˜˙ , x¨˜)T = (x −

k2 ˙ x ¨)T a , x, k1 ˙ x ¨)T a , x,

if X ∈ V1 if X ∈ V3 .

The characteristic eigenvalues of jerk system (5) are   √ √ c q q λ1 = − + 3 − + ∆ + 3 − − ∆ , 3 2 2

(6)

Fig. 3. (a) Bifurcation diagram of parameter k2 with k1 = −10; (b) maximal Lyapunov exponents.

and λ2,3

=

≡

   √ √ q q 3 3 c 1 −3 − 2 −2 + ∆ + −2 − ∆    √ √ √ q q 3 3 3 ± 2 i −2 + ∆ − −2 − ∆

Similarly, if X ∈ V2 , the jerk system is described by ... x + cx ¨ + b x˙ + (a − 1) x = 0 , (7)

α ± β i,

ac3 b 2 c2 abc b3 a2 1 where ∆ = − − + + , p = b − c2 , 27 108 6 27 4 3 2 3 1 c − bc + a. and q = 27 3 Our simulations show that jerk system (4) can generate chaotic behaviors under the conditions of λ1 < 0, α > 0, β = 0. Moreover, the solution of jerk system (5) is given by x ˜(t) = A1 eλ1 t + eαt (A2 cos(βt) + A3 sin(βt)) ,

(8)

where A1 = A2 = A3 =

(α2 + β 2 ) x ˜(0) − 2α y˜(0) + z˜(0) , (λ1 − α)2 + β 2 2 (λ1 − 2α λ1 ) x ˜(0) + 2α y˜(0) − z˜(0) , (λ1 − α)2 + β 2 (λ1 α2 −λ1 β 2 −λ21 α)˜ x(0)−(β 2 −α2 +λ21 )˜ y (0)+(α−λ1 )˜ z (0) β [(λ1 − α)2 + β 2 ]

Fig. 4. (a) Bifurcation diagram of parameter k1 with k2 = 1; (b) maximal Lyapunov exponents.

.

and its solution is given by

¯ ¯ ¯ + A¯3 sin(βt) ¯ x(t) = A¯1 eλ1 t + eαt A¯2 cos(βt) ,

(9)

(10)

where A¯1 = A¯2 = A¯3 =

(α ¯ 2 + β¯2 ) x(0) − 2α ¯ y(0) + z(0) , ¯ 1 − α) (λ ¯ 2 + β¯2 ¯ 2 − 2α ¯ 1 ) x(0) + 2α (λ ¯ λ ¯ y(0) − z(0) 1 , ¯ 1 − α) (λ ¯ 2 + β¯2 ¯ ¯ 1 β¯2 −λ ¯ 2 α)x(0)−( ¯ 2 )y(0)+(α− ¯ 1 )z(0) (λ1 α ¯ 2 −λ β¯2 −α ¯ 2 +λ ¯ λ 1¯ 1 ¯ 1 − α) β¯ [(λ ¯ 2 + β¯2 ]

,

¯ λ ¯ 1 are similarly defined by (6) and in which parameters α, ¯ β, (7) with (a − 1) to replace a. Based on the theoretical analysis, the dynamical behaviors of jerk system (4) are completely determined by (8) and (10). In essence, the function of threshold controller G0 is switching the dynamics between (8) and (10). Especially, the two thresholds k1 , k2 control the displacement transformation of state variable x. Jerk system (4) can generate various limit cycles, single-scroll attractor, and double-scroll attractor via adjusting the thresholds k1 , k2 of G0 .

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analysis shows that the dynamical behaviors of jerk system (11) are completely determined by (8) and (10). In particular, the four thresholds k1 , k2 , k3 , k4 control the displacement transformation of state variable x. System (11) can create various limit cycles, single-scroll attractor, double-scroll attractor, and 3−scroll attractor via controlling the thresholds k1 , k2 , k3 , k4 of G1 . V. C ONCLUSIONS This paper have introduced a novel threshold control method for generating multi-scroll chaotic attractors based on a general jerk circuit. The controlled jerk circuit can emerge various limit cycles and multi-scroll chaotic attractors by adjusting the thresholds. It indicates the potential engineering applications for various chaos-based information systems. ACKNOWLEDGMENT This work was supported by National Natural Science Foundation of China under Grant No.60304017 and Grant No.20336040, the Scientific Research Startup Special Foundation on Excellent PhD Thesis and Presidential Award of Chinese Academy of Sciences. R EFERENCES

Fig. 5. (a) Bifurcation diagram for the width of inner saturated plateau (k3 − k2 ) with k1 = −2 and k4 = 2; (b) Maximal Lyapunov exponents.

B. System (1) with threshold controller (3) Jerk system (1) with threshold controller (3) is described by ... x + cx ¨ + b x˙ + a x = G1 (x) . (11) Clearly, system (11) has five different subspaces: V1 = { X | x ≥ k4 }, V2 = { X | k3 ≤ x ≤ k4 }, V3 = { X | k2 ≤ x ≤ k3 }, V4 = { X | k1 ≤ x ≤ k2 }, V5 = { X | x ≤ k1 }, where X = (x, x, ˙ x ¨)T ∈ R3 . System (11) has a natural symmetry for k1 = −k4 and k2 = −k3 under the coordinate transform (x, x, ˙ x¨) → (−x, −x, ˙ −¨ x). Moreover, system (4) is dissipative in each subspace for c > 0. When X ∈ V1 , V3 , V5 , the jerk system is given by 3 ¨ (4), where (˜ x, x ˜˙ , x ˜)T = (x − k4 −k ˙ x ¨)T for X ∈ V1 , a , x, T T ¨ (˜ x, x ˜˙ , x ˜) = (x, x, ˙ x ¨) for X ∈ V3 and (˜ x, x ˜˙ , x¨ ˜)T = (x − k1 −k2 T ˙ x ¨) for X ∈ V5 . a , x, When X ∈ V2 , V4 , the jerk system is given by ... ¨ x ˆ + cx ˆ + bx ˆ˙ + (a − 1) x ˆ = 0, (12) where (ˆ x, x ˜˙ , x¨ ˜)T = (x + ka3 , x, ˙ x ¨)T for X ∈ V2 and k ¨ ˙ x ¨)T for X ∈ V4 . (ˆ x, x ˜˙ , x ˜)T = (x + a2 , x, Obviously, the solution of (12) is also given by (10) since (12) has the same algebraic form with (9). Our theoretical

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