Emerging multi-double-scroll attractor from variable-boostable chaotic system excited by multi-level pulse Ning Wang, Bo-Cheng Bao, Quan Xu, Mo Chen, Ping Ye Wu School of Information Science and Engineering, Changzhou University, Changzhou 213164, Jiangsu, People’s Republic of China E-mail:
[email protected] Published in The Journal of Engineering; Received on 29th September 2017; Revised on 6th November 2017; Accepted on 14th November 2017
Abstract: This study reports a simple approach to emerging multi-double-scroll attractor from a variable-boostable chaotic system via multi-level pulse excitation, which is achieved by replacing a constant controller with a multi-level pulse function. To intuitively show the simple and feasible approach, a four-double-scroll chaotic attractor coined in such a system with four-level pulse excitation is taken as a paradigm.
1
Introduction
In the past few decades, generating multi-scroll chaotic attractors from non-linear dynamical system has always been a research topic of chaos-based theories and applications due to higher topology complexity of multi-scroll chaotic attractors than that with few scrolls [1–4]. Most of these implementing approaches can be divided into two cases: autonomous and non-autonomous. However, these works achieved via autonomous approach often contain complex circuit implementations. From the point view of simplifying circuit implementation and practical application, non-autonomous approach has remarkable advantage due to the employment of signal generator [3, 4]. The signal generator can be easily realised by a ready-made signal voltage/current source. A new regime of chaotic systems called variable-boostable chaotic systems is explored in [5], whose compelling feature is that one of the variables has the freedom of offset boosting, leading to the excellent convenience for signal transformation and transmission [5, 6]. In this paper, a simple approach to generate multi-double-scroll attractor is introduced by replacing the constant controller in the variable-boostable chaotic systems with a multi-level pulse excitation. The newly proposed method is simple and feasible for circuit implementation and practical application. 2
c is substituted with a periodic piecewise function, the double-scroll chaotic attractor will shift to different locations within different time intervals, thereby resulting in the emergence of multi-double-scroll chaotic attractor. As a result, a non-autonomous approach to generating multi-double-scroll attractor can be easily constructed. A multi-level pulse function f (t) is taken as the periodic piecewise function, which can be described as [3, 4] f (t) =
x˙ = yz y˙ = 1 − z2
(1)
z˙ = (x + c) + yz where the real constant c is a boosting controller of the variable x along the x-axis. Positive c boosts the variable x in the negative direction; otherwise, negative c boosts the variable x in the positive direction. Note that the variation of c does not change the dynamics of system (1) [5]. With a pair of symmetry equilibrium points (−c, 0, ±1), system (1) can generate a double-scroll chaotic attractor, whose scroll locations are determined by c along the x-axis. If the constant
An sgn sin (2pfn t)
(2)
1
where An and fn stand for the amplitude and frequency of the n-th pulse excitation, respectively. Regarding n = 2 as a paradigm, then yields f (t) = A1 sgn sin (2pf1 t) + A2 sgn sin (2pf2 t)
(3)
When the parameters used in (3) are fixed as A1 = 16/15, A2 = 8/15, f1 = 0.001 Hz and f2 = 0.002 Hz, a four-level periodic pulse function can be built. By introducing (3) to substitute the constant boosting controller c in system (1), a novel variable-boostable chaotic system excited by four-level pulse is modelled as x˙ = yz
Constructing scheme of multi-double-scroll system
According to the definition given in [5], a variable-boostable chaotic system with the freedom for offset boosting the variable x is expressed as
n
y˙ = 1 − z2 z˙ = [x + f (t)] + yz
(4)
Obviously, system (4) has eight alternating-current (AC) equilibrium points with the time evolutions, which are calculated as (xe, ye, ze) = (−8/5, 0, ±1), (−8/15, 0, ±1), (8/15, 0, ±1), and (8/5, 0, ±1). The results indicate that the double-scroll attractor can be alternatively shifted to the different equilibrium point locations along the x-axis with the time evolution, leading to the emergence of a four-double-scroll chaotic attractor. Setting the initial condition (x(0), y(0), z(0))=(−1, 1, −1), the variable x(t) overlapped the time-domain waveforms of four-level periodic pulse are simulated, as shown in Fig. 1a. Correspondingly, the phase portrait of the four-double-scroll attractor in the x–z plane is given and plotted in Fig. 1b. From Fig. 1, it can be seen that the proposed approach is simple and feasible, which can be used to generate multi-double-scroll attractor indeed.
This is an open access article published by the IET under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0/)
J. Eng., 2018, Vol. 2018, Iss. 1, pp. 42–44 doi: 10.1049/joe.2017.0403
Fig. 1 Numerically simulated results a Time-domain waveforms of x(t) and −f(t) b Phase portrait of four-double-scroll chaotic attractor in the x–z plane
Fig. 3 Circuit simulated results a Time-domain waveforms of vx(t) and −f(t) b Phase portrait of four-double-scroll chaotic attractor in the vx–vz plane
Moreover, other non-autonomous functions, such as random bipolar sequences [7] and other different pulse-like sequences can be further extended to increase the topology complexity of the generating chaotic attractors. 3
Circuit implementation and verifications
The circuit implementation is necessary for the verifications of the dynamical behaviours. In order to verify the feasibility of the proposed method, a circuit schematic by using commonly off-the-shelf electronic components can be designed, as shown in Fig. 2. According to the designed circuit schematic given in Fig. 2, the circuit equation can be established as dvx = −10g1 vy vz dt dvy RC = 1 − 10g2 v2z dt dv RC z = [vx + f (t)] − 10g1 vy vz dt
RC
(5)
where t = tRC is the physical time; vx, vy, and vz are the voltages of three integrating capacitors; g1 and g2 are the gains of multipliers M1 and M2, respectively; f (t) represents the practical four-level periodic pulse excitation. It should be pointed that g1 = −1/10 and
g2 = 1/10 are chosen for simplifying circuit implementation and avoiding the output saturation of the two multipliers. With the circuit schematic given in Fig. 2, a circuit simulation model is built by NI Multisim 12.0 simulation and circuit design software. The AD711KN op-amps and AD633JN multipliers with ±15 V DC voltage supplies are utilised. The four-level periodic pulse can be implemented via series superposition of two function generators. When the element parameters in Fig. 2 are fixed as R = 10 kΩ, C = 10 nF, g1 = −1/10, and g2 = 1/10, and the amplitudes and frequencies of two function generators are set to A1 = 0.533 V, A2 = 0.266 V, f1 = 10 Hz, and f2 = 20 Hz, respectively, the timedomain waveforms of the four-level periodic pulse and state variable vx(t) are captured by using virtual mixed signal oscilloscope (Agilent 54622D), as shown in Fig. 3a. Correspondingly, the phase portrait of four-double-scroll chaotic attractor in the vx–vz plane is displayed in Fig. 3b. Comparing Fig. 3 with Fig. 1 it has been found that the circuit simulated results are consistent well with the numerical simulations. 4
In this paper, a simple approach to generating multi-double-scroll attractor from the variable-boostable chaotic system excited by non-autonomous function is proposed. As a paradigm, a four-level periodic pulse is employed as the non-autonomous function to generate four-double-scroll chaotic attractor. By means of mathematical model, numerical simulations and circuit simulations are performed. These results are well consistent, which demonstrate the feasibility of the proposed approach. Consequently, the proposed approach to generating multi-double-scroll attractor is simple and feasible, and may be suitable for chaos-based applications. 5
Fig. 2 Designed circuit schematic for system (4) J. Eng., 2018, Vol. 2018, Iss. 1, pp. 42–44 doi: 10.1049/joe.2017.0403
Conclusions
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Grant nos. 51777016, 61601062, and This is an open access article published by the IET under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0/)
51607013), and the Natural Science Foundation of Jiangsu Province, China (Grant no. BK20160282). 6
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This is an open access article published by the IET under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0/)
J. Eng., 2018, Vol. 2018, Iss. 1, pp. 42–44 doi: 10.1049/joe.2017.0403
