Novel two-directional grid multi-scroll chaotic attractors based on the Jerk system
Peng-Fei Ding(丁鹏飞)1,2, Xiao-Yi Feng(冯晓毅)1,†, and Cheng-Mao Wu(吴成茂)2
1School of Electronics and Information, Northwestern Polytechnical University, Xi’an 710072, China 2School of Electronics and Engineering, Xi’an University of Posts and Telecommunications, Xi’an 710121, China
A new method is presented to generate two-directional (2D) grid multi-scroll chaotic attractors via a specific form of the sine function and sign function series, which are applied to increase saddle points of index 2. The scroll number in the x-direction is modified easily through changing the thresholds of the specific form of the sine function, while the scroll number in the y-direction is controlled by the sign function series. Some basic dynamical properties, such as equilibrium points, bifurcation diagram, phase portraits, and Lyapunov exponents spectrum are studied. Furthermore, the electronic circuit of the system is designed and its simulation results are given by Multisim 10.
Peng-Fei Ding(丁鹏飞), Xiao-Yi Feng(冯晓毅)†, and Cheng-Mao Wu(吴成茂) Novel two-directional grid multi-scroll chaotic attractors based on the Jerk system 2020 Chin. Phys. B 29 108202
Fig. 1.
The waveform of the different sine functions: (a) f(x) with b = 0.5 and n1 = n2 = 2, (b) g(x) with b = 0.5, (c) h(x) with A = 6 and a = 0.5 π, (d) p(x) with a = 1, b = 0.1, c = 3, and d = 0. The a, c, and d are real constants, and x, y, and z are state variables of the system (3). The b, n1, and n2 are real constants in Eq. (4). The number of scrolls generated by the system (3) with suitable parameters can be adjusted by the parameters n1 and n2.
Fig. 2.
Different number of scroll chaotic attractors are generated by system (3) with a = c = d = 0.3 and b = 0.5: (a) 3-scroll chaotic attractor with n1 = 1 and n2 = 2, (b) 5-scroll chaotic attractor with n1 = 2 and n2 = 3.
Fig. 3.
Grid multi-scroll chaotic attractors for a = 1, b = 0.5, c = 0.3, d = 0.5, and A = 1: (a) 6 × 3 grid multi-scroll chaotic attractors with n1 = 3, n2 = 3, and M = 1; (b) 5 × 4 grid multi-scroll chaotic attractors with n1 = 2, n2 = 3, and M = 2.
Fig. 4.
The equilibrium point distribution of the 6 × 3 grid multi-scroll chaotic attractors.
Fig. 5.
The system (9) with Eqs. (4) and (10), and d ∈ (0,1): (a) Lyapunov exponents; (b) bifurcation diagram.
Fig. 6.
The specific form of the sine function f(x) with b = 0.5, n1 = n2 = 3. (a) Electronic circuit diagram; (b) simulation result with the unit of horizontal ordinate 2 s/Div, and the unit of vertical ordinate 500 mV/Div.
Fig. 7.
The specific form of the sine function f(x) with b = 0.5, n1 = 2, n2 = 3. (a) Electronic circuit diagram; (b) simulation result with the unit of horizontal ordinate 2 s/Div, and the unit of vertical ordinate 500 mV/Div.
Fig. 8.
The sign function f1(y) of Eq. (10) with A = 1, M = 1. (a) Electronic circuit diagram; (b) circuit simulation result with the unit of horizontal ordinate 1 V/Div, and the unit of vertical ordinate 1 V/Div.
Fig. 9.
The sign function f1(y) of Eq. (11) with A = 1, M = 1. (a) Electronic circuit diagram; (b) circuit simulation result with the unit of horizontal ordinate 1 V/Div, and the unit of vertical ordinate 2 V/Div.
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