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.

Fig. 10. Grid multi-scroll chaotic attractor circuit.

Fig. 11. Circuit simulation results: (a) 6 × 3 grid multi-scroll chaotic attractors, (b) 5 × 4 grid multi-scroll chaotic attractors.

Fig. 12. Hardware circuits experimental results. (a) Hardware circuits connection diagram; (b) experimental results of the 6 × 3 grid multi-scroll chaotic attractors; (c) experimental results of the 5 × 4 grid multi-scroll chaotic attractors.