Chaos control refers to purposefully eliminating or weakening chaotic behavior of a system through some control methods when the chaotic motion is harmful. Since the Ott–Grebogi–Yorke (OGY) method was proposed in 1990,^[1] much effort has been devoted to the study of controlling chaos. However, not all chaotic behaviors are harmful, and recent research has shown that chaos can actually be useful under certain circumstances, such as chaos-based multimedia encryption and decryption.^[2,3] Therefore, chaotification by making an originally non-chaotic dynamical system chaotic, enhancing existing chaos, or extending the number of saddle focus equilibria for generating multi-wing or multi-scroll chaotic and hyperchaotic systems has attracted some special attention in recent years.^[4–13]

Compared with the double wing chaotic system, the multi-wing chaotic system is a new type of generalized Lorenz system characterized by multiple wings which are distributed in a single direction or in multiple directions, the phase portrait for multi-wing chaotic attractor is formed with nested wing, the number and the grid distribution of the wing can be adjusted by parameters.^[10] In contrast to the double-wing chaotic system, the multi-wing chaotic system possesses the following features that have potential applications in secure communication: i) the phase trajectories of multi-wing chaotic attractors can randomly jump in all wings with ergodic property, so that the dynamical behavior is more complicated than those of Lorenz system and Chen system, and the statistical properties can be improved; ii) by introducing a nonlinear-function controller, several key parameters of multi-wing chaotic attractors can be easily adjusted, including the width of each segment, amplitude, slope, equilibrium, and turning points. As a result, one can easily control the system equilibrium, the number of scrolls, the shapes and sizes of the scrolls, the spatial distribution of the scrolls, and even the phase trajectories.

It is noticed that, up to now, multiple parameter nonlinear function switching controllers, such as duality-symmetric multi-segment linear function and multi-segment quadratic function, have usually been considered for generating multi-wing chaotic attractors from autonomous three-dimensional (3D) quadratic double-wing chaotic systems as shown in Fig. 1.^[10–12] However, there exist the following aspects for improving this kind of switching control method.

Fig. 1.

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	Fig. 1. Multiple parameter nonlinear functions: (a) multi-segment linear function and (b) multi-segment square function.

To cope with the above-mentioned issues, this paper introduces a new step function control method with a single parameter ε, which is characterized by the following two aspects.

Based on this single parameter ε step function control method, this paper constructs an absolute value piecewise linear 3D multi-wing chaotic system, a square 3D multi-wing chaotic system and a square four-dimensional (4D) multi-wing hyperchaotic system, respectively. According to the heteroclinic loop Shilnikov theorem,^[14] the chaos mechanism of absolute value piecewise linear 3D multi-wing chaotic system is analyzed, a circuit for both absolute value piecewise linear and square 3D multi-wing chaotic systems is designed and implemented.

Most reported chaotic systems, i.e., Lorenz system, Chen system, etc, have a limited number of equilibria.^[15–18] Therefore, chaos in these systems can be proved by using conventional Shilnikov theorem, in which at least one unstable equilibrium point for emergence of chaos is required. However, it is impossible to use the conventional Shilnikov theorem to verify the chaos in non-equilibrium systems. According to Shilnikov theorem, one or more saddle focus equilibria are keys to the formations of heteroclinic orbit and heteroclinic loop in a chaotic system.^[14] For example, double-wing chaotic attractors are formed from saddle or saddle focus equilibria in generalized Lorenz system family,^[15–18] and multi-wing chaotic attractors can be constructed by extending the number of saddle focus equilibria.^[10–12] As is well known, heteroclinic orbit and heteroclinic loop cannot be formed if there is no equilibrium point. It is noticed that a 4D multi-wing chaotic system was recently found to have no equilibrium point.^[19] However, to the best of the authors’ knowledge, there has been no report about a 4D multi-wing hyperchaotic system without equilibrium point so far. One may ask whether there is a possible way further to find a non-equilibrium 4D multi-wing hyperchaotic system. This paper gives a positive answer to the question.

In spite of the fact that a simple non-equilibrium chaotic system was found in 1994,^[20] studies on chaotic systems without equilibrium have only bloomed recently.^[21,22] It is worth noting that such a chaotic system without equilibrium is categorized as a chaotic system with hidden attraction because its basin of attraction does not intersect with small neighborhoods of any equilibrium.^[23] Owing to their academic significance and practical importance, studying systems with hidden attractors has received significant attention.^[24]

The rest of the paper is organized as follows. Construction of multi-wing chaotic and hyperchaotic systems via a unified single parameter ε step function control is introduced in Section 2. Some properties of a multi-wing system are discussed and analyzed in Section 3. Circuit design and implementation are demonstrated in Section 4. Finally, in Section 5 some conclusions are drawn.

In this section, two types of double-wing chaotic systems and one type of double-wing hyperchaotic system are first considered. Then, the corresponding multi-wing chaotic system and hyperchaotic system based on a unified single parameter ε step function control are introduced, and the construction criterion for the unified step function is also given.

2.1. Double-wing chaotic and hyperchaotic systems

According to the generalized Lorenz system family,^[13–16] two types of double-wing chaotic systems and one type of double-wing hyperchaotic system are given at first. From these systems, one can further obtain the corresponding multi-wing chaotic and hyperchaotic systems by a unified step function switching control.

The n (n = 3,4)-dimensional double wing chaotic or hyperchaotic system is in the form of

where is an even-symmetric function, ξ_n1 and ξ_n2 are constants, f₁(x₁,x₂, …,x_n−1) is a linear function, and fi(x₁,x₂, …,x_n) (i = 2,3, …,n) are piecewise linear function or quadratic polynomial. Furthermore, f₁(x₁,x₂, …,x_n), …, f_n−1 (x₁,x₂,…,x_n) can make Eq. (1) invariant under the transformation of (x,y,z) → (−x, −y,z).

According to Eq. (1), let n = 3, f₁ (x₁,x₂,x₃) is a linear function, f₂ (x₁,x₂,x₃) is a piecewise linear function, h(x₂) is an absolute value function, one can obtain the absolute value piecewise linear 3D double wing chaotic system as

where a₁₁ = 1.18, a₁₂ = 1.18, a₂₁ = 5.82, a₂₂ = 0.7, a₃₁ = 4, and a₃₂ = 0.168.

It is noticed that the third equation in Eq. (2) includes an even-symmetric absolute value term a₃₁|y| and a linear term a₃₂z, which can generate a multi-wing chaotic system by a unified step function switching control.

According to Eq. (1), let n = 3, f₁(x₁,x₂,x₃) is a linear function, f₂(x₁,x₂,x₃) is a quadratic polynomial, h(x₂) is a square function, one can obtain the square 3D double wing chaotic system as

where d₁₁ = 36, d₁₂ = 36, d₂₁ = 20, d₂₂ = 20, d₃₁ = 100, and d₃₂ = 3.

It is noticed that the third equation in Eq. (3) includes an even-symmetric square term d₃₁y² and a linear term d₃₂z, which can generate multi-wing chaotic system by a unified step function switching control.

According to Eq. (1), let n = 4, f₁ (x₁,x₂,x₃,x₄) is a linear function, f₂(x₁,x₂,x₃,x₄) and f₃(x₁,x₂,x₃,x₄) are both quadratic polynomials, h(x₂) is a square function, one can obtain the general form of a square 4D double wing hyperchaotic system as

where b₁₁ = 36, b₁₂ = 45, b₁₃ = 3.5, b₂₁ = 5.6, b₂₂ = 20, b₃₁ = 0.5, b₃₂ = 0.3, b₄₁ = 17.9, and b₄₂ = 3.

It is noticed that the fourth equation in Eq. (4) includes an even-symmetric square term b₄₁y² and a linear term b₄₂u, which can generate a multi-wing hyperchaotic system by a unified step function switching control.

Based on Eqs. (2)–(4), the numerical simulation results of double-wing chaotic and hyperchaotic attractors are shown in Fig. 2. The Lyapunov exponents of the absolute value piecewise linear 3D double wing chaotic system are LE₁ = 0.41, LE₂ = 0.00, and LE₃ = −1.05; the Lyapunov exponents of the square 3D double wing chaotic system are LE₁ = 1.36, LE₂ = 0.00, LE₃ = −20.36; the Lyapunov exponents of the square 4D double wing hyperchaotic system are LE₁ = 0.99, LE₂ = 0.35, LE₃ = 0.00, and LE₄ = −20.05.

Fig. 2.

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	Fig. 2. Double-wing attractors of chaotic and hyperchaotic system: (a) double-wing attractor of absolute value piecewise linear 3D chaotic system; (b) double-wing attractor of square 3D chaotic system; (c) double-wing attractor of square 4D hyperchaotic system.

2.2. Generation of multi-wing chaotic and hyperchaotic systems via a unified single parameter ε step function control

According to Eqs. (2)–(4), by adding a unified single parameter ε step function controller g_i(y, ε) (i = 1,2,3) to the last equation, one can obtain the corresponding three types of multi-wing chaotic and hyperchaotic systems.

Based on Eq. (2), an absolute value piecewise linear 3D multi-wing chaotic system controlled by g₁(y, ε) is given by

where parameters a_{i j} are the same as those in Eq. (2).

Based on Eq. (3), a square 3D multi-wing chaotic system controlled by g₂ (y, ε) is given by

where parameters d_{i j} are the same as those in Eq. (3).

Based on Eq. (4), a square 4D multi-wing hyperchaotic system controlled by g₃ (y, ε) is given by

where parameter b_{i j} are the same as those in Eq. (4).

Among Eqs. (5)–(7), let the general form of a unified single parameter ε even-symmetric step function controller be

where i = 1, 2, 3, H_k(ε) (k = 1,2, …,M) are altitudes of the step function, ε is a single parameter, W_k (k = 1,2, …,M) are jump point coordinates of the step function, step(y − W_k) = [sgn(y − W_k) + 1]/2, step(−y − W_k) = [sgn(−y − W_k) + 1]/2, sgn(y) is a sign function. Furthermore, from Eqs. (5)–(7) with expression (8), when M is determined, the generated maximum wing can reach N = 2M + 2. The unified single parameter ε step function controller is shown in Fig. 3.

Fig. 3.

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	Fig. 3. A unified single parameter ε even-symmetric step function switching controller gi(y, ε).

2.3. Construction criterion for unified single parameter ε even-symmetric step function

Given W_k (k = 1,2, …,M), the construction criterion for unified single parameter ε even-symmetric step function is described by the following two aspects. (i)

As for g₁(y, ε) in Eq. (5), the altitudes H_k(ε) in expression (8) are determined by

where H₀ = a₃₁, W₁ = 0.5, W₂ = 1.0, …, W_m − W_m−1 = 0.5 (m = 2,3, …,M), and ε = 0.32. When H_k(ε) are determined, to ensure that the absolute function a₃₁|y| and step function g₁(y, ε) as shown in Fig. 4 are not intersected, one can obtain the equations to determine the ε value range as

where m = 2,3,4, …,M.

(ii)

As for g₂(y, ε) and g₃ (y, ε)in Eqs. (6) and (7), the altitudes H_k(ε) in Eq. (8) are determined by

Fig. 4.

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	Fig. 4. Schematic diagram of disjunction between absolute value function a31|y| and step function g1(y, ε).

For g₂ (y, ε), let H₀ = d₃₁, W₁ = 0.3, W₂ = 0.45, …, W_m − W_m−1 = 0.15 (m = 2,3, …,M), and ε = 1.3. For g₃ (y, ε), let H₀ = b₄₁, W₁ = 1.0, W₂ = 1.5, …, W_m − W_m−1 = 0.5 (m = 2,3, …,M), and ε = 1.3. When H_k(ε) are determined, to ensure that there is an intersection between square function ky²(k ∈ {d₃₁,b₄₁}) and step function g_i (y, ε)(i = 1,2) in each interval (W_i−1, W_i) as shown in Fig. 5, one can obtain the equations to determine the ε value range as

where m = 2,3,4, …,M.

Fig. 5.

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	Fig. 5. Schematic diagram of intersection in each interval (Wi−1, Wi) between square function ky2 (k ∈ {d31,b41}) and step function gi(y, ε) (i = 2,3).

2.4. Simulation results of multi-wing chaotic and hyperchaotic attractors

Let M = 4, according to Eqs. (5)–(12), one obtains the numerical simulation results of absolute value piecewise linear 3D 10-wing chaotic attractor, square 3D 10-wing chaotic attractor and square 4D 10-wing hyperchaotic attractor as shown in Fig. 6, respectively. The Lyapunov exponents of the absolute value piecewise linear 3D double 10-wing chaotic system are LE₁ = 0.27, LE₂ = 0.00, LE₃ = −0.92, the Lyapunov exponents of the square 3D 10-wing chaotic system are LE₁ = 1.46, LE₂ = 0.00, and LE₃ = −20.45, and the Lyapunov exponents of the square 4D 10-wing hyperchaotic system are LE₁ = 1.28, LE₂ = 0.49, LE₃ = 0.00, and LE₄ = −20.46, respectively.

Fig. 6.

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	Fig. 6. 10-wing attractors of chaotic and hyperchaotic systems: (a) 10-wing attractor of absolute value piecewise linear 3D chaotic system; (b) 10-wing attractor of square 3D chaotic system; (c) 10-wing attractor of square 4D hyperchaotic system.

In this section, the dynamical properties of multi-wing chaotic and hyperchaotic systems generated by the unified step function control are analyzed, including the equilibria of three types of multi-wing systems, non-equilibrium square 4D multi-wing hyperchaotic system, the mechanism analysis for piecewise linear 3D double wing and multi-wing chaotic system, mirror symmetry of multi-wing chaotic system, and the solution of absolute value piecewise linear 3D multi-wing chaotic system.

3.1. Equilibrium points

According to Eq. (5), one obtains the equilibria of the absolute value piecewise linear 3D multi-wing chaotic system as

where q = 1,2,3, …, when .

According to Eq. (6), one obtains the equilibria of the square 3D multi-wing chaotic system as

where q = 1,2,3, …, when .

According to Eq. (7), one obtains the equilibria of the square 4D multi-wing hyperchaotic system as

where q = 1,2,3, …, when .

3.2. Non-equilibrium square 4D multi-wing hyperchaotic system

According to Eq. (13), since a_{i j} > 0, H_i > 0, and they are all real numbers, one can deduce that the coordinates of equilibria , , and are all real numbers, indicating that all of the equilibria really exist.

Similarly, according to Eq. (14), since d_{i j} > 0, H_i > 0, and they are all real numbers, one can deduce that the coordinates of equilibria , , and are all real numbers, which means that all of the equilibria really exist.

In Eq. (15), if the following inequality holds:

the coordinates of equilibria are all real numbers, which means that the equilibria really exist.

However, in Eq. (15), if the following inequality holds:

the coordinates of equilibria are all imaginary numbers, which means that the equilibria do not exist.

From inequality (17), when the equilibria do not exist, the corresponding range of parameter b₁₃ can be determined by

where q = 5, n = 4, b₁₁ = 36, b₁₂ = 45, b₂₁ = 5.6, b₂₂ = 20, b₃₁ = 0.5, b₃₂ = 0.3, b₄₁ = 17.9, b₄₂ = 3, H₀ = 17.9, W₁ = 1.0, W₂ = 1.5, W₃ = 2.0, W₄ = 2.5, and ε = 1.3. Also from Eq. (11), one obtains H₁ = 19.2, H₂ = 23.675, H₃ = 32.625, and H₄ = 41.575.

Substituting the above-mentioned parameters into inequality (18), one obtains

When b₁₃ > 8.1135, equation (7) is a non-equilibrium multi-wing hyperchaotic system. Let b₁₃ = 9.0 > 8.1135, the remaining parameters are the same as the above. The numerical simulation result of non-equilibrium 10-wing hyperchaotic attractor is shown in Fig. 7, and the corresponding Lyapunov exponents are LE₁ = 1.14, LE₂ = 0.17, LE₃ = 0.00, and LE₄ = −19.98. Furthermore, the Lyapunov exponents with parameter b₁₃ variation are also obtained as shown in Fig. 8. Therefore, when b₁₃ < 8.1135, equation (7) is an equilibrium multi-wing hyperchaotic system. However, when b₁₃ > 8.1135, it is a non-equilibrium multi-wing hyperchaotic system.

Fig. 7.

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	Fig. 7. Non-equilibrium 10-wing hyperchaotic attractor with b13 = 9.0.

Fig. 8.

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	Fig. 8. (a) Positive and zero Lyapunov exponents; (b) negative Lyapunov exponent.

As is well known, the equilibrium point is very important for a chaotic system. Based on the Shilnikov theorem, the key to forming heteroclinic orbit or heteroclinic loop is the existence of one or more saddle focus equilibria in the chaotic system. For example, in a generalized Lorenz system family, double-wing chaotic attractors can be formed from the saddle points or saddle focus points, and multi-wing chaotic attractors can be further formed from extending the number of saddle-focus equilibria. However, without equilibria, the homoclinic orbit or heteroclinic loop cannot be formed. The mechanism for such a kind of non-equilibrium hyperchaotic system needs to be further investigated in depth.

3.3. Chaotic mechanism analysis of double wing and multi-wing systems

In this section, according to the heteroclinic Shilnikov theorem, the chaotic mechanism analyses for absolute piecewise linear 3D double wing and multi-wing systems are given.

3.3.1. Chaotic mechanism analysis of double wing system

From Eqs. (2) and (13), one obtains the equilibria of absolute piecewise linear 3D chaotic system as follows:

The corresponding eigenvalues of the equilibria are γ = −1.18, σ ± jω = 0.266 ± j1.9523, which confirms that are saddle-focus equilibria with index 2. Furthermore, one can further obtain the eigenvectors (l_±1,m_±1,n_±1) for one-dimensional (1D) stable manifold and (A_±1,B_±1,C_±1) for two-dimensional (2D) unstable manifold at the equilibria , respectively.

According to heteroclinic Shilnikov theorem, and are two different equilibria of 3D system, and the system is chaotic in the sense of Smale horseshoe if it satisfies the following two conditions.^[12]

The 1D stable manifold E^S(P₊₁) and the 2D unstable manifold E^U(P₊₁) of are given by

where l₊₁, m₊₁, and n₊₁ are three components of eigenvector of 1D stable manifold, A₊₁, B₊₁, and C₊₁ are three components of eigenvector of 2D unstable manifold.

Similarly, the 1D stable manifold E^S(P₋₁) and the 2D unstable manifold E^U(P₋₁) of are given by

where l₋₁, m₋₁, and n₋₁ are three components of the eigenvector of 1D stable manifold, A₋₁, B₋₁, and C₋₁ are three components of the eigenvector of 2D unstable manifold.

According to Eqs. (20)–(21), one obtains the characteristic lines of 1D stable manifolds E^S(P₊₁) and E^S(P₋₁), and the characteristic planes of 2D unstable manifolds E^U(P₊₁) and E^U(P₋₁) as shown in Fig. 9, respectively, where S₀(y) = {(x,y,z)|y = 0} is switching plane. From Fig. 9, cross point Q₊₀ between E^S(P₊₁) and S₀(y) = {(x,y,z)|y = 0}, Q₋₀ between E^S(P₋₁) and S₀(y), cross lines L₊₀ and L₋₀ between E^U(P₊₁) and S₀(y), and L₋₀ between E^U(P₋₁) and S₀(y) can be further determined. In Fig. 9, switching plane S₀(y) and switching region V_±1(x,y,z) are obtained by

Fig. 9.

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	Fig. 9. Heteroclinic loop connecting two equilibria in absolute piecewise linear 3D double-wing chaotic system.

If Q₊₀ lies on L₊₀, there is a heteroclinic orbit HL₁ = E^U(P₊₁) ∪ Q₊₀ ∪ E^S(P₋₁) from P₊₁ to P₋₁. Similarly, if Q₋₀ lies on L₋₀, there is a heteroclinic orbit HL₂ = E^U(P₋₁) ∪ Q₋₀ ∪ E^S(P₊₁) from P₋₁ to P₊₁. By choosing a_{i j} in Eq. (2) and switching control parameters, if Q₊₀ lies in L₊₀ and Q₋₀ lies in L₋₀, then there is a heteroclinic loop HL = HL₁ ∪ HL₂ = E^U(P₊₁) ∪ Q₊₀ ∪ E^S(P₋₁) ∪ E^U(P₋₁) ∪ Q₋₀ ∪ E^S(P₊₁) connecting P₁ and P₂ together as shown in Fig. 9. Obviously, according to the heteroclinic Shilnikov theorem, equation (2) is chaotic in the sense of Smale horseshoe.

3.3.2. Chaotic mechanism analysis of multi-wing system

Equipped with the controller g₁ (y, ε), system (5) is a piecewise linear switching system, each switching plane S_±q(y) and each switching region V_±q (x,y,z) are given by

where q = 2,3, ….

According to Eqs. (5) and (13), one has the equilibrium regions of V_±q (x,y,z) for absolute value piecewise linear 3D multi-wing chaotic system as

In system (5), except for O(0,0,0), there are the three same eigenvalues γ = −1.18, σ ± jω = 0.266 ± j1.9523 of equilibria in each linear region V_±q. Therefore, these equilibria are saddle-focus equilibria with index 2. One has the 1D stable manifold E^S(P_±q) and 2D unstable manifold E^U(P_±q) of as follows:

where l_±q, m_±q, and n_±q are three components of the eigenvector of 1D stable manifold; A_±q, B_±q, and C_±q are three components of the eigenvector of 2D unstable manifold.

Based on Eq. (25), one further obtains the characteristic lines of 1D stable manifolds E^S(P_±q) and the characteristic planes of 2D unstable manifolds E^U(P_±q) as shown in Fig. 10. The cross points Q_±m (m = 0,1,2,3, …) between E^S(P_±q), S₀(y), and S_±q (y), and cross lines L_±m (m = 0,1,2,3, …) between E^U(P_±q), S₀(y), and S_±q (y) can be obtained.

Fig. 10.

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	Fig. 10. Multiple heteroclinic loops connecting multiple adjacent equilibria in absolute piecewise linear 3D multi-wing chaotic system.

By choosing a_{i j} in Eq. (5) and switching control parameters, all Q_±m lie on L_±m, then there is a heteroclinic loop connecting two adjacent equilibria together as shown in Fig. 10. Obviously, according to the heteroclinic Shilnikov theorem, equation (5) is chaotic in the sense of Smale horseshoe.^[25]

3.4. Mirror symmetry

Theorem 1 Assume that the general form of 3D multi-wing chaotic system family is

where h(y) ∈ {|y|, y²} is an even-symmetric function, ξ_n1 and ξ_n2 are constants, and g(y) is a step function switching controller.

If the corresponding conversion of Eq. (26) with respect to the z axis is given by

then the solution of Eq. (27) is of mirror symmetry with respect to the z axis. In Eq. (27), z₀ is a coordinate translation constant, which makes the solution of Eq. (26) always satisfy z ≥ 0 (or z ≤ 0).

Proof According to Eq. (27), one can obtain its solution of the integral form as follows:

Obviously, if z(τ) > 0, the solution of the integral form is given by

If z(τ) < 0, the solution of the integral form is given by

where

According to Eqs. (29) and (30), one can see that the solution of Eq. (27) is of mirror symmetry with respect to the z axis. Therefore, the proof of the theorem 1 is completed.

For example, by using the mirror symmetry conversion, system (5) is modified into

where z₀ = 5.25. From Eq. (31), the numerical simulation result of multi-wing chaotic attractor is shown in Fig. 11.

Fig. 11.

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	Fig. 11. Absolute value piecewise linear 3D 20-wing chaotic system after mirror symmetry conversion.

3.5. Solution of absolute value piecewise linear 3D chaotic system

3.5.1. Solution of absolute value piecewise linear 3D double wing chaotic system

In absolute value piecewise linear 3D double-wing chaotic equation (2), there are four linear regions, and the corresponding solutions of state equation can be obtained, respectively, as follows.

3.5.2. Solution of absolute value piecewise linear 3D multi-wing chaotic system

It is noticed that the absolute value piecewise linear 3D multi-wing chaotic equation (5) is obtained through linear transformation of Eq. (2) by switching control. From Eqs. (32)–(35), the corresponding solution of state equation (5) is given by

where are equilibria, q = 1,2,3,…, and symbol T is the transpose of matrix.

Based on Eqs. (5) and (6), the circuit diagrams are designed as shown in Figs. 12–14. When switch K₁ is at position 1 in Fig. 14, q(y) = q_C (y); when it is at position 2, q(y) = q_L (y). The values of resistance, comparative voltage, switch state, and number of wings, and also the types of chaotic system are given in Tables 1–3, respectively. Based on the circuits given in Figs. 12–14 and Tables 1–3, the experimental results are shown in Fig. 15.

Fig. 12.

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	Fig. 12. Circuit diagrams of multi-wing chaotic systems: (a) the circuit diagram of absolute value piecewise linear 3D multi-wing chaotic system and (b) the circuit diagram of square 3D multi-wing chaotic system.

Fig. 13.

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	Fig. 13. Circuit design for sgn(x) of Fig. 12.

Fig. 14.

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	Fig. 14. Circuit design for nonlinear functions qC (y) and qL (y) of Fig. 12.

Table 1.

Table 1.

Table 1. Resistance values and the types of chaotic system in Fig. 14. . R/kω R1kΩ R2/kΩ R3/kΩ R4/kΩ Type of chaotic system 2.5 12.5 11 11 11 Eq. (5) 0.1 20 16.67 12 11 Eq. (6)

Table 1. Resistance values and the types of chaotic system in Fig. 14. .

Table 2.

Table 2.

Table 2. Comparative voltages and the types of chaotic system in Fig. 14. . E1 E2 E3 E4 Type of chaotic system 0.5 1.0 1.5 2.0 Eq. (5) 0.3 0.45 0.6 0.75 Eq. (6)

Table 2. Comparative voltages and the types of chaotic system in Fig. 14. .

Table 3.

Table 3.

Table 3. On-off for switch linkages K2–K4 and numbers of wings in Fig. 14. . K2 K3 K4 Number of wing off off off 4 on off off 6 on on off 8 on on on 10

Table 3. On-off for switch linkages K2–K4 and numbers of wings in Fig. 14. .

Fig. 15.

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	Fig. 15. Circuit experimental results: (a) experimental result of absolute value piecewise linear 3D 10-wing chaotic attractor and (b) experimental result of square 3D 10-wing chaotic attractor.

In this paper, we introduce a novel method of constructing multi-wing butterfly chaotic and hyperchaotic systems via a unified single parameter ε even-symmetric step function control, to overcome the essential difficulties in iteratively adjusting multiple parameters for a conventional multi-parameter control method. By using the proposed strategy, we further find a non-equilibrium 4D multi-wing hyperchaotic system, which is valuable for future investigation due to the significance in basic research and importance in practical application. In addition, some properties for the multi-wing attractor and its chaos mechanism are also discussed and analyzed. The circuits for absolute piecewise linear 3D multi-wing system and for square 3D multi-wing system are designed and implemented for demonstration, respectively.