Secure communication and synchronizations in light of the stability theory of the hyperchaotic complex nonlinear systems

Abstract

This paper’s main objective is to reflect on hyperchaotic complex nonlinear systems’ phase synchronization (PS) and anti-phase synchronization (APS). In these complex systems, the number of state variables can be expanded by isolating the real and imaginative parts. In PS, while their amplitudes stay uncorrelated, the position between two coupled chaotic (or hyperchaotic) systems remains in step with each other. The absence of the sum of relevant variables can characterize APS. To study PS and APS with complex variables of hyperchaotic nonlinear systems, the active control technique on the basis of stability analysis is proposed. PS and APS investigations for high dimensional systems are demonstrated by studying a 6-dimensional Lü system. Phase synchronization concerns were also used to construct a straightforward and simple secure communication application. Numerical influences outlined for clarifying the phase synchronization of the hyperchaotic Lü model and for examining the gravity of scientific articulations’ control powers.

Keywords

Hyperchaotic system phase synchronization anti-phase synchronization active control lyapunov stability analysis complex

1 Introduction

The synchronization of chaos has displayed an area of great interest over the past three decades. In the event of two coupled real chaotic systems, different kinds of synchronization have been considered so far, e.g, complete synchronization, generalized synchronization, lag synchronization and phase synchronization [1 –22]. Phase synchronization (PS) alludes to an entire process where two chaotic real systems are illustrated via a wonderful relocation concerning the chaotic phases while the appropriateamplitudes with chaotic performance continue to stay uncorrelated [4 , 10–12]. In coupled chaotic systems, there’s a different kind of synchronization, anti-phase synchronization (APS), which does not hold enough consideration. The variables of two interconnected chaotic oscillators in APS are of the corresponding amplitude but differ or contradict in signs [22 –24]. Anti-synchronization (AS) may also be construed as APS. That is, for oscillators with identical amplitudes, there is no difference between AS and APS [[24] and references there in].

In numerical simulations, primarily for Rössler oscillators, the first searches of PS and associated phenomena were carried out. Parlitz et al. theorized phase synchronization in an electronic circuit of chaotic Rössler transistors [13]. Rosenblum et al.portrayed lag synchronization as an average and advanced arrangement between phase and full synchronization [14]. Pikovsky et al. discussed the synchronization of chaotic oscillators in a broad phase [15]. Schäfer et al. actually explained the synchronization of the heartbeat phase with the respiratory cycle [16]. Zhou and Kurths discovered phase synchronization due to popular noise [17]. PS proves to be a widespread event in coupled nonlinear oscillator systems and it was examined in laboratory experiments [5, 13], biomedical systems [16] and ecological population data [19]. The phenomenon of PS is also intimate associated with Phase-closed loops which are highly engineered [20] and neuroscience [21]. Also, see Boccaletti et al. for a comprehensive analysis of chaotic synchronization literature [18].

It is well known that APS is Huygens ’ first extrapolation on the synchrony of such two oscillators four hundred years ago. Huygens discovered that the pendulum clocks reinstated on each other’s side were swinging in actually exactly the corresponding frequency and π out of phase. APS can be used in practice areas such as secure communications [23 –25].

There are many other methods to analyze and study the phase of chaotic (or hyperchaotic) oscillations that apply to systems with a straightforward topology of chaotic attractors. First, the φ (t) (phase of a chaotic radio signal) can be characterized only as an angle in the (x, y) plane of the polar coordinate system [18, 22], where $φ (t) = {tan}^{- 1} (\frac{y}{x})$ . In this case, all the trajectories of the chaotic attractor projection onto the (x, y) plane will turn approximately the origin. Generally speaking, if the phase difference connecting couple chaotic (or hyperchaotic) systems is bound, the systems can be regarded as the synchronized phase [13]. Different approach to defining the phase of a chaotic dynamic system is by working the Hilbert transformation to introduce an analytical signal [18, 22]. Finally, the phase of a chaotic signal can be proposed by using the Poincaré cross-section [18].

A hyperchaotic attractor with at least possibly two positive Lyapunov exponents is generally described as hyperchaotic behavior. To ensure system dissipation, the total aggregate of exponents of Lyapunov shall be negative. The minimum size for a (continuous) hyperchaotic system is 4 (necessary condition but not sufficient) [26, 27]. Complex Lorenz, Chen and Lü systems which are 5-dimensional, they do not have hyperchaotic attractors [28, 29]. These systems, which involve complex variables, certainly appear in several crucial areas of physics and engineering, such as the disconnected laser, rotating fluids, disk dynamos, electronic circuits and social dynamics of particles in high-energy accelerators. [28–34 and references there in].

Mahmoud et. al. [35] introduced different forms of hyperchaotic complex Lü systems.

In this paper, we interest in investigating PS and APS of the autonomous hyperchaotic complex attractors with nonlinearity. The complex Lü system defined as an illustrative example to show the validity of our study. The complex Lü system is defined as: $\begin{matrix} \dot{x} & = & α (y - x) + iw, \\ \dot{y} & = & γ y - xz + iw, \end{matrix}$ (1) $\begin{matrix} \dot{z} & = & 1 / 2 (\bar{x} y + x \bar{y}) - β z, \\ \dot{w} & = & 1 / 2 (\bar{x} y + x \bar{y}) - σ w, \end{matrix}$ where α, γ and β are positive parameters that alter the system’s behavior, x = u₁ + iu₂, y = u₃ + iu₄ are complex function, $i = \sqrt{- 1}$ and z, w are real functions, σ is also a parameter for the control. Spots depict time-related derivatives and complex conjugate variables are signified by an overbar.

The hyperchaotic complex Lü system (1) is 6D real first order autonomous ordinary differential equations. We measured the exponents of Lyapunov as: λ₁ = 1.266, λ₂ = 0.303, λ₃ = 0, λ₄ = -1.484, λ₅ = -37.212, λ₆ = -45.366 when α = 14, β = 5, γ = 16.9 and σ = 10. Hence system (1) has a hyperchaotic actions since λ₁ and λ₂ are positive, for more dynamical properties, see Ref. [35].

This paper is structured into six sections: the proposed scheme for studying hyperchaotic complex nonlinear system PS and APS is addressed in Section 2, solely based on the active control strategy [22 , 36–38]. Section 3 investigates the PS of the autonomous hyperchaotic complex system using the proposed technique (1). The amplitudes, phases and phase difference regarding drive and response systems in state of PS are calculated. our investigations are depicted, in a good agreements between analytical and numerical results, to ensure the influence of these results. Section 4 indicates that APS is gained by using the same Section 2 technique. This technique’s analytical results are examined numerically (using the Mathematica 7 program) and there are outstanding and excellent agreements. We observe and indicate the variation between PS and APS in the amplitudes, phases and phase difference. Section 5 addresses the after-effects of phase synchronization to perform a fundamental application process for secure communications. Our closing argument is included in the last section.

2 The PS and APS scheme of nonlinear hyperchaotic complex systems

In this section, the PS plan is offered using Lyapunov stability analysis and active control technique [22 , 36–38] for hyperchaotic complex nonlinear system.

Consider the hyperchaotic complex nonlinear system: $\dot{w} = A w + f (w)$ (2) where w = w₁ + iw₂ = (w₁, w₂, . . . , w_n) ^t is a state complex vector with w_j = u_j + iu_j+1, w₁ = (u₁, u₃, . . . , u_2n-1) ^t, w₂ = (u₂, u₄, . . . , u_2n) ^t, t denotes transpose, j is the odd numbers (j = 1, 3, . . . , 2n - 1), $i = \sqrt{- 1},$ A ∈ R^n×n is actual (or complex) system parameter matrices, f = (f₁, f₂, . . . , f_n) ^t is a vector of complex nonlinear basic functions and the subscript 1, 2 is the real and imaginary parts,respectively.

We can consider the drive system as (we denote the drive system by the subscript d): ${\dot{w}}_{d} = {Aw}_{d} + f {(w}_{d}) .$ (3) By considering the additive controller L = L₁ + iL₂ = (L₁, L₂, . . . , L_n) ^t where L_j = v_j + iv_j+1, L₁ = (v₁, v₃, . . . , v_2n-1) ^t, L₂ = (v₂, v₄, . . . , v_2n) ^t, then the controlled response system (with subscript r) is given by: ${\dot{w}}_{r} = {Aw}_{r} + f {(w}_{r}) + L .$ (4) When the real and the imaginary parts of (3) and (4) are departed, one gets real systems of higher dimensions. Consequently, drive and response systems in real forms are defined as: ${\dot{u}}_{d} = {Mu}_{d} + g {(u}_{d}),$ (5) and ${\dot{u}}_{r} {= Mu}_{r} + g {(u}_{r}) + v,$ (6) where u = (u₁, u₂, . . . , u_2n) ^t, v = (v₁, v₂, . . . , v_2n) ^t $M \in ℝ^{2 n \times 2 n}$ are matrix and vector of system parameters and $g : ℝ^{2 n} \to ℝ^{2 n}$ is a nonlinear real function. Hyperchaotic systems’ active control problem is designing a controller v. We subtract (5) from (6) to get (7): ${\dot{e}}_{u} = M {[u}_{r} - u_{d}] + [g {(u}_{r}) - g {(u}_{d})] + v .$ (7) System (7) can be rewritten as: ${\dot{e}}_{u} = {M (e}_{u} {) + [g (u}_{r} {) - g (u}_{d})] + v,$ (8) where e_u = u_r - u_d (error functions) and g (u_r ) - g (u_d ) are a nonlinear real functions which could not form the error functions. Selecting controllers as s = (s₁, s₂, . . . , s_2n) ^t = v + [g (u_r ) - g (u_d )], the equation (8) turn out to be: ${\dot{e}}_{u} = {M (e}_{u}) + s .$ (9) Finally, the functions s must be determined. Lyapunov stability theory allows us to determine the functions s suitably, with the choice of s to make (9) in the form: $[\begin{matrix} {\dot{e}}_{u_{1}} \\ {\dot{e}}_{u_{2}} \\ ⋮ \\ ⋮ \\ {\dot{e}}_{u_{l}} \end{matrix}] = [\begin{matrix} μ_{1} & 0 & \dots & \dots & 0 \\ 0 & μ_{2} & 0 \\ ⋮ & ⋱ & ⋮ \\ ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & \dots & μ_{l} \end{matrix}] [\begin{matrix} e_{u_{1}} \\ e_{u_{2}} \\ ⋮ \\ ⋮ \\ e_{u_{l}} \end{matrix}],$ (10) where l = 1, 2, . . .2n . So, s take the form: $s = (μ_{l} I - M) e_{u},$ where $I \in ℝ^{2 n \times 2 n}$ is the unit matrix.

If all the eigenvalues μ_l = 0, e_u will clearly tend to constant values as t⟶ ∞, we might be suitable to get PS. As a result, the response system states and the drive system will be synchronized in step with each other asymptotically.

If we put the error functions in the form e_u = u_r + u_d, and repeat the previous steps but with choice μ_l < 0, e_{u
_l} will clearly tend to zero as t ⟶ ∞ , we may be able to achieve APS.

We can apply this scheme for hyperchaotic complex nonlinear systems in high dimensional.

3 Example: PS of system (1)

Now, we use Section 2 technique to review PS of the system (1) introduced in [35].

Therefore, systems of drive and response are specifically defined: $\begin{matrix} {\dot{x}}_{d} & = & α (y_{d} - x_{d}) + {iw}_{d}, \\ {\dot{y}}_{d} & = & γ y_{d} - x_{d} z_{d} + {iw}_{d}, \end{matrix}$ (11) $\begin{matrix} {\dot{z}}_{d} & = & 1 / 2 ({\bar{x}}_{d} y_{d} + x_{d} {\bar{y}}_{d}) - β z_{d}, \\ {\dot{w}}_{d} & = & 1 / 2 ({\bar{x}}_{d} y_{d} + x_{d} {\bar{y}}_{d}) - σ w_{d}, \end{matrix}$ and $\begin{matrix} \dot{x_{r}} & = & α (y_{r} - x_{r}) + {iw}_{r} + (v_{1} + {iv}_{2}), \\ \dot{y_{r}} & = & γ y_{r} - x_{r} z_{r} + {iw}_{r} + (v_{3} + {iv}_{4}), \end{matrix}$ (12) $\begin{matrix} \dot{z_{r}} & = & 1 / 2 ({\bar{x}}_{r} y_{r} + x_{r} {\bar{y}}_{r}) - β z_{r} + v_{5}, \\ \dot{w_{r}} & = & 1 / 2 ({\bar{x}}_{r} y_{r} + x_{r} {\bar{y}}_{r}) - σ w_{r} + v_{6}, \end{matrix}$ where x_d = u_1d + iu_2d, y_d = u_3d + iu_4d, z_d = u_5d, w_d = u_6d, x_r = u_1r + iu_2r, y_r = u_3r + iu_4r and z_r = u_5r, w_r = u_6r, “overbar” referrals complex conjugation, v₁ + iv₂, v₃ + iv₄ and v₅, v₆ are complex and real control functions, accordingly, which we need to determine. We can rewrite the complex system (11) in the form of six real first order ODEs: $\begin{matrix} {\dot{u}}_{1 d} & = & α (u_{3 d} - u_{1 d}), \\ {\dot{u}}_{2 d} & = & α (u_{4 d} - u_{2 d}) + u_{6 d}, \\ {\dot{u}}_{3 d} & = & γ u_{3 d} - u_{1 d} u_{5 d}, \end{matrix}$ (13) $\begin{matrix} {\dot{u}}_{4 d} & = & γ u_{4 d} - u_{2 d} u_{5 d} + u_{6 d}, \\ {\dot{u}}_{5 d} & = & u_{1 d} u_{3 d} + u_{2 d} u_{4 d} - β u_{5 d}, \\ {\dot{u}}_{6 d} & = & u_{1 d} u_{3 d} + u_{2 d} u_{4 d} - σ u_{6 d} . \end{matrix}$ And the response system (1) can be real-worldly rewritten as: $\begin{matrix} {\dot{u}}_{1 r} & = & α (u_{3 r} - u_{1 r}) + v_{1}, \\ {\dot{u}}_{2 r} & = & α (u_{4 r} - u_{2 r}) + u_{6 r} + v_{2}, \\ {\dot{u}}_{3 r} & = & γ u_{3 r} - u_{1 r} u_{5 r} + v_{3}, \end{matrix}$ (14) $\begin{matrix} {\dot{u}}_{4 r} & = & γ u_{4 r} - u_{2 r} u_{5 r} + u_{6 r} + v_{4}, \\ {\dot{u}}_{5 r} & = & u_{1 r} u_{3 r} + u_{2 r} u_{4 r} - β u_{5 r} + v_{5}, \\ {\dot{u}}_{6 r} & = & u_{1 r} u_{3 r} + u_{2 r} u_{4 r} - σ u_{6 r} + v_{6} . \end{matrix}$ We set the errors within the drive and the response systems to gain the control signs as:

$\begin{matrix} e_{u_{1}} + {ie}_{u_{2}} & = x_{r} - x_{d} = (u_{1 r} - u_{1 d}) + i (u_{2 r} - u_{2 d}), \\ e_{u_{3}} + {ie}_{u_{4}} & = y_{r} - y_{d} = (u_{3 r} - u_{3 d}) + i (u_{4 r} - u_{4 d}), \end{matrix}$ (15) $\begin{matrix} e_{u_{5}} & = z_{r} - z_{d} = u_{5 r} - u_{5 d}, \\ e_{u_{6}} & = w_{r} - w_{d} = u_{6 r} - u_{6 d} . \end{matrix}$

Subtracting (11) from (12) to get:

$\begin{matrix} {\dot{e}}_{u_{1}} + i {\dot{e}}_{u_{2}} = & α [(e_{u_{3}} - e_{u_{1}}) + i (e_{u_{4}} - e_{u_{2}})] \\ + {ie}_{u_{6}} + (v_{1} + {iv}_{2}), \\ {\dot{e}}_{u_{3}} + i {\dot{e}}_{u_{4}} = & γ (e_{u_{3}} + {ie}_{u_{4}}) - u_{5 r} (u_{1 r} + {iu}_{2 r}) \\ + u_{5 d} (u_{1 d} + {iu}_{2 d}) + {ie}_{u_{6}} + (v_{3} + {iv}_{4}), \end{matrix}$ (16) $\begin{matrix} {\dot{e}}_{u_{5}} = & - β e_{u_{5}} - (u_{1 d} u_{3 d} + u_{2 d} u_{4 d}) \\ + (u_{1 r} u_{3 r} + u_{2 r} u_{4 r}) + v_{5}, \\ {\dot{e}}_{u_{6}} = & - σ e_{u_{6}} - (u_{1 d} u_{3 d} + u_{2 d} u_{4 d}) \\ + (u_{1 r} u_{3 r} + u_{2 r} u_{4 r}) + v_{6} . \end{matrix}$

Equation (16) actually pictures a dynamic system through which "errors" adapt in time and ultimately this system’s ODEs become in the real form:

$\begin{matrix} {\dot{e}}_{u_{1}} = & α (e_{u_{3}} - e_{u_{1}}) + v_{1}, {\dot{e}}_{u_{2}} = α (e_{u_{4}} - e_{u_{2}}) + e_{u_{6}} + v_{2}, \\ {\dot{e}}_{u_{3}} = & γ e_{u_{3}} + u_{5 d} u_{1 d} - u_{5 r} u_{1 r} + v_{3}, \\ {\dot{e}}_{u_{4}} = & γ e_{u_{4}} + u_{5 d} u_{2 d} - u_{5 r} u_{2 r} + e_{u_{6}} + v_{4}, \end{matrix}$ (17) $\begin{matrix} {\dot{e}}_{u_{5}} = & - β e_{u_{5}} - (u_{1 d} u_{3 d} + u_{2 d} u_{4 d}) + (u_{1 r} u_{3 r} + u_{2 r} u_{4 r}) + v_{5}, \\ {\dot{e}}_{u_{6}} = & - σ e_{u_{6}} - (u_{1 d} u_{3 d} + u_{2 d} u_{4 d}) + (u_{1 r} u_{3 r} + u_{2 r} u_{4 r}) + v_{6} . \end{matrix}$

According to section 2 we define:

$\begin{matrix} s_{1} = & v_{1}, s_{2} = v_{2}, s_{3} = u_{5 d} u_{1 d} - u_{5 r} u_{1 r} + v_{3}, \\ s_{4} = & u_{5 d} u_{2 d} - u_{5 r} u_{2 r} + v_{4}, \end{matrix}$ (18) $\begin{matrix} s_{5} = & - (u_{1 d} u_{3 d} + u_{2 d} u_{4 d}) + (u_{1 r} u_{3 r} + u_{2 r} u_{4 r}) + v_{5}, \\ s_{6} = & - (u_{1 d} u_{3 d} + u_{2 d} u_{4 d}) + (u_{1 r} u_{3 r} + u_{2 r} u_{4 r}) + v_{6} . \end{matrix}$

So, the system (17) is going to take shape: $\begin{matrix} {\dot{e}}_{u_{1}} & = & α (e_{u_{3}} - e_{u_{1}}) + s_{1}, \\ {\dot{e}}_{u_{2}} & = & α (e_{u_{4}} - e_{u_{2}}) + e_{u_{6}} + s_{2}, \\ {\dot{e}}_{u_{3}} & = & γ e_{u_{3}} + s_{3}, \end{matrix}$ (19) $\begin{matrix} {\dot{e}}_{u_{4}} & = & γ e_{u_{4}} + e_{u_{6}} + s_{4}, \\ {\dot{e}}_{u_{5}} & = & - β e_{u_{5}} + s_{5}, {\dot{e}}_{u_{6}} = - σ e_{u_{6}} + s_{6} . \end{matrix}$ To form system (19) as (10) we find s_j,j = 1, . . . , 6 as: $[\begin{matrix} s_{1} \\ s_{2} \\ s_{3} \\ s_{4} \\ s_{5} \\ s_{6} \end{matrix}] = [\begin{matrix} {α + μ}_{1} & 0 & - α & 0 & 0 & 0 \\ 0 & {α + μ}_{2} & 0 & - α & 0 & - 1 \\ 0 & 0 & {- γ + μ}_{3} & 0 & 0 & 0 \\ 0 & 0 & 0 & {- γ + μ}_{4} & 0 & - 1 \\ 0 & 0 & 0 & 0 & {β + μ}_{5} & 0 \\ 0 & 0 & 0 & 0 & 0 & {σ + μ}_{6} \end{matrix}] [\begin{matrix} e_{u_{1}} \\ e_{u_{2}} \\ e_{u_{3}} \\ e_{u_{4}} \\ e_{u_{5}} \\ e_{u_{6}} \end{matrix}] .$ (20) Finally, we can determine and find v_j by using (18) and (20) as:

$\begin{matrix} v_{1} = & (μ_{1} + α) e_{u_{1}} - α e_{u_{3}}, v_{2} = (μ_{2} + α) e_{u_{2}} - α e_{u_{4}} - e_{u_{6}}, \\ v_{3} = & (μ_{3} - γ) e_{u_{3}} - u_{5 d} u_{1 d} + u_{5 r} u_{1 r}, \\ v_{4} = & (μ_{4} - γ) e_{u_{4}} - e_{u_{6}} - u_{5 d} u_{2 d} + u_{5 r} u_{2 r}, \end{matrix}$ (21) $\begin{matrix} v_{5} = & (μ_{5} + β) e_{u_{5}} + (u_{1 d} u_{3 d} + u_{2 d} u_{4 d}) - (u_{1 r} u_{3 r} + u_{2 r} u_{4 r}), \\ v_{6} = & (μ_{6} + σ) e_{u_{6}} + (u_{1 d} u_{3 d} + u_{2 d} u_{4 d}) - (u_{1 r} u_{3 r} + u_{2 r} u_{4 r}) . \end{matrix}$

When we pick μ_k = 0, the error statements are affirmed in support of the Lyapunov stability theory asymptotically stable (e_{u
_k} =constant values, k = 1, . . . , 6).

To ascertain the legitimacy of phrases (21), we are solving systems (11) and (12) with this controller (21) numerically (using e.g. Mathematica 7 software) for α = 14, β = 5, γ = 16.9 and σ = 10 with different initial conditions t₀ = 0, u_1d (0) = 1, u_2d (0) = 2, u_3d (0) = 3, u_4d (0) = 4, u_5d (0) = 5, u_6d (0) = 6 and u_1r (0) = -11, u_2r (0) = -12, u_3r (0) = -13, u_4r (0) = -14, u_5r (0) = -15, u_6r (0) = -16 and μ_k = 0 .

In Figure 1, the blue curves and the red curves respectively are the drive solutions and the response systems. The active control methodology is obviously a useful technique for achieving the PS of our system. Figure 2 shows PS errors and it is perfectly clear that the error shows constant values, j = 1, 2, . . . , 6 .

Fig.1

PS of systems (11) and (12) with (21) for α = 20, β = 5, γ = 40, σ = 13 with different initial conditions t₀ = 0, u_1d (0) = 1, u_2d (0) = 2, u_3d (0) = 3, u_4d (0) = 4, u_5d (0) = 5, u_6d (0) = 6 and u_1r (0) = -11, u_2r (0) = -12, u_3r (0) = -13, u_4r (0) = -14, u_5r (0) = -15, u_6r (0) = -16 and μ₁ = . . . = μ₆ = 0 . (a) u_1d (t) and u_1r (t) versus t, (b) u_2d (t) and u_2r (t) versus t, (c) u_3d (t) and u_3r (t) versus t, (d) u_4d (t) and u_4r (t) versus t, (e) u_5d (t) and u_5r (t) versus t, (f) u_6d (t) and u_6r (t) versus t (t =time/10).

Fig.2

PS errors (solutions of system (17)): (a) (e_{u
₁}, t) diagram, (b) (e_{u
₂}, t) diagram, (c) (e_{u
₃}, t) diagram, (d) (e_{u
₄}, t) diagram, (e) (e_{u
₅}, t) diagram, (f) (e_{u
₆}, t) diagram (t =time/10).

We sketch hyperchaotic drive and response attractors in Figures 3a, b and c. Figure 3a, the attractor in (u_5d, u_6d) plane of the drive system (11) and Figure 3b, the attractor in (u_5r, u_6r) plane of the response system (12). It is clear from Figures 3a,b that, the attractors of the drive and the response systems have the equivalent size and shape but different in abscissas. So, PS is achieved by using this technique and this is ensured by Figure 3c in (u₁, u₅, u₆) space.

Fig.3

PS with the same parameters, initial conditions and eigenvalues as in Fig. 1 (a) Phase space of hyperchaotic attractor of the drive system (11) in (u_5d, u_6d) plane (b) Phase space of the response system (12) in (u_5r, u_6r) plane (c) Phase spaces of the drive and the response systems in (u₁, u₅, u₆) space.

In Figures 4a, b, respectively, we plot the amplitudes of systems (11) and (12) (in u₅ - u₆ plane). Then one really can calculate the amplitudes in u₅ - u₆ plane of systems (11) and (12) by exploiting $A_{56} = \sqrt{u_{5}^{2} + u_{6}^{2}}$ . In Figure 4c we plotted the relation between $A_{56}^{d}$ (amplitude of the drive system in u₅ - u₆ plane) and $A_{56}^{r}$ (amplitude of the response system in u₅ - u₆ plane), this relation indicates that the amplitudes remain uncorrelated and hyperchaotic. The phases of systems (11) and (12) are shown in Figs. 4d, e (in u₅ - u₆ plane), respectively. We calculated $φ_{56}^{d}$ (phase of the drive system in u₅ - u₆ plane) and $φ_{56}^{r}$ (phase of the response system in u₅ - u₆ plane) by using $φ_{56} (t) = {tan}^{- 1} (\frac{u_{5}}{u_{6}})$ . In Fig. 4f, the phase difference between $φ_{56}^{d}$ and $φ_{56}^{r}$ is determined, and it is clear that $φ_{56}^{d} - φ_{56}^{r}$ is bounded, and this implies that PS is achieved. We can calculate the amplitudes, phases and phase difference of systems (11) and (12) in another planes as we didpreviously.

Fig.4

PS with the same parameters, eigenvalues and initial conditions as in Fig. 1 (a) $A_{d}^{56}$ (amplitude of the drive system in u₅ - u₆ plane) versus t (t =time/10) (b) $A_{r}^{56}$ (amplitude of the response system in u₅ - u₆ plane) versus t (t =time/10) (c) The relation between $A_{d}^{56}$ and $A_{r}^{56}$ (d) $φ_{d}^{56}$ (phase of the drive system in u₅ - u₆ plane) versus t (t =time/10) (e) $φ_{r}^{56}$ (phase of the response system in u₅ - u₆ plane) versus t (t =time/10) (f) Phase difference $(φ_{d}^{56} - φ_{r}^{56})$ versus t (t =time/10).

At long last, from Figs. 1-4 one can say that, in the phase synchronization of the hyperchaotic complex nonlinear frameworks, the contrast between the phase of the drive’s state variable and that of the reaction’s is steady amid cooperation while the amplitudes of their state variables advanceuninhibitedly.

4 APS of system (1) via proposed technique

Let us now study APS in the system (1) employing the process of active control method and Lyapunov stability examination as follows: we consider the same drive and response systems as Section 3, we define the error functions as:

$\begin{matrix} e_{u_{1}} + {ie}_{u_{2}} = & x_{r} + x_{d} = (u_{1 r} + u_{1 d}) + i (u_{2 r} + u_{2 d}), \\ e_{u_{3}} + {ie}_{u_{4}} = & y_{r} + y_{d} = (u_{3 r} + u_{3 d}) + i (u_{4 r} + u_{4 d}), \\ e_{u_{5}} = & z_{r} + z_{d} = u_{5 r} + u_{5 d}, \\ e_{u_{6}} = & w_{r} + w_{d} = u_{6 r} + u_{6 d} . \end{matrix}$ (22)

Using similar calculations as in Section 3, we choose again the active control function v_i as:

$\begin{matrix} v_{1} = & (μ_{1} + α) e_{u_{1}} - α e_{u_{3}}, v_{2} = (μ_{2} + α) e_{u_{2}} - α e_{u_{4}} - e_{u_{6}}, \\ v_{3} = & (μ_{3} + - γ) e_{u_{3}} + u_{5 d} u_{1 d} + u_{5 r} u_{1 r}, \\ v_{4} = & (μ_{4} - γ) e_{u_{4}} - e_{u_{6}} + u_{5 d} u_{2 d} + u_{5 r} u_{2 r}, \end{matrix}$ (23) $\begin{matrix} v_{5} = & (μ_{5} + β) e_{u_{5}} - (u_{1 d} u_{3 d} + u_{2 d} u_{4 d}) - (u_{1 r} u_{3 r} + u_{2 r} u_{4 r}), \\ v_{6} = & (μ_{6} + σ) e_{u_{6}} - (u_{1 d} u_{3 d} + u_{2 d} u_{4 d}) - (u_{1 r} u_{3 r} + u_{2 r} u_{4 r}) . \end{matrix}$

When we pick μ_j < 0, j = 1, 2, . . . , 6 and based on the sense of Lyapunov’s stability, error states are asymptotically stable, meaning e_{u
_j} (t)→ zero as t→ ∞. Next, the controller (23) with systems (11) and (12) are solved numerically for μ₁ = . . . = μ₆ = -1 but with the similar initial conditions and parameters as in Fig 1. The outcomes of the simulation are given in Figs. 5, 6 and 7. The explications in Fig. 5, the solutions of (11) and (12) with (23) are planned under various initial conditions. It reveals that for very small values of t (t = time/10) the APS is performed and we see that the drive variables and the response systems are symmetrical concerning the horizontal axis. Figure 6 shows that the APS errors meet to zero as t goes to infinity. Figure 7 shows that the drive and response systems ’ hyperchaotic attractors have the inverse pattern, and we demonstrate this in two and three dimensions.

Fig.5

APS of systems (11) and (12) with (23) for μ₁ = . . . = μ₆ = -1 and the same parameters and initial conditions as in Figure 1. (a) u_1d (t) and u_1r (t) versus t, (b) u_2d (t) and u_2r (t) versus t, (c) u_3d (t) and u_3r (t) versus t, (d) u_4d (t) and u_4r (t) versus t, (e) u_5d (t) and u_5r (t) versus t, (f) u_6d (t) and u_6r (t) versus t (t =time/10).

Fig.6

APS errors: (a) (e_{u
₁}, t) diagram, (b) (e_{u
₂}, t) diagram, (c) (e_{u
₃}, t) diagram, (d) (e_{u
₄}, t) diagram, (e) (e_{u
₅}, t) diagram, (f) (e_{u
₆}, t) diagram (t =time/10).

Fig.7

APS with the same parameters, initial conditions and eigenvalues as in Figure 5 (a) Phase space of the drive system (11) in (u_5d, u_6d) plane (b) Phase space of the response system (12) in (u_5r, u_6r) plane (c) Phase spaces of the drive and the response systems in (u₁, u₅, u₆) space.

As we did in Fig. 4, it is possible to determine the amplitudes, phases and phase difference of the drive and response systems in APS state. We remark the same shape of the drive amplitudes and response systems (Figs. 8 a, b). In Fig. 8 c, we plot the relation between A_d (amplitude of the drive system in u₅ - u₆ plane) and A_r (amplitude of the response system in u₅ - u₆ plane) and refined this relation is linear and this apprehension shows the equivalent amplitude of systems (11) and (12). Phase difference converge to zero as in Fig. 8f and this indicates that phases of systems (11) and (12) have the same values versus t (Figs. 8 d, e). So, we can say that, the amplitudes, phases and phase difference of the drive and the response systems in the situation of APS do not have the same peculiarities and features such as state of PS.

Fig.8

APS with the same parameters, eigenvalues and initial conditions as in Figure 5 (a) $A_{d}^{56}$ (amplitude of the drive system in u₅ - u₆ plane) versus t (t =time/10) (b) $A_{r}^{56}$ (amplitude of the response system in u₅ - u₆ plane) versus t (t =time/10) (c) The relation between $A_{d}^{56}$ and $A_{r}^{56}$ (d) $φ_{d}^{56}$ (phase of the drive system in u₅ - u₆ plane) versus t (t =time/10) (e) $φ_{r}^{56}$ (phase of the response system in u₅ - u₆ plane) versus t (t =time/10) (f) Phase difference $(φ_{d}^{56} - φ_{r}^{56})$ versus t (t =time/10).

5 Secure communication via phase synchronization results

In the field of safe and secure communication, various efforts were made to discuss the subject of information encryption and decoding in order to expand communication security [42, 43]. Considering that the dynamic appearance of a hyperchaotic model is sensitive to the initial values and parameters of a model, the advantage of a hyperchaotic model to ensure communication has been given significant consideration. Due to the fruitful use of chaos and hyperchaos synchronizations, various considerationsand techniques have been suggested in the field of secure communication over the last two decades [39]. These approaches depends on the signal hiding (message) transmitted hyperchaotic [40]. At the end of the collector, the hyperchaotic model is controlled to recover the message transmitted [41 , 44–47]. In secure communication, hyperchaotic signal secures the data to be transmitted. The after-effects of the new model’s phase synchronization were exploited in this section by using a basic plan (see Figure) to achieve an application in secure communications. The master model (11) was considered as the model of the transmitter. The slave model (12) as the model of the receiver. The transmitter model’s message r(t) and hyperchaotic signal are hurried by invertible nonlinear function methods Θ = ρ (r (t) , u_1d, u_2d, u_3d, u_4d, u_5d, u_6d) . Then join the signal or flag r (t) to (at least one) of the six factors u_1d, u_2d, u_3d, u_4d, u_5d, u_6d for example, injecting it in the u_5d variable. Thus, the successfully completed signal is $\bar{r} (t) = Θ + u_{5 d}$ . At this juncture, the transmitter model hyperchaotic signs and the combined signal is sent to the collector’s side. The controller can be formed in the recipient by equation (21). In addition to constant values, the phase synchronization among pair indistinguishable hyperchaotic structures (11), (12) will be achieved and the values of the slave model are the same as the master model evaluations with constant shift, for instance u_5s = u_5d - 20. These constant values are based on the selection for the initial values of hyperchaotic systems (11), (12). The collector starts recovering r(t) by a simple change $Θ = \bar{r} (t) -$ u_5s - 20. In obvious conclusion, the message signal can be recovered as the nonlinear function is invertible r^∗ (t) = ρ^-1 (u_1r - 12, u_2r - 14, u_3r - 16, u_4r - 18, u_5r - 20, u_6r - 22, Θ) .

In the numerical simulations below, The model parameters and initial requirements of transmitter and receiver descriptions shall be deemed to be indistinguishable from those set out in subsection (3.3). The invertible function as Θ = u_2d + r (t); r (t) = 2 sin(3πt) is observed and anticipated that the Θ flag or signal is actually joined to the u_5d variable $\Rightarrow \bar{r} (t) = u_{2 d} + r (t) + u_{5 d}$ . Figure 7 presents the numerical arrangements concerning the advantage of chaotic synchronization in secure and safe communication. The messages r (t) and $\bar{r} (t)$ are characterized in Figs. 7a and 7b, independently. Figure 7c depicts the recouped message $r^{*} (t) = \bar{r} (t) - u_{2 d} - 14 - u_{5 d} - 20$ . Figure 7d highlights the error within the first and the recovered message. The data signal r(t) is recovered after a short transient is shown from Fig. 7d.

We can declare that secure communication has associated interesting features in full information of phase synchronization:

1- High data transfer security since this type of synchronization is based on constant values resulting from the selection of the initial values of the transmitter and receiver models. In previous investigations, this segment was not presented in secure or safe communications.

2- Earlier communication strategies for security are based on the transmitter and receiver models’ state variables. As it may be, here transmitter and collector models’ initial values have an effective part to play in performing a secure connection. This case leads to increased data transmission security.

Fig.9

A plan to realize secure communication based on PS results.

6 Conclusions

Over the past 20 years, extensive research has been carried out into PS and APS aspects reported by real variables on chaotic or hyperchaotic systems. Our main objective in this paper is to analyze the nonlinear hyperchaotic systems PS and APS phenomena including complex variables that have not been explored as active. A scheme is planned to explore these phenomena. This scheme implements as an example of two identical hyperchaotic complexes Lü systems of and other complex examples can be studied similarly. Hyperchaotic complex systems PS and APS are defined. We also defined the hyperchaotic attractor’s amplitude and phase on the projection level. The Lü hyperchaotic complex system is a 6-dimensional real first-order differential equation, therefore its solutions and attractors hold multiple level projections. In our Figures, we outlined fascinating of these projections and the other concluded the same. The excellent arrangement has been obtained with complex and actual control function in equations (21), (23) and PS and APS were achieved quickly.

Fig.10

(a) The original or initial message r (t). (b) The transmitted signal $\bar{r} (t)$ . (c) The recovered message r^∗ (t). (d) The error signal r (t) - r^∗ (t).

The events of PS were employed for the first time to create a secure application for communications. Secure communication in the information of phase synchronization has some highlights and does not live in writing, for instance, organizational security depends on selecting the master and slave models’ initial values.

Lastly, our events show that this paper’s scheme is effective and widely applicable to complex nonlinear systems displaying hyperchaotic (or chaotic) dynamics.

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