# Complex Modified Hybrid Projective Synchronization of Different Dimensional Fractional-Order Complex Chaos and Real Hyper-Chaos

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

_{i}. Liu et al. [14] introduced modified generalized projective synchronization (MGPS) of fractional-order chaotic systems with different structures, where the drive and response systems could be asymptotically synchronized up to a desired non-diagonal transformation matrix.

^{jγ}η, where ρe

^{jγ}= ρ(cos γ + j sin γ), ζ and η denote the complex state variables of drive and response systems, respectively, ρ > 0 denotes the zoom rate and γ ∈ [0, 2π) denotes the rotate angle. In addition, different dimensional drive and response systems could be synchronized in practical applications [19]. Modified hybrid projective synchronization with complex state transformation matrix Θ = Θ

^{r}+ jΘ

^{i}(CMHPS) considers both different dimensions and the complex scaling factors. By means of complex state transformation, every state variable in a response system will be involved in multiple state variables of the drive system, which will increase the complexity of the synchronization and further increase the diversity and the security of communications [20]. Therefore, it is interesting and significant to study CMHPS of different dimensional fractional-order complex chaos and real hyper-chaos. Up till now, to the best of my knowledge, all of the works involved in complex scaling factors focus on the integer-order complex chaotic systems, and there is almost no paper about this type of CMHPS for fractional-order chaotic systems.

^{n}and ℂ

^{n}stand for n dimensional real and complex vector space, respectively. If z is a complex vector (or complex number), then $z={z}^{r}+j{z}^{i},j=\sqrt{-1}$ is the imaginary unit, superscripts r and i stand for the real and imaginary parts of z, z

^{T}and $\overline{z}$ are the transpose and the complex conjugate of z, respectively. ‖z‖ implies the two-norm of z, defined by $\Vert z\Vert =\sqrt{{z}^{T}\overline{z}}$.

^{α}denotes the Riemann–Liouville-type fractional integral of order α, D

^{α}denotes the Riemann–Liouville-type fractional derivative of order α, ${D}_{*}^{\alpha}$ denotes the Caputo-type fractional derivative of order α, Γ(·) and denotes the gamma function $\mathrm{\Gamma}(x)={\displaystyle {\int}_{0}^{\infty}{t}^{x-1}{e}^{-t}dt,x>0}$.

## 2. Preliminaries

#### 2.1. The Definition of Fractional Derivative

^{β}is the β-order Riemann-Liouville integral operator as described by:

#### 2.2. The Stability of Fractional-Order Systems

_{0}, where 0 < α < 1 and x ∈ ℝ

^{n}, A is a constant matrix.

**Lemma 1.**System (4) is:

- Asymptotically stable if and only if:$$\left|\mathrm{arg}({\lambda}_{\ell}(A))\right|>\frac{\alpha \pi}{2},\phantom{\rule{0.5em}{0ex}}(\ell =1,2,\cdots ,n),$$
_{ℓ}(A)) denotes the argument of the eigenvalue λ_{ℓ}of A. In this case, each component of the states decays toward zero like t^{−α}. - Stable if and only if:$$\left|\mathrm{arg}({\lambda}_{\ell}(A))\right|\ge \frac{\alpha \pi}{2},\phantom{\rule{0.5em}{0ex}}(\ell =1,2,\cdots ,n),$$
_{i}that satisfy |arg(λ_{ℓ}(A))| = απ/2 (ℓ = 1, 2,⋯, n), have geometric multiplicity one.

## 3. CMHPS Scheme of a Different Dimensional Fractional-Order Real Hyper-Chaotic (Chaotic) Drive System and Complex Chaotic Response System

#### 3.1. Mathematical Model and Problem Descriptions

_{1}, y

_{2},⋯, y

_{n})

^{T}∈ ℝ

^{n}is a real state vector, z = z

^{r}+ jz

^{i}ℂ

^{m}is a complex state vector, C ∈ ℝ

^{n}

^{×}

^{n}and P ∈ ℝ

^{m}

^{×}

^{m}are the coefficient matrices of y and z, while h = (h

_{1}, h

_{2},⋯, h

_{n})

^{T}and Φ = (ϕ

_{1}, ϕ

_{2},⋯, ϕ

_{m})

^{T}are the nonlinear parts, respectively, and v = (v

_{1}, v

_{2},⋯, v

_{m})

^{T}is the controller to be designed.

**Definition 1.**For the fractional-order complex chaotic drive system (7) and response system (8), it is said to be CMHPS with Θ = Θ

**+ jΘ**

^{r}^{i}between z(t) and y(t), if there exists a complex controller v = v

^{r}+ jv

^{i}∈ ℂ

^{m}, such that:

^{m}

^{×}

^{n}is defined as a complex transformation matrix of the fractional-order real hyper-chaotic (chaotic) drive system (7).

**Remark 1.**Lots of classical fractional-order real hyper-chaotic (chaotic) systems can be formed as system (7), such as the fractional-order real Chua system [7], the fractional-order real hyper-chaotic Rössler system [8], the fractional-order real Liu system [9] and other fractional-order real Lorenz-like systems [10]. Lots of classical fractional-order complex chaotic systems can be formed as system (8), such as fractional-order complex Lorenz system [15] and fractional-order complex Chen system [16].

**Remark 2.**Several types of synchronization are special cases of CMHPS, such as complex generalized projective synchronization (CGPS), complex projective synchronization (CPS), modified hybrid projective synchronization (MHPS), modified generalized projective synchronization (MGPS), generalized projective synchronization (GPS), projective synchronization (PS), complete synchronization (CS), anti-synchronization (AS); see Table 1.

#### 3.2. General Method of CMHPS

**Theorem 1.**Given complex transformation matrix Θ = Θ

^{r}+ jΘ

^{i}and initial conditions y(0), z(0), if the complex controller is designed as:

^{m}

^{×}

^{m}is the control gain matrix.

**Proof.**Equation (10) can be written as:

^{n}

^{×}

^{n}is the control gain matrix. That is, $\underset{t\to +\infty}{\mathrm{lim}}\Vert {\delta}^{r}(t)\Vert =0$, and $\underset{t\to +\infty}{\mathrm{lim}}\Vert {\delta}^{i}(t)\Vert =0$. Therefore, $\underset{t\to +\infty}{\mathrm{lim}}\Vert \delta (t)\Vert =0$; CMHPS between the systems (8) and (7) is realized. This completes the proof. □

## 4. CMHPS Scheme of Different Dimensional Fractional-Order Complex Chaotic Drive Systems and Real Hyper-Chaotic (Chaotic) Response Systems

#### 4.1. Mathematical Model and Problem Descriptions

^{r}+ jw

^{i}∈ ℂ

^{n}is the complex state vector, x = (x

_{1}, x

_{2},⋯, x

_{m})

^{T}∈ ℝ

^{m}is the real state vector, Q ∈ ℝ

^{n}

^{×}

^{n}and B ∈ ℝ

^{m}

^{×}

^{m}are the coefficient matrices of w and x, while Ψ = (h

_{1}, h

_{2},⋯, h

_{n})

^{T}and p = (p

_{1}, p

_{2},⋯, p

_{m})

^{T}are the nonlinear parts and v = (v

_{1}, v

_{2},⋯, v

_{m})

^{T}is the controller to be designed, respectively.

^{r}+ jΘ

^{i}between systems (17) and (16) is defined as:

#### 4.2. General Method of CMHPS

**Theorem 2.**Given complex transformation matrix Θ = Θ

^{r}+ jΘ

^{i}and initial conditions w(0), x(0), if the designed controller is real as:

^{m}

^{×}

^{m}is the control gain matrix.

**Proof.**Substituting Equation (16) and Equation (17) into Equation (18), one can get the derivative of the error system:

^{n}

^{×}

^{n}is the control gain matrix. That is, $\underset{t\to +\infty}{\mathrm{lim}}\Vert \delta (t)\Vert =0$; CMHPS between the fractional-order real hyper-chaotic (chaotic) response system (17) and fractional-order complex chaotic drive system (16) is realized. This completes the proof. □

## 5. CMHPS Scheme of Different Dimensional Fractional-Order Complex Chaotic Systems

^{r}+ jΘ

^{i}is defined as:

**Theorem 3.**Given complex transformation matrix Θ = Θ

^{r}+ jΘ

^{i}and initial conditions w(0), z(0), if the complex controller is designed as:

^{m}

^{×}

^{m}is the control gain matrix.

**Proof.**This is similar to the proof in Theorem 1 and, thus, is omitted. □

## 6. Numerical Examples

#### 6.1. Reduced Order CMHPS

_{1}, y

_{2}, y

_{3}, y

_{4})

^{T}∈ ℝ

^{4}is real state vector. The system (25) is hyper-chaotic when c

_{1}= 0.32, c

_{2}= 3, c

_{3}= 0.5, c

_{4}= 0.05, α = 0.95 in Figure 1; see [8] for more details.

_{3}is a real state variable. The system (26) is chaotic when p

_{1}= 35, p

_{2}= 28, p

_{3}= 3, α = 0.95 and in the absence of the controller v = v

^{r}+ jv

^{i}in Figure 2; see [16] for more details.

_{y}(t) is obtained as:

_{1}= 0.32, c

_{2}= 3, c

_{3}= 0.5, c

_{4}= 0.05 and p

_{1}= 35, p

_{2}= 28, p

_{3}= 3, respectively. The initial values are randomly chosen as y

_{0}= (−10, −6, 0, 10)

^{T}and ${z}_{0}={z}_{0}^{r}+j{z}_{0}^{i}={(7+4j,1+6j,2)}^{T}$, respectively. Therefore, all of the eigenvalues of P − K are λ

_{1}= −1 − j, λ

_{2}= −1 + j, λ

_{3}= −3, which satisfies $\left|\mathrm{arg}({\lambda}_{\ell}(P-K))\right|>\frac{\alpha \pi}{2}$, (ℓ = 1, 2, 3). The simulation results are demonstrated in Figure 3, where the blue line presents the states of drive system (25) and the red (pink) line presents the real (imaginary) parts of the states in the response system (26). The errors of CMHPS converge asymptotically to zero as in Figure 4. Hence, CMHPS has been achieved between fractional-order real hyperchaotic Rössler drive system (25) and fractional-order complex chaotic Chen response system (26).

#### 6.2. Increased Order CMHPS

_{3}is real state variable. The system (29) is chaotic when q

_{1}= 10, q

_{2}= 180, ${q}_{3}=\frac{8}{3}$, α = 0.95 in Figure 5; see [15] for more details.

_{1}, x

_{2}, x

_{3}, x

_{4})

^{T}∈ ℝ

^{4}is real state vector and v = (v

_{1}, v

_{2}, v

_{3}, v

_{4})

^{T}is the controller to be designed.

^{r}w

^{r}(t) + Θ

^{i}w

^{i}(t) is obtained as:

_{1}= 10, q

_{2}= 180, ${q}_{3}=\frac{8}{3}$ and b

_{1}= 0.32, b

_{2}= 3, b

_{3}= 0.5, b

_{4}= 0.05, respectively. The initial values are randomly selected as ${w}_{0}={w}_{0}^{r}+j{w}_{0}^{i}=(2+3j,5+6j,9)$ and x

_{0}= (−10, −6, 0, 10)

^{T}, respectively. Therefore, all of the eigenvalues of B − K are λ

_{1}= −1 + j, λ

_{2}= −1 − j, λ

_{3}= −2, λ

_{4}= −3, which satisfies $\left|\mathrm{arg}({\lambda}_{\ell}(B-K))\right|>\frac{\alpha \pi}{2}$, (ℓ = 1, 2, 3, 4). The simulation results are demonstrated in Figure 6, where the blue line presents the states of response system (30) and the red (pink) line presents the real (imaginary) parts of the states in the drive system (29). The errors of CMHPS converge asymptotically to zero as in Figure 7. Hence, CMHPS has been achieved between three-dimensional fractional-order complex chaotic Lorenz drive system (29) and four-dimensional fractional-order real hyper-chaotic Rössler response system (30).

## 7. Conclusions

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The hyper-chaotic attractor of the fractional-order real Rössler system (25) for c

_{1}= 0.32, c

_{2}= 3, c

_{3}= 0.5, c

_{4}= 0.05, α = 0.95.

**Figure 2.**Chaotic attractor projections of fractional-order complex Chen system (26) for p

_{1}= 35, p

_{2}= 28, p

_{3}= 3, α = 0.95.

**Figure 3.**Reduced order synchronization-CMHPS between four-dimensional fractional-order real hyper-chaotic Rössler drive system (25) and three-dimensional fractional-order complex chaotic Chen response system (26) with the controller (28). (

**a**) ${z}_{1}^{r}$ synchronizes y

_{1}; (

**b**) ${z}_{1}^{i}$ anti-synchronizes y

_{1}; (

**c**) ${z}_{2}^{r}$anti-synchronizes y

_{1}; (

**d**) ${z}_{1}^{i}$synchronizes 2y

_{2}; (

**e**) z

_{3}synchronizes y

_{3}− y

_{4}.

**Figure 5.**The chaotic attractor projections of fractional-order complex Lorenz system (29) for q

_{1}= 10, q

_{2}= 180, ${q}_{3}=\frac{8}{3}$, α = 0.95.

**Figure 6.**Increased order synchronization-CMHPS between three-dimensional fractional-order complex chaotic Lorenz drive system (29) and four-dimensional fractional-order real hyper-chaotic Rössler response system (30) with the controller (32). (

**a**) x

_{1}synchronizes ${w}_{1}^{i}$; (

**b**) x

_{2}anti-synchronizes ${w}_{2}^{i}$; (

**c**) x

_{3}synchronizes 2w

_{3}; (

**d**) x

_{4}anti-synchronizes w

_{3}.

Settings the Matrix Θ | Synchronization Type |
---|---|

Θ=Θ^{r}+jΘ^{i}∈ℂ^{m}^{×}^{n},m≠n | CMHPS |

Θ=diag{θ_{1},θ_{2},⋯,θ_{n}}∈ℂ^{n}^{×}^{n},m=n | CGPS |

Θ=diag{θ,θ,⋯,θ}∈ℂ^{n}^{×}^{n},m=n | CPS |

Θ∈ℝ^{m}^{×}^{n},m≠n | MHPS |

Θ∈ℝ^{m}^{×}^{n},m=n | MGPS |

Θ=diag{θ_{1},θ_{2},⋯,θ }∈ℝ^{n}^{×}^{n},m=n | GPS |

Θ=diag{θ,θ,⋯,θ}∈ℝ^{n}^{×}^{n},m=n | PS |

Θ=diag{−1, −1,⋯, −1},m=n | CS |

Θ=diag{−1,−1,⋯,−1},m=n | AS |

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**MDPI and ACS Style**

Liu, J.
Complex Modified Hybrid Projective Synchronization of Different Dimensional Fractional-Order Complex Chaos and Real Hyper-Chaos. *Entropy* **2014**, *16*, 6195-6211.
https://doi.org/10.3390/e16126195

**AMA Style**

Liu J.
Complex Modified Hybrid Projective Synchronization of Different Dimensional Fractional-Order Complex Chaos and Real Hyper-Chaos. *Entropy*. 2014; 16(12):6195-6211.
https://doi.org/10.3390/e16126195

**Chicago/Turabian Style**

Liu, Jian.
2014. "Complex Modified Hybrid Projective Synchronization of Different Dimensional Fractional-Order Complex Chaos and Real Hyper-Chaos" *Entropy* 16, no. 12: 6195-6211.
https://doi.org/10.3390/e16126195