Complex Rayleigh-Van Der Pol-Duffing Oscillators: Dynamics, Phase and Anti-Phase Synchronization

1. Dynamics of complex RVDOs (3-4)

This section presents the characteristics and behavior of our suggested autonomous RVDO (3) and nonautonomous RVDO (4). These models are dissipative under the condition $eb<0$. It is clear that model (3) is not symmetric and model (4) is symmetric under the transformation $(z_1,z_2,z_3,z_4)$$⟶(-z_1,-z_2,-z_3,-z_4)$. Model RVDO (3) has not fixed points, while model (4) has three ones as: $E_0={(0,0,0,0)}^T$ and $E_{1,2}={(\pm \frac{1}{\sqrt{a}},0,0,0)}^T$ for $a>0$. To study the stability of $E_0={(0,0,0,0)}^T$, we calculate the Jacobian matrix of the model (4) at $E_0$ as:

$$\left(\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& 1\\ 1& 0& eb& 0\\ 0& 1& 0& eb\end{array}\right),$$

(5)

and its eigenvalues are: ${\mu }_{1,2}={\mu }_{3,4}=\frac{be\pm \sqrt{b^2{e}^2+4}}{2}$. It is clear that $E_0$ is not stable for any values of the parameters e and b. The fixed points $E_{1,2}$ can be similarly study as we did for $E_0$.

For the choice, $a=1.25$, $b=0.5$, $c=0.085$, $e=1,$$h=1$, $m=0.5$, $l=0.05,$$w=1$, $\mathsf{\Omega }=1$ and calculating the Lyapunov exponents [23] for the real autonomous RVDO (2), the complex autonomous RVDO(3) and the complex nonautonomous RVDO (4). We found those values are $({\lambda }_1=0.1556,{\lambda }_2=0,{\lambda }_3=-0.4198)$, $({\lambda }_{1,2}=0.028,{\lambda }_3=0,{\lambda }_4=-0.7289,{\lambda }_5=-0.6709)$, and $({\lambda }_{1,2}=0.0878,{\lambda }_3=-0.3291,{\lambda }_4=-5.1644)$, respectively. This means that the real autonomous RVDO (2) has chaotic solution, the complex autonomous RVDO(3) has hyperchaotic solution of order 2 and the complex nonautonomous RVDO (4) has hyperchaotic solution of order 2 as shown in Figure 1, Figure 2 and Figure 3.

In the following remarks, we will present the differences between models (2-4). Model (3) is considered a generalization of model (2). Model (3) is not symmetric and model (4) is symmetric. Model (4) has 3 fixed points, while model (3) does not. It is observed that the count of Lyapunov exponents in non-autonomous models is higher by one compared to autonomous models with the same dimensions. This difference arises due to the inclusion of time as an additional dimension in the non-autonomous models.

2. A scheme for PS and APS with different dimensions

In our plan, we make the assumption that there is a single master model and a single slave system. The master system represents:

$$\begin{array}{c}\hfill \dot{z}=f(z,t),\end{array}$$

(6)

while the slave model is:

$$\begin{array}{c}\hfill \dot{\zeta }=g(\zeta ,t)+u,\end{array}$$

(7)

where $z\in R^n$, $\zeta \in R^m$ are the state vectors of models and $u\in R^m$ is the control function which will be calculated. PS can be observed in the master system (6) as well as the slave system (7), provided that there exist two constant matrices $A\in R^{n\times n}$, and $B\in R^{n\times m}$ in the real matrix space, if

$$\begin{array}{c}\hfill \underset{x\to \infty }{lim}\parallel E\parallel =\underset{x\to \infty }{lim}\parallel B\zeta -Az\parallel =C,\end{array}$$

(8)

where $E={(e_1,e_2,...,e_n)}^T\in R^n$ is the error vector, and C is a constant vector. The APS for different dimensions can be obtained if one takes $A=-A$ in Eq. (8). If $\zeta in{R}^n$ and $A=B=I\in R^{n\times n}$, then the PS for different dimensions converted to PS with the same dimensions [14]. The modified PS with the same dimensions can be get from Eq. (8), if one choose $\zeta in{R}^n$ and $B\in R^{n\times n}$ [15].

We can achieve PS among two distinct models (6-7), if we design the vector of the control functions $U\in R^n$ in the following way:

$$\begin{array}{c}\hfill U\equiv Bu=Af(z,t)-Bg(\zeta ,t)-KE,\end{array}$$

(9)

where $K=diag(k_1,k_2,...,k_n)$ is the control gain diagonal matrix.

Proof. 

From Eq. (8), we have

$$\begin{array}{c}\hfill E=B\zeta -Az,\end{array}$$

(10)

by taking the derivative of Equation (10) with respect to time, we derive the following expression:

$$\begin{array}{c}\hfill \dot{E}=B\dot{\zeta }-A\dot{z},\end{array}$$

(11)

using Eqs. (6-7) and (9), we get

$$\begin{array}{c}\hfill \dot{E}=-KE,\end{array}$$

(12)

by utilizing Lyapunov stability analysis, if we assign K to be zero, Eq. (12) can be expressed in the following form $\dot{E}=0$. This means that $E⟶C$ as $t⟶\infty$, then PS is achieved. □

For the choice $k_i<0$ and using the stability theory, then $E⟶0$ as $t⟶\infty$. According to Remark 2, the APS between the models (6-7) can be achieved if the control functions take the form:

$$\begin{array}{c}\hfill U\equiv Bu=-Af(z,t)-Bg(\zeta ,t)-KE.\end{array}$$

(13)

The phase differences ${\Delta }_{ij}$ between the $\left({\displaystyle \sum _{k=1}^n}{A}_{ik}{z}_k,{\displaystyle \sum _{k=1}^n}{A}_{jk}{z}_k\right)$ projection for the master model and the $\left({\displaystyle \sum _{l=1}^m}{B}_{il}{\zeta }_l,{\displaystyle \sum _{l=1}^m}{B}_{jl}{\zeta }_l\right)$ projection for the slave one are described as follows:

$$\begin{array}{c}\hfill {\Delta }_{ij}={\psi }_{ij}^d-{\psi }_{ij}^r,\end{array}$$

(14)

where $i,j=1,2,3,...,n$, $i\ne j$, ${\psi }_{ij}^d=ta{n}^{-1}\left(\frac{{\displaystyle \sum _{k=1}^n}{A}_{jk}{z}_k}{{\displaystyle \sum _{k=1}^n}{A}_{ik}{z}_k}\right)$ and ${\psi }_{ij}^r=ta{n}^{-1}\left(\frac{{\displaystyle \sum _{l=1}^m}{B}_{jl}{\zeta }_l}{{\displaystyle \sum _{l=1}^m}{B}_{il}{\zeta }_l}\right)$. The occurrence of the PS is dependent on the bounded nature of these phase differences.

The definitions of the amplitudes in the $\left({\displaystyle \sum _{k=1}^n}{A}_{ik}{z}_k,{\displaystyle \sum _{k=1}^n}{A}_{jk}{z}_k\right)$ projection for the master model and the $\left({\displaystyle \sum _{l=1}^m}{B}_{il}{\zeta }_l,{\displaystyle \sum _{l=1}^m}{B}_{jl}{\zeta }_l\right)$ projection for the slave one are as follows:

$$\begin{array}{c}\hfill A_{ij}^d=\sqrt{{\left({\displaystyle \sum _{k=1}^n}{A}_{ik}{z}_k\right)}^2+{\left({\displaystyle \sum _{k=1}^n}{A}_{jk}{z}_k\right)}^2},\\ \hfill A_{ij}^r=\sqrt{{\left({\displaystyle \sum _{l=1}^m}{B}_{il}{\zeta }_l\right)}^2+{\left({\displaystyle \sum _{l=1}^m}{B}_{jl}{\zeta }_l\right)}^2}.\end{array}$$

(15)

Subsequently, we utilize this approach to examine the occurrence of PS in Section 3 and APS in Section 4 by applying it to hyperchaotic attractors (2-4).

3. Illustrative example for PS

In this section, we test the PS technique using an example. We consider the model (3) as the master model and the slave one is model (4). The slave model can be written after adding the control functions as:

$$\begin{array}{cc}\hfill {\dot{\zeta }}_1=& {\zeta }_3+u_1,\hfill \\ \hfill {\dot{\zeta }}_2=& {\zeta }_4+u_2,\hfill \\ \hfill {\dot{\zeta }}_3=& {\zeta }_1-a({{\zeta }_1}^3-3{\zeta }_1{{\zeta }_2}^2)+e\{b{\zeta }_3-c[({\zeta }_1^2-{\zeta }_2^2){\zeta }_3-2{\zeta }_1{\zeta }_2{\zeta }_4]-d[{{\zeta }_3}^3-3{\zeta }_3{{\zeta }_4}^2]\}+u_3,\hfill \\ \hfill {\dot{\zeta }}_4=& {\zeta }_2-a(-{{\zeta }_2}^3+3{\zeta }_1^2{\zeta }_2)e\{b{\zeta }_4-c[({\zeta }_1^2-{\zeta }_2^2){\zeta }_4+2{\zeta }_1{\zeta }_2{\zeta }_3]-d[-{{\zeta }_4}^3+3{\zeta }_3^2{\zeta }_4]\}+u_4,\hfill \end{array}$$

(16)

where $u={(u_1,u_2,u_3,u_4)}^T$ is the control functions.

For the choice $A=B=I_{4\times 4}$ and applying Theorem 2, the control functions (9) can be written as:

$$\begin{array}{cc}\hfill U_1\equiv u_1=& z_3-{\zeta }_3-k_1{e}_1,\hfill \\ \hfill U_2\equiv u_2=& z_4-{\zeta }_4-k_2{e}_2,\hfill \\ \hfill U_3\equiv u_3=& z_1-a({z_1}^3-3z_1{z_2}^2)+e\{b{z}_3-c[(z_1^2-z_2^2)z_3-2z_1{z}_2{z}_4]-d[{z_3}^3-3z_3{z_4}^2]\}-el\{[1-a(z_1^2-z_2^2)]z_1\hfill \\ \hfill & +2a{z}_1{z}_2^2\}cos\left(2wt\right)+e(h+2msin\left(\mathsf{\Omega }t\right))cos\left(wt\right)-{\zeta }_1+a({{\zeta }_1}^3-3{\zeta }_1{{\zeta }_2}^2)-e\{b{\zeta }_3-c[({\zeta }_1^2-{\zeta }_2^2){\zeta }_3\hfill \\ \hfill & -2{\zeta }_1{\zeta }_2{\zeta }_4]-d[{{\zeta }_3}^3-3{\zeta }_3{{\zeta }_4}^2]\}-k_3{e}_3,\hfill \\ \hfill U_4\equiv u_4=& z_2-a(-{z_2}^3+3z_1^2{z}_2)e\{b{z}_4-c[(z_1^2-z_2^2)z_4+2z_1{z}_2{z}_3]-d[-{z_4}^3+3z_3^2{z}_4]\}-el\{[1-a(z_1^2-z_2^2)]z_2\hfill \\ \hfill & -2a{z}_1^2{z}_2\}cos\left(2wt\right)-{\zeta }_2+a(-{{\zeta }_2}^3+3{\zeta }_1^2{\zeta }_2)-e\{b{\zeta }_4-c[({\zeta }_1^2-{\zeta }_2^2){\zeta }_4+2{\zeta }_1{\zeta }_2{\zeta }_3]-d[-{{\zeta }_4}^3+3{\zeta }_3^2{\zeta }_4]\}-k_4{e}_4.\hfill \end{array}$$

(17)

During the numerical simulation, we use the same parameters and initial conditions of the master and slave models (3) and (16) of Figure 2 and Figure 3. For the choice $K=0$, the results of PS are shown in Figure 4, Figure 5 and Figure 6. Figure 4 and Figure 5 show the phase differences for six plane projections versus t. The confirmation of achieving PS is supported by the observation that the phase differences remain within a bounded range. The amplitudes of the master and the slave models in $(z_1,z_3)$ and $({\zeta }_1,{\zeta }_3)$ spaces ($A_{13}^d$ and $A_{13}^r$) are shown in Figure 6 (a) and (b), and Figure 6 (c) displays $A_{13}^d$ versus $A_{13}^r$. Figure 6 demonstrates that these amplitudes exhibit hyperchaotic behavior and are uncorrelated. Similar results are observed for the other amplitudes of the master and slave systems in different plane projections.

While for $K=diag(0.5,2,1.5,1)$, the state variables for the master model (3) and the slave model (16) are shown in Figure 7. Figure 8 depicts the synchronization errors which are approach zero.

4. Illustrative example for APS

In this section, we introduce an example for APS with different dimensions. We consider the model (2) as the 2D master model and model (16) is the 4D slave one.

For the choice $A=I_{2\times 2}$ and $B=\left(\begin{array}{cccc}1& 0& 1& 0\\ 0& 1& 0& 1\end{array}\right)$ and applying Corollary 2, the control functions (13) can be written as:

$$\begin{array}{cc}\hfill U_1\equiv u_1+u_3=& -v_2-{\zeta }_3-({\zeta }_1-a({{\zeta }_1}^3-3{\zeta }_1{{\zeta }_2}^2)+e\{b{\zeta }_3-c[({\zeta }_1^2-{\zeta }_2^2){\zeta }_3-2{\zeta }_1{\zeta }_2{\zeta }_4]-d[{{\zeta }_3}^3-3{\zeta }_3{{\zeta }_4}^2]\}+u_3)-k_1{e}_1,\hfill \\ \hfill U_2\equiv u_2+u_4=& -(v_1-a{v_1}^3+e(b-c{v_1}^2-d{v_2}^2)v_2-el(1-a{v_1}^2)v_{1}cos\left(2wt\right)+e(h+2msin\left(\mathsf{\Omega }t\right))cos\left(wt\right))\hfill \\ & -{\zeta }_4-({\zeta }_2-a(-{{\zeta }_2}^3+3{\zeta }_1^2{\zeta }_2)e\{b{\zeta }_4-c[({\zeta }_1^2-{\zeta }_2^2){\zeta }_4+2{\zeta }_1{\zeta }_2{\zeta }_3]-d[-{{\zeta }_4}^3+3{\zeta }_3^2{\zeta }_4]\})-k_2{e}_2.\hfill \end{array}$$

(18)

In the numerical simulation, we use the same parameters and initial conditions of the master and slave models (2) and (16) of Figure 1 and Figure 3. For the choice $K=0$, the results of PS are shown in Figure 9 and Figure 10. Figure 9 depicts the phase differences for one plane projections versus t. The fact that the phase differences are bounded confirms the achievement of PS. The amplitudes of the master and the slave models in $(v_1,v_2)$ and $({\zeta }_1+{\zeta }_3,{\zeta }_2+{\zeta }_4)$ spaces ($A_{12}^d$ and $A_{12}^r$) are shown in Figure 10 (a) and (b), and Figure 6 (c) displays $A_{12}^d$ versus $A_{12}^r$. Figure 10 demonstrates that these amplitudes exhibit hyperchaotic behavior and are uncorrelated (linearly independent).

While for $K=diag(0.5,2)$, the state variables for the master model (2) and the slave model (16) are shown in Figure 11. Figure 12 depicts the synchronization errors which are approach zero.

5. Conclusion

In conclusion, this paper has introduced the hyperchaotic non-autonomous and autonomous complex Rayleigh-van der Pol-Duffing oscillators (RVDOs) (3) and (4). We studied into the fundamental dynamics of these oscillators, such as stability, fixed points, and dissipation. We demonstrated the hyperchaotic solutions of models (3) and (4) using the Lyapunov exponents and are shown in Figure 1, Figure 2 and Figure 3. The hyperchaotic nature of these models may be used in numerous significant engineering and scientific domains. The proposed scheme in Section 2, utilizing an active control technique based on Lyapunov stability analysis, aims to achieve PS and APS for different dimensional models. Through the analytical determination of control functions (9) and (13), the scheme successfully achieved PS and APS. These types of synchronization are considered a generalization of many types in the literature [14,15]. The scheme is applied to investigate PS of hyperchaotic attractors (3) and (4). As well as to achieve APS, we used the models (2) and (4) including those with different dimensions. The presented numerical results illustrating the phases and amplitudes of these chaotic and hyperchaotic attractors, and provide evidence of the effective accomplishment of PS and APS. These results are shown in Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12.