Symmetric Oscillator: Special Features, Realization, and Combination Synchronization

Abstract

Researchers have recently paid significant attention to special chaotic systems. In this work, we introduce an oscillator with different special features. In addition, the oscillator is symmetrical. The features and oscillator dynamics are discovered through different tools of nonlinear dynamics. An electronic circuit is designed to mimic the oscillator’s dynamics. Moreover, the combined synchronization of two drives and one response oscillator is reported. Numerical examples illustrate the correction of our approach.

1. Introduction

Several studies have provided important results on chaos [1,2,3]. Chaos was observed in a memristive system [4], glucose–insulin regulatory system [5], financial system [6], and modified logistic map [7], etc. Panahi et al. proposed a chaotic network model for epilepsy [8], while a chaotic model for HIV virus was introduced in [9]. Chaos is useful for developing applications [10,11]. Potential applications of chaos were reported in wireless communication [12], robot motion [13], authenticated Hash function [14], and pseudo-random generators [15]. Adeyemi et al. introduced the FPGA realization of chaos-encrypted transmission via the parameter-switching technique [16]. Muhammad and Ozkaynak combined chaos, optimization algorithms and physical unclonable functions to design the novel image encryption [17]. Many studies focused on random number generators [18,19,20]. Moreover, Tutueva et al. improved the random number generators [21].

More recent attention has been concentrated on chaos in systems with special features [22]. Some noticeable features are the absence of linear terms, appearance of many equilibrium points, and multistability. Most studies in the field of chaotic systems have been focused on systems with linear terms. However, results based upon systems without linear terms are limited. Xu and Wang were mentioned that there was much less information about chaotic attractors without a linear term [23]. Therefore, the authors constructed a system with natural logarithmic, exponential and quadratic terms. Using six quadratic terms, a system with eight equilibrium points was proposed in [24]. Zhang et al. applied a fractional derivative to obtain a new system with six quadratic terms [25]. The authors found four equilibrium points and twin symmetric attractors in Zhang’s system. Previously published studies on nonlinear systems paid particular attention to saddle point equilibrium in [26]. The existence of saddle point equilibrium is critical to the design of chaotic systems [27]. Relatively recent research has been found, concerned with different equilibia [28,29]. Recently, investigators have examined chaos in systems with infinite equilibrium [30,31]. Another special feature observed in nonlinear systems is multistability [32]. Depending on the initial conditions, coexisting attractors can be seen. Multistability has emerged as a powerful approach for investigating asymmetric and symmetric attractors [33,34,35]. Interestingly, multiple attractors attract new research on memristor circuits [36,37].

In this paper, we study a oscillator with nonlinear terms (quadratic and cubic ones). In contrast with conventional systems, there are infinite equilibria in our oscillator. The features and dynamics of the oscillator are presented in Section 2. Section 3 discusses the oscillator’s implementation. A combination synchronization of the oscillator is reported in Section 4, while conclusions are provided in the last section.

2. Features and Dynamics of the Oscillator

We consider an oscillator described by

$$\left\{\begin{array}{c}\dot{x}=yz\hfill \\ \dot{y}=x^3-y^3\hfill \\ \dot{z}=a{x}^2+b{y}^2-cxy\hfill \end{array}\right.$$

(1)

with parameters $a,b,c>0$. By solving the following equations:

$$\left\{\begin{array}{c}yz=0\hfill \\ x^3-y^3=0\hfill \\ a{x}^2+b{y}^2-cxy=0\hfill \end{array}\right.$$

(2)

we get the equilibrium points of oscillator (1):

$$E^{*}(0,0,z^{*})$$

(3)

Therefore, oscillator (1) has an equilibrium line. Oscillator (1) is invariant under the transformation

$$(x,y,z)\to (-x,-y,z)$$

(4)

and oscillator (1) is symmetric. Note that the Jacobian matrix at $E^{*}$ is

$$J_{E^{*}}=\left[\begin{array}{ccc}0& z^{*}& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$$

(5)

Therefore, the characteristic equation is ${\lambda }^3=0$ and the eigenvalue $\lambda =0$.

We fix $a=0.2$, $b=0.1$ and the initial conditions $(0.1,0.1,0.1)$ while c is varied. The Lypunov exponents (Figure 1a) and bifurcation diagram (Figure 1b) for c are presented. As seen from Figure 1, the oscillator can generate periodical signals and chaotic signals. For $c=0.5$, chaotic attractors are displayed in Figure 2. We used the Runge–Kutta method for simulations and the Wolf’s algorithm for Lypunov exponent calculations [38]. Interestingly, the oscillator displays symmetric attractors, as illustrated in Figure 3. Symmetric attractors coexist with the same parameters ($a=0.2$, $b=0.1$, $c=0.68$) but under different initial conditions. This means that there is multistability in the oscillator. When varying c, multistability is reported in Figure 4.

Oscillator (1) displays offset boosting dynamics because of the presence of z. Consequently, the amplitude of z is controlled by adding a constant k in oscillator (1), which becomes

$$\left\{\begin{array}{c}\dot{x}=y(k+z)\hfill \\ \dot{y}=x^3-y^3\hfill \\ \dot{z}=a{x}^2+b{y}^2-cxy\hfill \end{array}\right.$$

(6)

The bifurcation diagram and phase portraits of system (6) in planes $(z-x)$ and $(z-y)$ with respect to parameter c and some specific values of constant parameter k are provided in Figure 5 for $a=0.2$, $b=0.1$, $c=0.5$.

From Figure 5, we observe that the amplitude of z is easily controlled through the constant parameter k. This phenomenon of offset boosting control has been reported in some other systems [39,40].

3. Oscillator Implementation

The electronic circuit of mathematical models displaying chaotic behavior can be realized using basic modules of addition, subtraction, and integration. The electronic circuit implementation of such models is very useful in some engineering applications. The objective of this section is to design a circuit for oscillator (1). The proposed electronic circuit diagram for a system oscillator (1) is provided in Figure 6.

By denoting the voltage across the capacitor $V_v$, $V_y$ and $V_z$, the circuit state equations are as follows:

$$\left\{\begin{array}{c}\frac{d{V}_x}{dt}=\frac{1}{10R_{1}C}{V}_y{V}_z\hfill \\ \frac{d{V}_y}{dt}=\frac{1}{100R_{2}C}{V}_x^3-\frac{1}{100R_{3}C}{V}_y^3\hfill \\ \frac{d{V}_z}{dt}=\frac{1}{10R_{a}C}{V}_x^2+\frac{1}{10R_{b}C}{V}_y^2-\frac{1}{10R_{c}C}{V}_x{V}_y\hfill \end{array}\right.$$

(7)

For the system oscillator parameters (1) $a=0.2$, $b=0.1$, $c=0.5$ and initial voltages of capacitor $(V_x,V_y,V_z)$ = $(0.1\mathrm{V},0.1\mathrm{V},0.1\mathrm{V})$, the circuit elements are $C=10\mathrm{nF}$, $R_1=1\mathrm{k}\mathsf{\Omega }$, $R_2=R_3=100\mathsf{\Omega }$, $R_a=5\mathrm{k}\mathsf{\Omega }$, $R_b=10\mathrm{k}\mathsf{\Omega }$, and, $R_c=2\mathrm{k}\mathsf{\Omega }$. The chaotic attractors of the circuit implemented in PSpice are shown in Figure 7. Moreover, the symmetric attractors of the circuit are reported in Figure 8. As seen from Figure 7 and Figure 8, the circuit displays the dynamical behaviors of special oscillator (1). The real oscillator is also implemented, and the measurements are captured (see Figure 9).

4. Combination Synchronization of Oscillator

One of the successful applications of the synchronization phenomenon is in secure communication systems. Different methods have been developed for secure communications. To increase security in communication systems, some new synchronization techniques have been proposed in [41,42,43]. Based on the great advantages of such methods, the combination synchronization is designed. This is the combination of two drives and one response oscillator (1). The drive systems are

$$\left\{\begin{array}{c}\frac{d{x}_m}{dt}=y_m{z}_m\hfill \\ \frac{d{y}_m}{dt}=x_m^3-y_m^3\hfill \\ \frac{d{z}_m}{dt}=a{x}_m^2+b{y}_m^2-c{x}_m{y}_m\hfill \end{array}\right.$$

(8)

where $m=1,2$. The response system is:

$$\left\{\begin{array}{c}\frac{d{x}_s}{dt}=y_s{z}_s+u_1\hfill \\ \frac{d{y}_s}{dt}=x_s^3-y_s^3+u_2\hfill \\ \frac{d{z}_s}{dt}=a{x}_s^2+b{y}_s^2-c{x}_s{y}_s+u_3\hfill \end{array}\right.$$

(9)

Controllers $u_i(i=1,2,3)$ guarantee synchronization among the three systems. We express the error

$$e=Ax+By-Cz$$

(10)

where $x={(x_1,y_1,z_1)}^T$, $y={(x_2,y_2,z_2)}^T$, $z={(x_s,y_s,z_s)}^T$, $e={(e_x,e_y,e_z)}^T$ and A, B, C $\in R^{3\times 3}$. The controllers $u_i$ are designed to asymptotically stabilize error (10) at the zero equilibrium. Assuming that $A=diag({\eta }_1,{\eta }_2,{\eta }_3)$, $B=diag({\gamma }_1,{\gamma }_2,{\gamma }_3)$ and $C=diag({\epsilon }_1,{\epsilon }_2,{\epsilon }_3)$, system (10) becomes

$$\left\{\begin{array}{c}{e}_x={\eta }_1{x}_1+{\gamma }_1{x}_2-{\epsilon }_1{x}_s\hfill \\ e_y={\eta }_2{y}_1+{\gamma }_2{y}_2-{\epsilon }_2{y}_s\hfill \\ e_z={\eta }_3{z}_1+{\gamma }_3{z}_2-{\epsilon }_3{z}_s\hfill \end{array}\right.$$

(11)

The differentiation of system (11) leads to the error of dynamical system, expressed as

$$\left\{\begin{array}{c}\frac{d{e}_x}{dt}={\eta }_1\frac{d{x}_1}{dt}+{\gamma }_1\frac{d{x}_2}{dt}-{\epsilon }_1\frac{d{x}_s}{dt}\hfill \\ \frac{d{e}_y}{dt}={\eta }_2\frac{d{y}_1}{dt}+{\gamma }_2\frac{d{y}_2}{dt}-{\epsilon }_2\frac{d{y}_s}{dt}\hfill \\ \frac{d{e}_z}{dt}={\eta }_3\frac{d{z}_1}{dt}+{\gamma }_3\frac{d{z}_2}{dt}-{\epsilon }_3\frac{d{z}_s}{dt}\hfill \end{array}\right.$$

(12)

Replacing system (8), (9) and (11) into system (12) yields

$$\left\{\begin{array}{c}\frac{d{e}_x}{dt}={\eta }_1{y}_1{z}_1+{\gamma }_1{y}_2{z}_2-{\epsilon }_1{y}_s{z}_s-{\epsilon }_1{u}_1\hfill \\ \frac{d{e}_y}{dt}={\eta }_2(x_1^3-y_1^3)+{\gamma }_2(x_2^3-y_2^3)-{\epsilon }_2(x_s^3-y_s^3)-{\epsilon }_2{u}_2\hfill \\ \frac{d{e}_z}{dt}={\eta }_3(a{x}_1^2+b{y}_1^2-c{x}_1{y}_1)+{\gamma }_3(a{x}_2^2+b{y}_2^2-c{x}_2{y}_2)-{\epsilon }_3(a{x}_s^2+b{y}_s^2-c{x}_s{y}_s)-{\epsilon }_3{u}_3\hfill \end{array}\right.$$

(13)

From system (13), the controllers can be deduced as follows:

$$\left\{\begin{array}{c}{u}_1=\left({\eta }_1{y}_1{z}_1+{\gamma }_1{y}_2{z}_2-{\epsilon }_1{y}_s{z}_s-v_1\right)/{\epsilon }_1\hfill \\ u_2=\left({\eta }_2(x_1^3-y_1^3)+{\gamma }_2(x_2^3-y_2^3)-{\epsilon }_2(x_s^3-y_s^3)-v_2\right)/{\epsilon }_2\hfill \\ u_3=\left({\eta }_3(a{x}_1^2+b{y}_1^2-c{x}_1{y}_1)+{\gamma }_3(a{x}_2^2+b{y}_2^2-c{x}_2{y}_2)-{\epsilon }_3(a{x}_s^2+b{y}_s^2-c{x}_s{y}_s)-v_3\right)/{\epsilon }_3\hfill \end{array}\right.$$

(14)

where $v_i(i=1,2,3)$ are specific linear functions. Define

$$\left(\begin{array}{c}{v}_x\hfill \\ v_y\hfill \\ v_z\hfill \end{array}\right)=\overline{A}\left(\begin{array}{c}{e}_x\hfill \\ e_y\hfill \\ e_z\hfill \end{array}\right)$$

(15)

with $3\times 3$ real matrix $\overline{A}$.

For $\overline{A}=\left(\begin{array}{ccc}-1& 0& 0\\ 0& -1& 0\\ -1& -2& -3\end{array}\right)$ the error dynamical system is:

$$\left\{\begin{array}{c}\frac{d{e}_x}{dt}=-e_x\hfill \\ \frac{d{e}_y}{dt}=-e_y\hfill \\ \frac{d^q{e}_z}{d{t}^q}=-e_x-2e_y-3e_z\hfill \end{array}\right.$$

(16)

The error dynamical system is asymptotically stable.

Numerical results (see Figure 10) verified the combination synchronization among the two drive systems (8) and the response one. Here, system (8) is chaotic for $a=0.2$, $b=0.1$, and $c=0.5$. We set the initial conditions $x_1\left(0\right)=y_1\left(0\right)=z_1\left(0\right)=0.1$, $x_2\left(0\right)=2$, $y_2\left(0\right)=-1$, $z_2\left(0\right)=0.1$ for two drive systems (8). The response system (9) has $x_s\left(0\right)=1$, $y_s\left(0\right)=0.3$, and $z_s\left(0\right)=2$.

5. Conclusions

By using nonlinear dynamic tools, we investigated an oscillator with noticeable features. The oscillator has no linear term but infinite equilibria. Chaos and symmetrical co-existing attractors were displayed by the oscillator. The results show the complex dynamics of the oscillator, which are useful for varied applications. This work contributes to special systems with different noticeable features. In our future studies, we would like to use this oscillator to generate random numbers in cryptography.