A Chaotic Quadratic Oscillator with Only Squared Terms: Multistability, Impulsive Control, and Circuit Design

Abstract

Here, a chaotic quadratic oscillator with only squared terms is proposed, which shows various dynamics. The oscillator has eight equilibrium points, and none of them is stable. Various bifurcation diagrams of the oscillator are investigated, and its Lyapunov exponents (LEs) are discussed. The multistability of the oscillator is discussed by plotting bifurcation diagrams with various initiation methods. The basin of attraction of the oscillator is discussed in two planes. Impulsive control is applied to the oscillator to control its chaotic dynamics. Additionally, the circuit is implemented to reveal its feasibility.

1. Introduction

Chaotic flows have attracted lots of attention recently [1,2]. Many systems with various features have been proposed to study the chaotic dynamics [3,4]. Some of the proposed oscillators are discussed from the viewpoint of their quadratic or cubic terms [5,6]. Some other studies have focused on the equilibrium points [7]. Oscillators with no equilibria [8], with stable equilibria [9], with curves of equilibria [10], and with a peanut-shaped equilibrium curve [11] are some examples. A hyperjerk oscillator has been investigated in [12]. A multi-dimensional chaotic system was discussed in [13]. In [14], a multi-scroll chaotic circuit was analyzed. Various dynamics of the Sprott B system were studied in [15]. The dynamics of coupled neurons were investigated in [16]. Chaotic dynamics have many applications, such as encryption [17,18,19]. In [20,21], a discrete chaotic dynamic was used in image encryption. A chaotic encryption method and its application in the internet of things were studied in [22]. A plain-text-related image encryption method using Chen oscillator was proposed in [23].

Multistability is an exciting feature of dynamical systems [24,25,26]. Multistable oscillators have various applications [27]. A multistable oscillator with various attractors was discussed in [28]. In [29,30], the multistability of the series hybrid electric vehicle was investigated. Extreme multistability is a particular case of multistability [31]. In [32], the extreme multistability of a fractional-order oscillator was studied. The multistability of a Chua system was discussed in [33]. Memristive chaotic oscillators are very interesting [34]. A memristive neural system was studied in [35]. A memristive Chua system was discussed in [36]. In [37], a memristive oscillator with a unique attractor was investigated. A memristive oscillator with a fractional-order difference was studied in [38]. In [39], the multistability of a five-value memristive oscillator was investigated.

Control and synchronization of dynamical systems have been a hot topic [40,41,42]. Many algorithms have been proposed to control chaotic oscillators [43]. Control of the Chen oscillator was investigated in [44]. A fuzzy-based controller was studied in [45]. In [46], control of a piecewise linear oscillator was discussed. Adaptive control of a chaotic oscillator was discussed in [47]. Delayed feedback control of chaotic oscillators was investigated in [48]. Impulsive control is a valuable method for this purpose [49]. Event-triggered impulsive control was studied in [50,51,52]. Synchronization of the Chen oscillator was discussed in [53]. Impulsive control for synchronization of a chaotic network was investigated in [54].

Chaotic circuits show the feasibility of these dynamics [55,56]. Implementing circuits for chaotic oscillators has been an exciting topic [57]. A memristive circuit was studied in [1]. Various dynamics of the system were discussed. A jerk circuit with the arcsinh function was proposed in [58], and its behaviors were investigated. The application of a chaotic circuit on image encryption was discussed in [59]. The circuit design of a 5D hyperchaotic oscillator was studied in [60], and its multistability was discussed. In [61], the circuit design of a 3D system was studied.

Here, a quadratic oscillator with only squared terms is presented. The comparison of the proposed system with some previous important literature is presented in

Table 1. Comparison of the proposed system with the previous ones.

Ref.	Dimension	Type of Terms	Number of Terms	Number of Equilibrium Points	Multistability	Circuit
[62]	4	Cubic	9	Infinite	✓	✓
[63]	3	Cubic & tanh(.)	7	5	✓	✓
[64]	4	Quadratic	9	0	✓	✓
[16]	3	Linear & Tanh	10	3–7	✓	✓
This work	3	Quadratic with only squared terms	9	8	✓	✓

to highlight the distribution of this paper. The proposed system is a simple quadratic system with only squared terms, and some important properties of the system are investigated. The chaotic behavior of the oscillator is discussed in Section 2. Additionally, its equilibrium points and their stabilities are investigated. In Section 3, various oscillator dynamics are discussed by changing its three parameters. 1D and 2D bifurcation diagrams are used to investigate the dynamics of the oscillator. Lyapunov exponents (LEs) help to investigate the type of dynamics in various parameters. Then by plotting bifurcation diagrams with various initial conditions, the multistability of the oscillator is investigated. The basin of attraction for the oscillator shows the variation of dynamics by changing the initial values of its three variables. In Section 4, the impulsive control of the oscillator is designed, and its results are discussed. Then, the circuit of the oscillator is designed in Section 5. In Section 6, the conclusion of the paper is discussed.

2. The Proposed Oscillator

A quadratic oscillator is presented as:

$$\begin{array}{c}\dot{x}=-y^2+z^2+C_2{x}^2+C_1\\ \dot{y}=-z^2+x^2+2\\ \dot{z}=z^2+C_3{x}^2\end{array}$$

(1)

To propose the system, a parametric quadratic system with only squared terms and constants in each variable is designed. Then a computer search is applied to compute the value of parameters and initial values for chaotic solutions. The oscillator only has squared terms and not a multiplication of two variables. It is important to investigate if this system is symmetric or has offset boosting properties. Offset boosting and symmetry experiments are significant features of chaotic systems [65,66]. We examine the existence of offset boosting and symmetry by adding a constant excitation force to all the right-hand sides of equations one by one. However, there was no offset boosting. The system shows chaotic dynamics in $C_1=-0.6, C_2=-0.7,C_3=-2$ and initial conditions $\left(0,0,0\right).$ Figure 1 presents the time series of chaotic dynamics in part (a), the 3D chaotic attractor in part (b), and its three 2D projections in $X-Y$, $Y-Z$, and $X-Z$ planes with gray color.

The equilibrium points of the system are calculated by setting zeros on the right-hand side of Equation (1). The system has eight equilibrium points, as shown in Table 1. To investigate the stability of equilibrium points, the characteristic equation and eigenvalues should be computed for each of them. The corresponding eigenvalues of the equilibrium points are presented in

Table 2. Equilibrium points and eigenvalues of Oscillator (1).

#	Equilibrium	Eigenvalues
E1	$\left(\sqrt{2}, \sqrt{2},2\right)$	${\lambda }_1=-1.2036, {\lambda }_{2,3}=1.6118\pm 4.8979i$
E2	$\left(-\sqrt{2}, \sqrt{2},2\right)$	${\lambda }_1=8.9257, {\lambda }_{2,3}=-1.4729\pm 1.1899i$
E3	$\left(\sqrt{2},-\sqrt{2},2\right)$	${\lambda }_1=3.1356, {\lambda }_{2,3}=-0.5577\pm 3.1455i$
E4	$\left(-\sqrt{2},-\sqrt{2},2\right)$	${\lambda }_1=6.2323, {\lambda }_2=-2.3957, {\lambda }_3=2.1432$
E5	$\left(\sqrt{2}, \sqrt{2},-2\right)$	${\lambda }_1=-6.2323, {\lambda }_2=2.3957, {\lambda }_3=-2.1432$
E6	$\left(-\sqrt{2},\sqrt{2},-2\right)$	${\lambda }_1=-3.1356, {\lambda }_{2,3}=0.5577\pm 3.1455i$
E7	$\left(\sqrt{2},-\sqrt{2},-2\right)$	${\lambda }_1=-8.9257, {\lambda }_{2,3}=1.4729\pm 1.1899i$
E8	$\left(-\sqrt{2},-\sqrt{2},-2\right)$	${\lambda }_1=1.2036, {\lambda }_{2,3}=-1.6118\pm 4.8979i$

. All of the equilibrium points have at least one positive real part of the eigenvalue, so they are unstable.

3. Dynamical Properties

The oscillator has three crucial parameters that significantly affect its dynamics. The first studied parameter is $C_1$. Figure 2 presents the bifurcation diagram of Oscillator (1) by varying $C_1$. The other parameters are kept constant as $C_2=-0.7,C_3=-2$. The maximum values of three variables of the oscillator with the forward continuation method are plotted in parts (a–c). The oscillator shows a period-doubling route to chaos. Part (d) of the Figure 2d shows the oscillator’s LEs by changing $C_1$. A positive LE can prove the existence of chaos. Additionally, one LE approaches zero by approaching bifurcation points.

The bifurcation diagram is discussed by changing $C_2$ in Figure 3. The diagram is plotted using a forward continuation method in constant parameters $C_1=-0.6, C_3=-2$. A period-doubling route to chaos can be seen by changing the parameter. LEs of the oscillator confirm the chaotic dynamics in small $C_2$.

To better investigate various dynamics of the oscillator, the 2D bifurcation diagrams are discussed by changing parameters $C_1$ and $C_2$ in Figure 4. A classic bifurcation diagram presents the dynamics by changing one parameter. The 2D bifurcation diagram is helpful since it shows the variations by changing two parameters. The bifurcation diagram by changing $C_1$ is plotted for nine values of parameter $C_2$. The diagram helps to investigate various dynamics by changing these two parameters. The results show that in the studied interval of $C_1$, increasing $C_2$ causes a decrease in the complexity of dynamics.

The multistability of the oscillator can be revealed by plotting a bifurcation diagram with different initial conditions. In Figure 5, various bifurcations are plotted using different colors in parameters $C_1=-0.6$ and $C_2=-0.7$. The magenta color shows a bifurcation diagram in $C_3\in \left[-2,-1.953\right]$. It is plotted by the forward continuation method and the first initial conditions at the origin. The blue one is the forward continuation bifurcation diagram with origin as the first initial conditions. The green color is the forward bifurcation in $C_3\in \left[-1.9836,-1.983\right]$ and the first initial conditions as $\left(0,0,0\right)$. Comparison of the magenta color diagram with blue and green ones reveals the coexisting attractors in various intervals of $C_3$.

The basin of attraction of the oscillator in $C_1=-0.6, C_2=-0.7,C_3=-1.9722$ is discussed to investigate the initial conditions that result in chaotic and periodic dynamics as presented in Figure 5. In Figure 6, the basin of attractions is plotted in two planes as $z_0=0$ and $z_0=1$. The pink color shows chaotic regions, and the white one presents the periodic regions. The gray color depicts unbounded regions. In each plane, the dynamics in the intervals $x_0\in \left[-3,7\right]$, $y_0\in \left[-4,4\right]$ are computed for constant $z_0$. So the variations of dynamics by changing $x_0$ and $y_0$ can be seen in each plane. The effect of $z_0$ can be seen by comparing the two planes. Four sets of initial conditions are selected from the two planes to show the coexisting chaotic and periodic attractors (Figure 7).

4. Impulsive Control

In this section, impulsive control [67,68] is applied to stabilize the proposed oscillator. As was discussed in Table 2, the system does not have an equilibrium point in origin. So in the first step, the change of variables $x_{new}=x_{old}-\sqrt{2}, y_{new}=y_{old}-\sqrt{2}, z_{new}=z_{old}-2$ is used to move the equilibrium point $\left(x^{*},y^{*},z^{*}\right)=\left(\sqrt{2},\sqrt{2},2\right)$ to the origin. So the transformed oscillator is as follows:

$$\begin{array}{c}\dot{x}=-{\left(y+y^{*}\right)}^2+{(z+z^{*})}^2-0.7{(x+x^{*})}^2-0.6\\ \dot{y}=-{\left(z+z^{*}\right)}^2+{\left(x+x^{*}\right)}^2+2\\ \dot{z}={(z+z^{*})}^2-2{\left(x+x^{*}\right)}^2\end{array}$$

(2)

where $x_{new}, y_{new},z_{new}$ are called $x, y, z$ in Equation (2). Then Oscillator (2) can be rewritten as:

$$\dot{P}=A\times P+\varphi \left(P\right)$$

(3)

where $P$ is the vector of variables ${\left[x,y,z\right]}^T$, $A\times P$ is the linear term of Equation (2), and $\varphi \left(P\right)$ is the nonlinear term. From Equation (2), we have:

$$\begin{array}{cc}\begin{array}{cc}A=& \left[\begin{array}{ccc}-1.4\times x^{*}& -2\times y^{*}& 2\times z^{*}\\ 2\times x^{*}& 0& -2\times z^{*}\\ -4\times x^{*}& 0& 2\times z^{*}\end{array}\right]\end{array},& \varphi \left(P\right)=\left[\begin{array}{c}-y^2+z^2-0.7x^2\\ -z^2+x^2\\ z^2-2x^2\end{array}\right]\end{array}$$

(4)

Now the controlled oscillator can be written as:

$$\begin{array}{cc}\dot{P}=\mathrm{g}\left(t,P\right)=A\times P+\varphi \left(P\right)& t\ne {\tau }_i\\ P\left(t^{+}\right)-P\left(t^{-}\right)=B\times P& t={\tau }_i,i=1,2,\dots \end{array}$$

(5)

So $B, {\tau }_i$ should be calculated for this control method.

Definition 1.

If$VV: R_{+}\times R^n \to R_{+},$then$VV$belong to the class$V{V}_0$, if:

(1) $VV$ is continuous in $\left({\tau }_{i-1},{\tau }_i\right]\times R^n$ and for each $P\in R^n$, i $=1, 2, . . . , \left(t,Y\right) \to \left({\tau }_i^{+},P\right)$ $lim$ $VV\left(t, Y\right)=VV\left({\tau }_i^{+},P\right)$ exists;

(2) $VV$ is locally Lipschitzian in $P$.

Definition 2.

For$\left(t,P\right)\in \left({\tau }_{i-1},{\tau }_i\right]\times R^n$, we have,

$$D^{+}VV\left(t,P\right)\triangleq h\to 0+\lim \sup \frac{1}{h}[VV\left(t+h,P+hg\left(t,P\right)-VV\left(t,P\right)\right]$$

Definition 3.

Comparison system:$VV \in VV0$and consider:

$$D^{+}VV\left(t,X\right)\le a\left(t, VV\left(t,P\right)\right),t \ne {\tau }_i; \mathrm{and} VV\left(t,P+UU\left(i,P\right)\right)\le \Psi i\left(VV\left(t, P\right)\right), t={\tau }_i,$$

where $a: R_{+}\times R_{+} \to R$is continuous and $\Psi i: R_{+} \to R_{+}$is non-decreasing. Then the following system is the comparison system:

$$\begin{gathered} \omega \omega =a\left(t, \omega \right), t \ne {\tau }_i \\ \omega \omega \left({\tau }_i^{+} \right)=\Psi i\left(\omega \omega \left({\tau }_i\right)\right) \\ \omega \omega \left({\tau }_0^{+}\right)=\omega \omega 0 \ge 0 \end{gathered}$$

(6)

Theorem 1.

The following conditions are considered:

(1) $VV:R{ }^n\times R{ }^n\to R_{+}, VV\in VV0, KK\left(t\right)D^{+}VV\left(t,P\right)+D^{+}KK\left(t\right)VV\left(t,P\right)\le a\left(t, KK\left(t\right)VV\left(t,P\right)\right), t\triangleq li$, when $a$ is continuous in $\left({\tau }_{i-1},{\tau }_i\right]\times R{ }^n$ for each $P \in R^n, i=1, 2, . . . , \left(t, y\right)\to \left({\tau }_i^{+},P\right)\lim a\left(t, y\right)=a\left({\tau }_i^{+},P\right)$ exists. $KK\left(t\right)\ge mm>0, t \to {\tau }_i^{+}\lim KK\left(t\right)=KK\left({\tau }_i\right), t \to {\tau }_i^{+}\lim KK\left(t\right)$ exists, $i=1, 2, . . . ,D^{+}KK\left(t\right)=h \to 0+\lim \sup \left(\frac{1}{h}\right)\left[KK\left(t+h\right)– KK\left(t\right)\right];$

(2)$KK\left({\tau }_i+0\right)VV\left({\tau }_i+0,P+UU\left(kk,P\right)\right)\le \Psi i\left(KK\left({\tau }_i\right)VV\left({\tau }_i,P\right)\right), i=1, 2, . . . ;$

(3) $VV\left(t, 0\right)=0$ and $\alpha \left(\left|\left|P\right|\right|\right) \le VV\left(t,P\right)$ on $R_{+}\times R{ }^n$, when $\alpha (·) \in \aleph$ (continuous strictly increasing function class $\alpha : R_{+} \to R_{+}$ so that $\alpha \left(0\right)=0$) are satisfied. The global asymptotic stability for the solution $\mathsf{\omega }\mathsf{\omega }=0$ of the comparison system implies global asymptotic stability of impulsive system trivial solution.

Theorem 2.

Consider$a\left(t, \omega \omega \right)=\dot{\lambda }\left(t\right)\omega \omega , \Psi i\left(\omega \omega \right)=d_i\omega \omega ,$$d_i\ge 0$for all i ≥ 1. Consequently, the system origin is global asymptotically stable if Theorem 1 conditions and the following conditions are kept:

(1) $\lambda \left(t\right)$ is non-decreasing, $t \to {\tau }_i^{+} lim \lambda \left(t\right)=\lambda \left({\tau }_i\right), t\to {\tau }_i^{+} lim \lambda \left(t\right)=\lambda \left({\tau }_i^{+}\right)$ exists, for all $=1, 2, \dots$;

(2) $su{p}_i$ $\left[d_i exp\left(\lambda \left({\tau }_{i+1}\right)-\lambda \left({\tau }_i{}^{+} \right)\right)\right]=\epsilon {\epsilon }_0<\infty$;

(3) There is a $r>1$ such that $\lambda \left({\tau }_{2i+3}\right)+\lambda \left({\tau }_{2i+2}\right)+ln\left(r{d}_{2i+2}+d_{2i+1}\right) \le \lambda \left({\tau }_{2i+2}{}^{+}\right)+\lambda \left({\tau }_{2i+2}{}^{+}\right)$ is held for all $d_{2i+2}{d}_{2i+1}$ $\ne$ $0, i=1, 2, \dots$, or there is a $r>1$ so that $\lambda \left({\tau }_{i+1}\right)+ln\left(r{d}_i\right)$ ≤ $\lambda \left({\tau }_i{}^{+}\right)$ for all $i$;

(4) $VV\left(t, 0\right)=0$, and we have $\alpha (·)$ in class $N$ such that $\alpha \left(\left|\left|P\right|\right|\right) \le VV\left(t,P\right).$

Theorem 3.

The origin is an asymptotically stable equilibrium for the proposed oscillator if there is a$\xi >1$and a differentiable at$t\ne {\tau }_i$and non-increasing function$KK\left(t\right)$which satisfies:

$$\begin{gathered} -\frac{\dot{KK\left(t\right)}}{KK\left(t\right)}\le q+r\le \frac{1}{\left(1+\epsilon \epsilon \right){\Delta }_2}\ln \left(\frac{KK\left({\tau }_{2i}^{+}\right)KK\left({\tau }_{2i-1}^{+}\right)}{KK\left({\tau }_{2i+1}\right)KK\left({\tau }_{2i}\right)\xi d^2}\right) \\ \mathrm{or} \\ -\frac{\dot{KK\left(t\right)}}{KK\left(t\right)}\le q+r\le \frac{1}{\max \left({\Delta }_1,{\Delta }_2\right)}\ln \left(\frac{KK\left({\tau }_i^{+}\right)}{KK\left({\tau }_{i+1}\right)\xi d}\right) \\ r=\{\begin{array}{c}0, if K=I\\ 2M\sqrt{\frac{{\lambda }_2}{{\lambda }_1}} if K\ne I\end{array} \end{gathered}$$

(7)

where$q$is defined as the largest eigenvalue of$\left(A+K^{-1}{A}^{T}K\right)$,$K$is a positive definite matrix, and${\lambda }_1>0$and${\lambda }_2>0$are the smallest and the largest eigenvalues of$K$, respectively.$\rho \left(V\right)$is the spectral radius of$V$and$d={\rho }^2\left(I+B\right).$ $M$is considered as$\left|x_{\left(t\right)}\right|<M,$ $\left|y_{\left(t\right)}\right|<M$,$\left|z_{\left(t\right)}\right|<M.$ $KK\left(t\right)$is as in Theorem 1,${\tau }_i: i=1, 2,\dots$should satisfy:

$$\begin{gathered} {\Delta }_1=su{p}_{1\le i<\infty } \left(T{ }_{2i+1}-T{ }_{2i}\right)<\infty \\ {\Delta }_2=su{p}_{1\le i<\infty } \left(T{ }_{2i}-T_{2i-1}\right)<\infty \end{gathered}$$

(8)

For a constant $\epsilon$, we have:

$$T_{2i+1}-T_{2i}\le \epsilon \left(T_{2i}-T_{2i-1}\right), \forall i\in 1, 2, \dots ,\infty$$

(9)

The theorem’s proof can be seen in [67].

Remark 1.

Theorem 3 estimates the upper bound${\Delta }_1{}_{max}$and${\Delta }_2{}_{max}$of impulsive intervals.

$$\begin{gathered} {\Delta }_1=\frac{1}{\left(1+\epsilon \epsilon \right)\left(q+2\left|a\alpha \right|\right)}\ln \left(\frac{KK\left({\tau }_{2i}^{+}\right)KK\left({\tau }_{2i-1}^{+}\right)}{KK\left({\tau }_{2i+1}\right)KK\left({\tau }_{2i}\right)\xi d^2}\right) \\ {\Delta }_2=\epsilon \epsilon {\Delta }_1 \end{gathered}$$

(10)

For controlling the proposed oscillator, the matrix $B$ is considered as:

$$B=\left[\begin{array}{ccc}-1.1& 0& 0\\ 0& -1& 0\\ 0& 0& -1\end{array}\right]$$

(11)

Here, $q$ is defined as the largest eigenvalue of $\left(A+K^{-1}{A}^{T}K\right)$, where $K$ is a positive definite matrix. By considering $K=I$, $q$ is calculated as the maximum eigenvalue of $\left(A+A^T\right)$. For the Oscillator (2), $q=9.8272$.

Another parameter in this control method is $d$ which is defined as:

$$d={\rho }^2\left(I+B\right)$$

(12)

where $\rho \left(V\right)$ is the spectral radius of $V$. So, we have $d={\left(-1.1+1\right)}^2=0.01$. Then the intervals of applying the controller are computed as:

$$\begin{array}{c}{T}_{2j+1}-T_{2j}=T_{2j}-T_{2j-1}=\mathsf{\Delta }<\left({\mathsf{\Delta }}_1={\mathsf{\Delta }}_2\right)\\ {\mathsf{\Delta }}_1={\mathsf{\Delta }}_2=\frac{-\ln \left(\xi d\right)}{q}\end{array}$$

(13)

where $\xi$ is considered 1.1; we have ${\mathsf{\Delta }}_1=0.4589$. So $\mathsf{\Delta }$ is considered $0.45$. Figure 8 presents the results of applying the discussed controller. Part (a) of the figure shows the controlled system (2), while in part (b), the original time series of the oscillator for various variables are plotted in the same time interval. So the controller makes all variables approach zero.

5. Circuit Design

Here the circuit of the oscillator is investigated in $C_1=-0.6, C_2=-0.7,C_3=-2$, as its schematic is shown in Figure 9. The circuit is implemented with OrCAD-Pspice. The values of resistors are considered as $Re{s}_1=Re{s}_2=Re{s}_5=Re{s}_6=Re{s}_8=70 \mathrm{k}\mathsf{\Omega }, Re{s}_3=100 \mathrm{k}\mathsf{\Omega },Re{s}_4=17500 \mathrm{k}\mathsf{\Omega }, Re{s}_7=5250 \mathrm{k}\mathsf{\Omega }, Re{s}_9=35 \mathrm{k}\mathsf{\Omega },Re{s}_{10}=Re{s}_{11}=Re{s}_{12}=Re{s}_{13}=Re{s}_{14}=Re{s}_{15}=100 \mathrm{k}\mathsf{\Omega }$. The capacitors are selected as $Ca{p}_1=Ca{p}_2=Ca{p}_3=10 \mathrm{nF}$. Here, AD633 was used as a multiplier, and OPA404 was used as the operational amplifier. The positive voltage source is set to $15 \mathrm{V}$. The initial values of voltage in capacitors are considered as $\left(0,0,0\right)$. Figure 10 presents the results of the designed circuit for the Oscillator (1). Part (a) of the figure shows the time series of the chaotic circuit, while the other parts show the projection of its dynamics in three different planes. The results are wholly matched with the dynamics of Figure 1. In other words, the chaotic oscillator was completely implemented without any issue, and its feasibility was realized. So the oscillator with only squared terms is physically realizable.

6. Conclusions

A novel quadratic chaotic oscillator was proposed here. The attractors of the oscillator were studied. Investigating the oscillator has shown the existence of eight equilibrium points, and none of them are stable; 1D and 2D bifurcation diagrams were studied to investigate the various dynamics of the oscillator. The results have shown the rich dynamics of the oscillator. LEs have revealed the types of dynamics. Studying bifurcation diagrams of the system by different initial values has shown coexisting attractors in different parameter regions. The basin of attractions was discussed in two planes. In addition, some of the multistable attractors were shown. Impulsive control was applied to the oscillator to force the chaotic dynamics approaches to the origin. The results have shown the high potential of this controller. Then the circuit of the oscillator was designed, which presents the feasibility of the chaotic dynamics. The complex dynamics of the oscillator make it a proper choice for random number generators and encryption applications.