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

: 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.


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].

The Proposed Oscillator
A quadratic oscillator is presented as: .
x = −y 2 + z 2 + C 2 x 2 + C 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 (0, 0, 0). 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. (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 = −0.6, = −0.7, = −2 and initial conditions (0,0,0). 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 − , − , and − 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. All of the equilibrium points have at least one positive real part of the eigenvalue, so they are unstable.   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. All of the equilibrium points have at least one positive real part of the eigenvalue, so they are unstable. Table 2. Equilibrium points and eigenvalues of Oscillator (1).

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.

Dynamical Properties
The oscillator has three crucial parameters that significantly affect its dynamics. The first studied parameter is . Figure 2 presents the bifurcation diagram of Oscillator (1) by varying . The other parameters are kept constant as = −0.7, = −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 . 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 in Figure 3. The diagram is plotted using a forward continuation method in constant parameters = −0.6, = −2. A period-doubling route to chaos can be seen by changing the parameter. LEs of the oscillator confirm the chaotic dynamics in small .
To better investigate various dynamics of the oscillator, the 2D bifurcation diagrams are discussed by changing parameters and 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 is plotted for nine values of parameter . The diagram helps to investigate various dynamics by changing these two parameters. The results show that in the studied interval of , increasing causes a decrease in the complexity of dynamics. The bifurcation diagram is discussed by changing C 2 in Figure 3. The diagram is plotted using a forward continuation method in constant parameters 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 ∈ [−2, −1.953]. 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 ∈ [−1.9836, −1.983] and the first initial conditions as (0, 0, 0). Comparison of the magenta color diagram with blue and green ones reveals the coexisting attractors in various intervals of C 3 .  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 = −0.6 and = −0.7. The magenta color shows a bifurcation diagram in ∈ [−2, −1.953]. 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 ∈ [−1.9836, −1.983] and the first initial conditions as (0,0,0). Comparison of the magenta color diagram with blue and green ones reveals the coexisting attractors in various intervals of . The basin of attraction of the oscillator in = −0.6, = −0.7, = −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  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 ∈ [−3, 7], y 0 ∈ [−4, 4] 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). 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).

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 − √ 2, y new = y old − √ 2, z new = z old − 2 is used to move the equilibrium point (x * , y * , z * ) = √ 2, √ 2, 2 to the origin. So the transformed oscillator is as follows: where x new , y new , z new are called x, y, z in Equation (2). Then Oscillator (2) can be rewritten as: where P is the vector of variables [x, y, z] T , A × P is the linear term of Equation (2), and φ(P) is the nonlinear term. From Equation (2), we have: Now the controlled oscillator can be written as: So B, τ i should be calculated for this control method.

Theorem 3.
The origin is an asymptotically stable equilibrium for the proposed oscillator if there is a ξ > 1 and a differentiable at t = τ i and non-increasing function KK(t) which satisfies: where q is defined as the largest eigenvalue of A + K −1 A T K , K is a positive definite matrix, and λ 1 > 0 and λ 2 > 0 are the smallest and the largest eigenvalues of K, respectively. ρ(V) is the spectral radius of V and d = ρ 2 (I + B). M is considered as x (t) < M, y (t) < M, z (t) < M. KK(t) is as in Theorem 1, τ i : i = 1, 2, . . . should satisfy: For a constant ε, we have: The theorem's proof can be seen in [67].

Remark 1.
Theorem 3 estimates the upper bound ∆ 1max and ∆ 2max of impulsive intervals. For controlling the proposed oscillator, the matrix B is considered as: Here, q is defined as the largest eigenvalue of A + K −1 A T K , where K is a positive definite matrix. By considering K = I, q is calculated as the maximum eigenvalue of A + A T . For the Oscillator (2), q = 9.8272.
Another parameter in this control method is d which is defined as: where ρ(V) is the spectral radius of V. So, we have d = (−1.1 + 1) 2 = 0.01. Then the intervals of applying the controller are computed as: where ξ is considered 1.1; we have ∆ 1 = 0.4589. So ∆ is considered 0.45. Figure Here, is defined as the largest eigenvalue of ( + ), where is a positive definite matrix. By considering = , is calculated as the maximum eigenvalue of ( + ). For the Oscillator (2), = 9.8272.
Another parameter in this control method is which is defined as: where ( ) is the spectral radius of . So, we have = (−1.1 + 1) = 0.01. Then the intervals of applying the controller are computed as: where is considered 1.1; we have Δ = 0.4589. So Δ is considered 0.45. Figure 8 presents the results of applying the discussed controller. Part (a) of the figure shows the controlled system (2)

Circuit Design
Here the circuit of the oscillator is investigated in = −0.6, = −0.7, = −2, as its schematic is shown in Figure 9.  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.

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 Res 1 = Res 2 = Res 5 = Res 6 = Res 8 = 70 kΩ, Res 3 = 100 kΩ, Res 4 = 17500 kΩ, Res 7 = 5250 kΩ, Res 9 = 35 kΩ, Res 10 = Res 11 = Res 12 = Res 13 = Res 14 = Res 15 = 100 kΩ. The capacitors are selected as Cap 1 = Cap 2 = Cap 3 = 10 nF. Here, AD633 was used as a multiplier, and OPA404 was used as the operational amplifier. The positive voltage source is set to 15 V. The initial values of voltage in capacitors are considered as (0, 0, 0). 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.

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.