A Novel Highly Nonlinear Quadratic System: Impulsive Stabilization, Complexity Analysis, and Circuit Designing

This work introduces a three-dimensional, highly nonlinear quadratic oscillator with no linear terms in its equations. Most of the quadratic ordinary diﬀerential equations (ODEs) such as Chen, Rossler, and Lorenz have at least one linear term in their equations. Very few quadratic systems have been introduced and all of their terms are nonlinear. Considering this point, a new quadratic system with no linear term is introduced. This oscillator is analyzed by mathematical tools such as bifurcation and Lyapunov exponent diagrams. It is revealed that this system can generate diﬀerent behaviors such as limit cycle, torus, and chaos for its diﬀerent parameters’ sets. Besides, the basins of attractions for this system are investigated. As a result, it is revealed that this system’s attractor is self-excited. In addition, the analog circuit of this oscillator is designed and analyzed to assess the feasibility of the system’s chaotic solution. The PSpice simulations conﬁrm the theoretical analysis. The oscillator’s time series complexity is also investigated using sample entropy. It is revealed that this system can generate dynamics with diﬀerent sample entropies by changing parameters. Finally, impulsive control is applied to the system to represent a possible solution for stabilizing the system.


Introduction
Chaos is a complex behavior that has been investigated in nature and mathematics [1]. It refers to the systems' sensitivity to their initial conditions and parameters [2]. Nonlinear systems such as ordinary differential equations (ODEs) can generate chaotic behavior [3]. erefore, they can have applications in modeling natural systems with chaotic behaviors such as neurons [4]. Besides, using coupling methods, these systems can investigate collective behaviors of neuronal networks [5]. It is good to mention that sometimes these dynamical systems are used in their nonchaotic mode to model some behaviors of natural systems like central pattern generators (neurons that make rhythms for locomotions) [6]. ODEs chaotic systems have been categorized based on their equilibrium points types and locations [7]. In this way, chaotic systems' attractors can be divided into two groups: ones that have at least one equilibrium point in their basin of attraction (self-excited attractors) [8] and ones that have no equilibrium point in their basin of attraction (hidden attractor(s)) [9]. Besides, the systems' equilibrium points are interesting nonlinear dynamics properties for researchers. For example, systems have been introduced and investigated with a line of equilibria [9] or just one stable equilibrium [10]. Besides, two main groups of systems can be defined based on the equations' time dependency: autonomous systems in which no term as a function of time exists in their equations [11] and nonautonomous systems in which a term dependent to the time can be found in their equations (forced systems) [12]. Besides, forcing systems can lead nonchaotic oscillators to systems that have the capability of generating chaos. For instance, using this technique, two-dimensional systems that cannot generate chaotic time series can demonstrate chaotic attractors [13]. ese nonautonomous systems are also capable of generating hidden attractors [14]. Another method to make a system chaotic is considering delays in the equations of the system. For instance, a system with just one equation can make chaos if it has a time delay in its equation [15]. On the other hand, systems with four dimensions or more can generate hyperchaotic behaviors. For instance, a four-dimensional jerk system implicated with memristors has shown the potential of demonstrating hyperchaos [16]. Besides these features, multistability refers to the existence of several attractors (at least two ones) for different initial values for a system without parameters' changing [17]. In addition to the mentioned properties, some other features are defined for chaotic systems based on the topology and shapes of attractors [18]. Some strange attractors with different symmetries' types were reported [19]. Besides, the systems with several wings' attractors (multiscrolls) have grabbed researchers' interest [20]. For instance, it is investigated how the strange multiscrolls attractor for a system can emerge and how its shape can be preserved [21]. In addition, the systems that their attractors look like known objects were also reported. For instance, chaotic systems have been introduced to look like a Persian carpet [22] or a peanut [23].
Among different ODE systems, quadratic ones are mainly focused on by some researchers interested in finding elegant systems [2]. One reason is that these systems can have simpler equations [24]. Lorenz equations, the first introduced chaotic system, are one of these classes and have just quadratic terms. Some quadratic systems were introduced whose equations' terms are lower than that of the Lorenz equations [24]. Various dynamics of a quadratic system were studied in [25]. Most of the quadratic systems have at least one linear term in one of their equations [26]. Few systems have been introduced with no linear term in their equations [27]. Xu and Wang introduced such a system built by just nonlinear quadratic terms for the first time [28]. As another example of the pure nonlinear systems category, a multistable system can be mentioned with heterogeneous attractors [29]. Here, an oscillator with absolute nonlinear terms is introduced to generate various types of nonlinear dynamics' behaviors such as torus and chaos. e chaotic feasibility of nonlinear ODEs systems always has been a question. Designing analog circuits for chaotic systems has been a hot topic recently. Electrical circuits simulated with PSpice or implemented physically are tools to assess ODE systems' chaotic behaviors. For example, an electrical circuit was introduced to regenerate the chaotic signals with a multiscroll dynamic [30]. In another instance, analog electrical circuits of a system with multistability were impacted [23]. Using memristors to model chaotic dynamics is one of the hot topics; for instance, a five-dimensional system with three linear dimensions was implicated using two memristors [1]. Besides, chaotic systems' implication using digital circuits like field-programmable gate array (FPGA) has been carried out to assess the possibility of implicating chaotic systems. For instance, a jerk system feasibility with strange coexisting attractors was assessed with FPGA [31]. In another example, the chaotic time series of a system with coexisting attractors and strange fixed points' curves was regenerated using FPGA [23]. One of the applications of these circuits is random number generation [32]. Other applications can be secure communications [33] and image encryption [34]. In this work, the system's analog circuit is designed with PSpice, and the results of simulations are reported. e complexity of chaotic systems' signals has recently become an exciting subject for researchers [35]. For instance, the complexities of a system with hidden attractors (for time series of its different parameters' values) were calculated and discussed [36]. Sample entropy is a feature for comparing the complexity of time series repetitively [37]. In this method, the philosophy of calculating complexity is based on the possibility of predicting the future of the signals based on their previous samples [37].
is method has some advantages in comparison with other methods of measuring complexity. For instance, it is less dependent on the length of time series than approximated entropy [37]. Here, sample entropy is used for calculating the complexities of the oscillator's signals for different ranges of the introduced system's parameters.
Controlling chaotic oscillators has been an interesting topic [38]. Various methods have been proposed to control the chaotic dynamics [39,40]. Impulsive control is a method of stabilizing nonlinear systems such as the ones with infinite [41,42] or finite delays [43], delayed neural networks [44] (that also includes exponentially stabilization [45] and fixed time control [46]), stochastic delayed systems [47], or singularly perturbed models [48]. For instance, it was used for stabilizing systems whose states are not measurable [49]. In another example, an event-based version of this method was used for controlling Chua-coupled systems [50]. is method also has been used for synchronization among nonlinear systems [51], switched complex networks [52,53], high-dimensional Kuramoto systems [54], and fuzzy neural networks [55]. Some advanced methods of impulsive control have been introduced, for instance, versions with adaptable frequencies [56]. In this paper, an impulsive-based method for controlling the introduced pure nonlinear system is implicated as a possible solution for stabilizing its equilibrium points.
In the next section, the system's equations whose terms are all nonlinear quadratic are presented (Section 2). Also, the oscillator's bifurcation and Lyapunov diagrams for different parameters' values are analyzed. Besides, the basin of attractions of the pure nonlinear oscillator is plotted and discussed. Section 3 explains the design of the introduced pure nonlinear oscillator's analog circuit and its 2 Complexity simulations with PSpice. e next part assesses the complexity of the oscillator's signals for various parameter values (Section 4). Applying the impulsive control method (Section 5) to the proposed system helps to enhance its applications. e simulations' results are concluded in the final part (Section 6).

The Highly Nonlinear System: Analytical and
Numerical Analysis e construction of chaotic dynamics is an unknown subject that attracted lots of attention [3,57]. After revealing some counterexamples for the hypothesis of a relation between saddle equilibrium points and chaotic attractors [58,59], many works have been focused on studying chaotic flows with unique properties [60,61]. ey have tried to understand the construction of chaotic attractors. Some examples are chaotic flows with different equilibrium points [62,63] and special attractors [64]. So, a pure nonlinear chaotic flow is proposed here, and its various dynamics are investigated. e oscillator can be described by three-dimension equations that are coupled as follows: where x, y, and z are the system's variables when a and b are considered parameters. e system is symmetric with the change of variables (x, y, . So any attractor of system (1) has a twin in reversed time and is symmetric to the origin of the main attractor. e system's equilibrium points are as follows: Considering these eight fixed points, the system's Jacobian and eigenvalues are as follows: e types of equilibria when a � 0.5 and b � − 2.3 are shown in Table 1 (considering eigenvalues for each equilibrium). e system's attractors for different parameters' values have been presented in Figure 1. e Lyapunov exponent and bifurcation diagrams for different parameters' set are calculated to investigate more about possible behaviors that the introduced system can present. Firstly the b parameter is fixed (b � − 2.3), and Lyapunov and bifurcation diagrams for a range of a are plotted ( Figure 2). Figure 2(a) demonstrates two Lyapunov exponents that have higher values than the rest. e third Lyapunov exponent's values are always negative and have a higher absolute value than the two others. For the two Lyapunov with higher values, the system's behavior is periodic when one is zero and the other is negative. For the situation that one of them is zero and another is positive, the system's behavior is chaotic. When both are zero, the system's behavior is the torus. Figure 2(a) demonstrates all of the mentioned situations; therefore, the system has the capability of having limit cycles, torus, and chaotic solutions. Figure 2(b) is the bifurcation diagram for the same range of a . Period windows can be seen in Figure 2(b). In the bifurcation diagram, a period-doubling route to chaos can be observed by decreasing parameter a.
In the next step, parameter a is fixed, and the oscillator's behaviors for various b are investigated. Figure 3 reveals the Lyapunov exponent and bifurcation diagrams when a � 0.5 and the b's value changes. For better visualization, the Lyapunov exponent with the largest absolute value (its value is always negative) is not plotted in Figure 3(a). e system's different behaviors from different limit cycles' periods to torus and chaos can be seen based on the previously explained situations of the two larger Lyapunov exponents (Figure 3(a)). An inverse route of the period-doubling route to chaos can be observed in the bifurcation diagram by increasing b (Figure 3(b)). e basin of attractions when the oscillator's parameters are set a � 0.5 and b � 2.3 are plotted for a range of initial values ( Figure 4). Two surfaces each containing four equilibrium points are plotted. Studying the basin of attraction of the oscillator shows that the oscillator has only one attractor.
) are located at the edge of the unstable region and the basin of attraction. e type of both of them is unstable (spiral). It can be seen that some equilibrium points exist in the system's attractor's basin of attraction. erefore, the system's attractor is self-excited.
In the next section, an analog circuit of the system is implicated for the system when it is in its chaotic mode.

Analog System's Circuit, Design, and
PSpice Implication e pure nonlinear system's analog circuit in the chaotic mode is designed. Simple elements such as resistors and Op-Amps are used in its designed circuit. Its circuit's schematic is demonstrated in Figure 5. AD633/AD as an analog device is used for multiplying variables together. e values of capacitors and resistors are tuned to compensate for the mentioned coefficient. To avoid the analog devices' saturation, Table 1: e pure nonlinear system's equilibrium points and their related eigenvalues when a � 0.5 and b � − 2.3 are set. e equilibria types are determined based on their eigenvalues.

Equilibrium points
Eigenvalues 4 Complexity x � 10X, y � 10Y, z � 10Z, and t � 0.1T are considered. erefore, the system's equations can be rewritten as follows: e new version of the system's equation (Eq. (4)) is designed by analog elements ( Figure 5). e values of the analog elements are as follows: and C3 � 10nF. Finally, the implicated system's equation to simulate in PSpice can be written as follows: e circuit simulation in PSpice when a � 0.5 and b � − 3.7 is demonstrated in Figure 5. All elements that are used are analog. e outputs of the designed analog circuit compared to Matlab simulations are demonstrated in Figure 6.

The Pure Nonlinear System's Complexity Analysis
Defining the complexity of the time series based on their predictability results in the definition of sample entropy  (SaEn). Accepting this definition, SaEn is applied to estimate the complexity of the system's time series, as reviewed in the Introduction section. SaEn tries to measure the predictability of (t + 1)th samples of time series when the previous samples (1, 2, . . . , t) are observed. e algorithm of calculating SaEn can be read in [37]. To calculate the algorithm of SaEn, m � 2 and r � 0.2 are considered. e algorithm is applied to the oscillator's attractors (the x variable time series) for ranges of the parameters (a and b). e initial conditions are considered (0, 0, 0) and the transient time parts of the time series are emitted before calculating SaEn. e results of SaEn values can be observed in Figure 7. e attractor is a fixed point in parameters that SaEn values are zero. A trend can be seen that increasing a, at first, causes an increase in SaEn and then decreases it. In comparison with Figure 2, generally chaotic states of the system have more sample entropy values than periodic ones. Besides, a trend also can be observed that decreasing b parameter values increases SaEn values. Comparing this trend with the bifurcation diagram reveals that the chaotic regions generally have more complexity.

Impulsive Control
Here, the pure nonlinear oscillator is controlled using impulsive control. In the first step, the system under impulsive control can be described as follows [65][66][67]: When f: R + × R n ⟶ R n is continuous, U: R n × R n ⟶ R n is continuous; X ∈ R n is the vector of state variables; and 0 < T 1 < T 2 < ... < T K < T K+1 < . . . , T K⟶∞ as k ⟶ ∞. AX, in general, represents linear terms of systems when Φ(X) contains nonlinear terms.
Definition 3 (comparison system). Let V ∈ V0 and assume that where g : R + × R + ⟶ R is continuous and Ψk : R + ⟶ R + is nondecreasing. en the following system is the comparison system of Eq. (6): Theorem 1. ese three conditions are assumed:  (6) origin is global asymptotically stable if eorem 1 conditions and the following conditions are held: Theorem 3. e origin of the introduced pure nonlinear chaotic system is asymptotically stable if there exists a ξ > 1 and a differentiable at t ≠ T k , and nonincreasing function K(t) which satisfies the following: q is the (A + P − 1 A T P) largest eigenvalue assuming P is a positive definite symmetric matrix and λ 1 > 0 and λ 2 > 0 are the smallest and the largest eigenvalues of P, respectively. ρ(A) denote the spectral radius of A: d � ρ 2 (I + B). M for the pure nonlinear chaotic system considered so that |x (t) | < M, |y (t) | < M, |z (t) | < M.K(t). It is as in eorem 1, T i : i � 1, 2, . . . are varying but satisfy the following: Furthermore, for a given constant ε, is theorem's proof can be seen in [66]. Remark 1. eorem 3 also gives an estimate for the upper bound. Δ1 max and Δ2 max of impulsive intervals are given by e introduced pure nonlinear system when a � 0.5 and b � 2.3 are set is considered. According to the second section, this system has eight equilibrium points. Assuming (x * , y * , z * ) as an equilibrium point of the system, the system equations considering x 1 � x − x * , y 1 � y − y * , z 1 � z − z * can be rewritten as follows: Equilibrium is considered to be stabilized. Without losing generosity for stabilizing equilibrium points of the system, the same method can be applied to the other equilibria of the pure nonlinear system. In this way, considering (6) and (12), for the equilibrium point ( , equations can be rewritten as follows: Considering considered 0.1)). e stabilized system numerical simulations are plotted in Figure 8. Figures 8(a)-8(c) demonstrate the time series of x 1 , y 1 , and z 1 , respectively, for the oscillator described in Eq. 12. e time series of x 1 , y 1 , and z 1 for the stabilized system using the impulsive controller (based on Eq. 13) are demonstrated in Figures 8(d)-8(f ), respectively.

Conclusion
Here, a pure nonlinear 3D system was presented. It was observed that the system could generate periodic, torus, and chaotic time series. Analytical analysis revealed that the oscillator has eight unstable equilibrium points for a set of parameters. e basin of attraction diagrams showed for this set of parameters that the system attractor is self-excited. e pure nonlinear system's feasibility was investigated with an analog circuit built by simple elements like capacitors and Op-Amps. Changing parameters' values revealed that the system could generate time series with a wide range of complexities. A possible solution to system stabilization was described by using the impulsive controller on the system. For this system, when both constants (parameters a and b) were equal to zero, the system had an unbounded solution. According to the authors' best knowledge, no pure nonlinear quadratic system has been introduced before with no constant values in its equations. erefore, searching for such a system can be interesting for future research.

Data Availability
All the numerical simulation parameters are mentioned in the respective text part, and there are no additional data requirements for the simulation results.

Conflicts of Interest
e authors declare that there are no conflicts of interest.