Coexistence of Multiple Attractors in a Novel Simple Jerk Chaotic Circuit With CFOAs Implementation

A novel, simple Jerk chaotic circuit with three current feedback operational amplifiers included (CFOA-JCC) is proposed, which has a simpler circuit structure, fewer components, but higher frequency characteristics. The dynamic behaviors of CFOA-JCC are analyzed, including equilibrium, stability, Lyapunov exponent, bifurcation diagram, offset boosting, and phase diagram. Furthermore, the frequency spectrum characteristic of the ordinary op-amps Jerk chaotic circuit was compared with CFOA-JCC under the same circuit parameters, and the chaotic attractor frequency of CFOA-JCC can reach about 650 kHz, much better than that of ordinary op-amps (12 kHz). Numerical simulation shows that CFOA-JCC has coexisting attractors, verified by hardware circuit experiments.


INTRODUCTION
Chaos is a complex nonlinear phenomenon with special dynamical properties, and various chaotic systems are found and widely used in the fields of electronic communication systems, control systems, and so on [1][2][3][4][5][6]. Chaotic circuits have become an important tool for studying chaos theory due to easy observation. Proposed by Sprott based on the method of computer exhaustive, the Jerk chaotic system (a class of third-order autonomous chaotic systems) has attracted much attention because it is easy to achieve complex dynamical behavior with a simple structure. The general mathematical expression is x ··· J(x ·· , x · , x) [7][8][9][10]. In 2011, Sprott proposed a novel chaotic circuit of Jerk, which realizes a chaotic system with only resistance, capacitance, operational amplifier, and diode. The three first-order differential equation is as follows [11]: The nonlinear term in the chaotic circuit can be nonlinear functions, piecewise linear functions, and mersister generation [12][13][14]. However, most Jerk circuit implementations use traditional voltage mode op-amps as active devices, and the gain of the designed circuit must be decreased to increase bandwidth. For the current feedback operational amplifier (CFOA), there is almost no such relationship between gain and bandwidth. Therefore, it is not necessary to design a circuit to balance the gain and bandwidth as a voltage feedback amplifier [15][16][17]. Therefore, using CFOA to realize chaotic circuits has become a hotspot because CFOA has better frequency gain characteristics [18][19][20][21][22][23][24]. It makes the circuit structure simpler and more flexible due to its good port characteristics [25][26][27][28][29]. There have been many reports about the coexistence attractors in recent years, but all the reports did not pay much attention to the simpler topology of the chaotic circuit. For example, in 2011, Sprott proposed a Jerk chaotic circuit with traditional voltage feedback operational amplifiers (VFOA). He did not discuss the coexistence of attractors of his system, and the circuit was not the simplest one. In 2016, Kengene et al. proposed a novel chaotic Jerk circuit, which can realize the coexistence of multiple attractors [30]. Five amplifiers and two diodes had been used in the circuit, and they discussed the frequency of the circuit. In their circuit, the center frequency can be obtained at about 4 Hz. In 2020, Xu et al. proposed a memristive Chua's circuit with attractors with two stable foci [14]. A onedimensional multi-scroll chaotic circuit is designed with CFOA [26], but the circuit structure is complex, and there are many active devices. A grid multi-scroll chaotic circuit is designed using CFOA [27,28], but more active devices are used with lower operating frequency. Wu proposed a chaotic circuit only using CFOA [29], but the center frequency of the chaotic signal is 250 kHz.
In this article, a novel Jerk chaotic circuit with three CFOAs (called CFOA-JCC) was designed to realize a chaotic system in Description of the CFOA-JCC. In Equilibrium Point Analysis, based on the equation of state of CFOA-JCC, the equilibrium point and the system dissipation are analyzed by solving the dimensionless equation. In Dynamical Behaviors, numerical analysis and description were done using MATLAB R2018b to perform the numerical simulation. Lyapunov exponent spectra and bifurcation diagram are plotted in adjusting the parameter regions to reveal the dynamical behaviors. The occurrence of coexisting attractors at deferent initial conditions is also discussed using phase diagrams and bifurcation. Single parametercontrolled offset boosting for variable x is simulated. In Circuit Implementation and Spectrum Characteristics, hardware experiments are performed, the phase and offset boosting diagrams in the x-y plane captured by the UNI-T2102CM oscilloscope strongly confirm the previous theoretical analysis, and then the spectrum of CFOA-JCC is compared with that of the VFOA circuit in adjusting the parameter regions of capacitance under the same value of the resistor. The conclusions are summarized in the last section.

DESCRIPTION OF THE CFOA-JCC
AD844AN is a typical CFOA with better frequency properties. Its circuit symbol is shown in Figure 1A. Under ideal conditions, the port characteristics of AD844AN are satisfied: .
With the current output port, the current state of CFOA is easy to observe. As can be seen from the characteristic curve in Figure 1B, the gain of both op-amps is stable when the frequency is within 100 kHz. AD844AN has a better high-frequency performance than VFOA. The closed-loop bandwidth frequency at −3 dB can be expressed as f cl 1 2πC P R f . The closed-loop bandwidth of the CFOA is determined by the feedback resistor R f and the internal circuit C p . It illustrates the importance of the feedback resistance R f of the CFOA, so R f can be used to adjust the frequency response of the amplifier [31]. This designed circuit configuration is shown in Figure 2, which includes three CFOAs; three capacitors C 1 , C 2 , C 3 ; four resistors R 1 , R 2 , R 3 , R 4 , and a silicon diode named D 1 . This circuit is derived from the CFOAs-based sinusoidal oscillator [32], and a nonlinear function is obtained according to volt-ampere characteristics of the diode.

EQUILIBRIUM POINT ANALYSIS
According to Kirchhoff's Current Law and Kirchhoff's Voltage Law, the port characteristics of CFOA, and the current-voltage relations of each circuit component of the novel circuit in Figure 2, in terms of the three-node voltage of u 1 , u 2 , u 3 , three coupled firstorder autonomous nonlinear differential equations can be attained to express the nonlinear dynamics of the circuit. The differential equations of the proposed system can be written as .
The voltage signals u 1 , u 2 , u 3 and time variable t are nondimensional to obtain x, y, z through the following relationship [33]: When it is at room temperature 300 K, the diode reverse saturation current I S = 10 −9 A and the voltage equivalent of temperature V T = 26 mV [34]. Therefore, Eq. 3 can be written as Assuming x · 0, y · 0, and z · 0 to analyze the equilibrium point and stability, Eq. 3 has a zero equilibrium point P 0 = (0,0,0) [35].
The Jacobin matrix at the equilibrium point can be derived as The characteristic equation at the equilibrium is yielded as The Routh-Hurwitz conditions for the above cubic polynomial Eq. 8 are given by [36][37][38][39] Obviously, the first and second conditions are satisfied in this system, but the third condition is only satisfied when ε and δ take very large values. Then, the system can be stable. Nevertheless, the system is unstable when the value of ε and δ is small, and the above characteristic equation has both positive and negative roots.
The circuit parameters in Figure 2 are given as R 1 = R 2 = R 3 = R 4 = 1 kΩ and C 1 = C 2 = C 3 = 100nF. Therefore, the normalized parameters can be calculated by Eq. 4 as α = 1, β = 1, γ = 1, δ = 1, and ε = 1, so that the characteristic equation at the equilibrium is yielded as Three eigenvalues can be obtained from Eq. 10: λ1 = −1.4665571, λ2 = 0.232786 + 0.792552i, and λ3 = 0.232786 + 0.792552i. Among them, λ2 and λ3 are complex roots, and λ1 is a negative real root. From the three eigenvalues, it can be seen that P0 (0, 0, 0) is an unstable saddle point. Consequently, system (3) may produce chaotic behaviors [40][41][42]. System (3) is a three-dimensional system, and the divergence of its vector field can be described as It shows that the system is dissipative. When t→∞, each volume element of the trajectory of the system shrinks to 0 at an exponential rate of −1, and its gradual motion will be fixed on an attractor [43][44][45][46].
The system can maintain a chaotic state when the parameter ε changes within a certain range. The value of the max LE of the system is approximately 0 when ε ∊ [0.001, 0.394), which means the system is in a period state. The bifurcation diagram shown in Figure 3D is represented by one or several curves. When ε ∊ [0.0.39, 0.394), the system produces a bifurcation, which evolves from period 1 to period 2, respectively. When ε ∊ [0.394, 1.2), there is one positive LE, which means the system is in a chaotic state. For instance, as Figure 3C shows, when ε = 0.5, the LEs can be calculated as LE 1 = 0.14215, LE 2 = −0.63976, LE 3 = −0.0023916. LE 1 is positive, and LE 2 and LE 3 are negative, meaning the system's phase volume is exponentially shrinking. According to the Kaplan-York dimension (D KY ), an integer j must satisfy the following requirement, so D KY of the chaotic state can be calculated as follows: .
In Figure 3D, the bifurcation diagram also proves the system is chaotic when ε ∊ [0.394, 1.2). With the increase in ε, the system returns to the periodic state when ε ≥ 1.2. From Figure 3D, we also can obtain that there are two different attractors when the initial value is fixed (x 0 , y 0 , z 0 ) = (0, 0.4, 0) and (x 0 , y 0 , z 0 ) = (0, −0.4, 0). That means there are coexisting attractors in this system. Figure 4 shows the phase diagram of the system corresponding to different ε values. The process of the system from period bifurcation to chaos and then back to the periodic state can be clearly seen from the diagrams. The trajectories colored in red start from the initial condition (x 0 , y 0 , z 0 ) = (0, 0.4, 0) and those colored in blue correspond to (x 0 , y 0 , z 0 ) = (0, −0.4, 0). For the two initial conditions, it can be observed from Figure 4 that when ε = 0.5, the chaotic attractor is obtained, as shown in Figures 4A1, A2. Then as ε increases, the system returns to the periodic state, but the period attractors are different under two initial conditions, which are shown in Figures 4B1,B2,C1,C2.
The state variable x appears once in the second equation of Eq. 6. Consequently, the state variable x can be easily boosted. In system (5) [47,48], we fixed the parameters as α = 1, β = 1, γ = 1, δ = 1, and ε = 1 and added a parameter k, which can realize the offset boosting without changing the basic dynamics as the substitution of x → x + k transforms it to its original form at k = 0. Then, the equation can be written as To better clarify the variable-boostable phenomenon, different k was selected to plot the phase diagrams. For the dimensionless system (13), the fixed time step of 0.001 was used for numerical simulations. The projections of attractors in the x-y plane are given in Figures 5A,B, where the blue, red, and yellow orbits correspond to k = 2 for (x 0 , y 0 , z 0 ) = (−2, 0.4, 0), k = 0 for (x 0 , y 0 , z 0 ) = (0, 0.4, 0),  and k = −2 for (x 0 , y 0 , z 0 ) = (2, 0.4, 0). It is found that the attractor is linearly boosted towards the x direction. Besides, we choose a relatively long time interval τ ∊ [1000,5000] to compute the average and amplitude of each variable [49][50][51]. The average and amplitude (maximum and minimum size) of x, y, and z are plotted in Figures  5C,D, respectively. The results show that both the average and amplitude of the variable x are boosted when adjusting k, whereas the other three variables almost keep unchanging.

Circuit Implementation
To verify the authenticity of the simulation results, the real circuit was made to physically implement the system shown in Figures 6A,B, in which one precision potentiometer, three monolithic ceramic capacitors, one silicon diode 1N4148, three carbon film resistors with the error of ±5%, and three CFOAs AD844AN with operating voltages of ±12 V     Table 1. The experiment results of x-y plane phase diagrams and spectrum can be captured by the UNI-T UTD2102CM and UNI-T UPO9504Z, respectively.
According to the previous normalization formula Eq. 4, when the component parameters in the circuit are fixed, the value of ε depends on R 1 . When R 1 = 34.48 kΩ, the value of ε is 0.29, and the system is in a period one state. Moreover, the value of R 1 continues to decrease when R 1 is adjusted to correspond to the value of ε in Figure 4, respectively, and the phase diagrams are captured by UNI-T UTD2102CM as shown in Figure 7. According to the phase diagrams, it is found that when R 1 is adjusted to a certain fixed value, the phase diagrams can be consistent with the numerical simulation results. For example, R 1 = 19.8 kΩ and ε = 0.5, and its phase diagram is shown in Figure 7C. The system is in a chaotic state and is in the same state in Figure 4A. Meanwhile, it is found that when R 1 is in the range of [10.1, 12.58] kΩ, the system shows periods corresponding to another period attractor, which also fully proves that the system has the phenomenon of attractors coexistence, as shown in Figures 7D,F.

Offset Boosting Implementation
Rewrite Eq. 13 as follows: According to the circuit diagram of Figure 2 and Eq. 14, the simple Jerk circuit is built in NI Multisim 14.0, as shown in Figure 8A. A circuit in the red dashed box, containing a precision potentiometer, a common discharge constituting a voltage follower, and a resistor forming DC current I k , is incorporated into the reverse port of u 1 . Adjusting for R7 to 35, 50, and 65%, respectively, when V k = −4.5 V, V k = 0 V, and V k = 4.5 V, its chaotic attractors correspond to the blue, red, and yellow attractors in Figure 8B.
To verify the implementation ability of the simulation, we performed the experimental test as shown in Figure 9A. Adjusting the precision potentiometer to correspond the voltage of V k to 4.5 , 0 , and −4.5 V, respectively, the attractor phase diagrams can be captured by the oscilloscope as shown in

Frequency Spectrum Characteristics
The frequency spectrum is one of the most important characteristics of the chaotic signal. The frequency-domain characteristics of chaos are of great significance in chaotic encryption communication and automatic control [52]. In the integrator circuit, the time constant τ = RC, when R is determined to be unchanged, changing the value of the capacitor C, different time constants can be obtained [53].
For comparison, an experimental circuit using four ordinary op-amps to realize the chaotic system [11] was made as Figure 10 shows which includes four VFOAs AD711AN; three capacitors C 1 , C 2 , C 3 ; five resistors R 1 , R 2 , R 3 , R 4 , R 5 ; one precision potentiometer R 6 ; and one silicon diode D 1 1N4148, which provides the nonlinear function.
The specific parameters are that all the resistances are 10 kΩ in both of the two kinds of the chaotic circuit, and all capacitors are changed according to the experimental requirements at the same time. The frequency spectrums of the chaotic attractor under different time constants are obtained, as shown in Figure 11, during changing the value of integrating capacitors. It can be seen that when C = 100 nF, the center frequency of the two chaotic circuits is about 1.5 kHz. When C = 10 nF, the center frequency is about 12 kHz. Continuously, when C = 1 nF, the circuit with VFOAs is no longer chaotic. It can be seen from the frequency spectrum and the phase diagram that it is in a periodic state. However, CFOA-JCC is still in chaos and the center frequency of the chaotic attractor is about 120 kHz. What is more, the center frequency can be improved to about 650 kHz when C = 200 pF in the CFOA-JCC. Compared with the chaotic circuit based on VFOA, it can generate a highfrequency chaotic signal. Consequently, the CFOA-JCC is not only simple in structure but also can increase the operating frequency of the circuit and broaden its spectrum range. In ideal conditions, the dominant frequency of oscillation is expected to be f ω 2πRC and ω is the imaginary part of λ 2 and February 2022 | Volume 10 | Article 835188 10 λ 3 11 . The values of f can be obtained as shown in Table 2 in the case of different capacitances when all resistance values are 10 kΩ. Compering with Figure 8, the value of fin Table 2 is extremely similar to the value in Figure 8 experimentally observed.

CONCLUSION
The dynamics of CFOA-JCC and the electronic circuit implementation have been discussed extensively in terms of its parameters using standard nonlinear analysis techniques such as bifurcation diagrams, Lyapunov exponent plots, offset boosting, and frequency spectra. The bifurcation analysis suggests that chaos that arises in the CFOA-JCC follows the classical period-doubling when adjusting the bifurcation control parameters slowly. It is also found that the proposed CFOA-JCC exhibits the asymmetrical and striking feature of multiple coexistence attractors for a wide range of circuit parameters. A numerical simulation and circuit experimental test verified the coexistence of multiple attractors of this system. The voltage slew rate of the feedback amplifier is almost independent of frequency and has better high-frequency characteristics. Consequently, the CFOA-JCC has better high-frequency performance than the chaotic circuit designed by ordinary VFOAs, so that higher frequency chaotic attraction can be obtained. At the same time, compared with the VFOA chaotic circuit, the CFOA-JCC has a simpler topology and fewer components, conducive to system integration.

DATA AVAILABILITY STATEMENT
The original contributions presented in the study are included in the article/Supplementary Material. Further inquiries can be directed to the corresponding author.