A Chaotic Jerk Oscillator with Complete Control via Fractional Exponentiation and Its Experimental Analog Circuit Realization

Abstract

By introducing fractional exponentiation into a three-dimensional chaotic system, a jerk system with only six terms is designed. It has the property of total amplitude control, where a single non-bifurcation parameter can directly rescale all system variables without affecting the dynamics. It also features two-dimensional offset boosting, where a single parameter can realize direct offset boosting while another provides interlocked cross-dimensional offset boosting. Furthermore, this jerk system has a parameter-dominated symmetric attractor, which means that symmetric attractors appear successively as the parameter changes from positive to negative. Circuit experiments confirm the feasibility of analog fractional exponentiation using the 444 circuit and the complete control, including amplitude control and offset boosting, of the resulting system. The proposed circuit may facilitate applications of chaotic signal generators where signal versatility is important and exemplifies the generative potential of analog fractional exponentiation.

1. Introduction

Chaos is a peculiar form of motion that is ubiquitous in natural and artificial nonlinear dynamic systems across scales of time and space, representing an interface between determinism and randomness, regularity, and complexity. To date, a plethora of chaotic systems have been discovered, such as the Lorenz system [1,2], the Chua system [3,4], and the Rössler system [5,6]. Based on the computational enumeration method, Sprott proposed a class of simple third-order autonomous chaotic systems known as jerk systems. Such systems find widespread applications in control and signal processing systems and are currently the subject of active research. Their equation takes the general form $\stackrel{⃛}{x}=J\left(x, \dot{x}, \ddot{x}\right)$, where represents $\dot{x}$ velocity, $\ddot{x}$ is acceleration, and $\stackrel{⃛}{x}$ is jerk. In 1997, Sprott proposed some standard forms and the simplest quadratic form of jerk systems [7]. Subsequently, notable results have been obtained by modifying chaotic systems based on jerk equations and uncovering different types of dynamics, such as coexisting attractors [8], multi-scroll attractors [9], and hyperchaos [10].

In addition, much research has been conducted on shaping chaotic dynamics through amplitude control and offset boosting. A chaotic system that possesses both of these controls is said to offer complete control [11]. Several studies have shown that in suitable chaotic systems, the amplitude and direction of the chaotic orbits and corresponding signals can be freely controlled by independent non-bifurcation parameters. In particular, amplitude control provides a convenient means of rescaling signals, allowing the generation of chaotic signals with the desired amplitude adjusted by a single knob [12,13]. On the other hand, offset boosting allows chaotic signals to arbitrarily switch between bipolar and unipolar states by introducing an independent constant that can move the variables to any position in phase space [14,15]. When chaotic systems are used in large-scale networks, such as networks that replicate neural connectomes, or in practical engineering applications, such as mixing and steering control, the simple control of the size and location of the attractor, independent of its geometry, provides maximum flexibility [16]. In this paper, we construct a chaotic system that provides full amplitude control and two-dimensional offset boosting, where non-bifurcation parameters realize full control over the size and location of the attractor.

As the properties of chaotic dynamics have been progressively revealed, the diligent performance of circuit experiments has become an indispensable step, not only to confirm the viability of numerical simulations but also to discover entirely new phenomena. In addition, experiments are of paramount importance to reveal the effects of component non-idealities and tolerances, which are ubiquitous in physical devices but difficult to represent in numerical simulations. Circuit experiments are also a prerequisite for demonstrating the applicability of chaotic systems in engineering practice. For example, the Chua circuit [17], which is well-known as a basis for implementing chaos via analog circuits, is structurally simple and requires only a handful of components. If the state equations of a system are known, a modular design approach can be used to design a corresponding analog circuit, using basic elements such as adders, subtractors, multipliers, integrators, and inverters to implement prescribed algebraic relations [18,19]. This method is versatile, but it often results in circuits that are significantly larger than those discovered by serendipity or from first principles, which may be as simple as requiring only a single transistor. In recent years, many researchers have pursued simplifications of chaotic circuits and systems [20,21].

For example, a jerk circuit based on the absolute value function was implemented by Li et al. [22] using diodes and operational amplifiers, while Wu et al. utilized the coupling of capacitors and resistors along with the inherent characteristics of multipliers to realize compact quadratic computation and generate chaos [23]. In some circumstances, chaos can be produced in fractional order systems [24,25] or even resort to the feedback of fractional exponentiation. In this paper, the circuit implementation of a three-dimensional jerk system has been optimized by exploiting the characteristics of a positive-phase integrator with only a minimal number of operational amplifiers and linear components while incorporating a novel circuit that realizes fractional exponentiation. This circuit structure is highly advantageous for advancing future research on memristive chaotic systems as well as in-depth studies of neural dynamics [26]. The novelty concerns the following aspects:

(1)

The overall amplitude control parameter is discovered in the designed chaotic oscillator, allowing one to directly influence the amplitude of all system variables;

(2)

Two offset parameters are identified, one realizing offset amplification along one dimension and the other providing cross-dimensional offset amplification;

(3)

Fractional exponentiation is achieved in the analog domain using logarithmic and exponential operation modules as building blocks, exploiting the nonlinear response of bipolar transistors, according to a newly introduced circuit known as the 444 circuit;

(4)

A detailed experimental characterization is undertaken.

The remainder of this paper is organized as follows. In Section 2, the model and dissipative analysis are provided. In Section 3, the dynamic behavior is analyzed. In Section 4, the possibilities of amplitude control and offset boosting are introduced. In Section 5, the system is implemented using a simplified analog electronic circuit, yielding experimental results consistent with theoretical analysis and numerical simulations. Finally, Section 6 provides a concluding summary.

2. System Definition

Based on the notion that nonlinear feedback is the fundamental prerequisite for chaos generation in dynamic systems, a modification of the jerk system MO₄ [27] was made by introducing fractional exponentiation, namely,

$$\left\{\begin{array}{l}\dot{x}=y, \\ \dot{y}=az, \\ \dot{z}=-z-cy+|y{|}^b-x.\end{array}\right.$$

(1)

The original system of MO₄ has only a single nonlinear quadratic term x², and it includes another linear term of x, which provides the possibility to replace the nonlinear term with a fractional exponentiation of ${\left|x\right|}^b$. For the controllability of two-dimensional offset boosting, we further revise the feedback ${\left|x\right|}^b$ with ${\left|y\right|}^b$, which happens to be satisfied by another linear feedback of y. In this system, a, b, and c are the system parameters, assumed to be constant, while x, y, and z are the state variables. The system comprises five linear terms and one nonlinear term. When its parameters are set to a = 1, b = 2.4, and c = 0.6, given the initial condition (0, 0.1, 0), the Lyapunov exponents (LEs) are found to be (0.0527, 0, −1.0527), and the Kaplan–Yorke dimension (D_KY) is consequently 2.05. The corresponding chaotic attractor obtained by integrating the system using a variable-step fourth-order Runge–Kutta method is shown in Figure 1.

As Equation (1) represents a jerk system, it can be rewritten more compactly as a third-order differential equation, namely,

$$\stackrel{⃛}{x}=-\ddot{x}-ac\dot{x}+a{\left|\dot{x}\right|}^b-ax.$$

(2)

3. Dynamical Analysis

3.1. Equilibrium Points and Stability

Let ẋ = 0, ẏ = 0, ż = 0, yielding the following:

$$\left\{\begin{array}{l}y=0, \\ az=0, \\ -z-cy+{\left|y\right|}^b-x=0.\end{array}\right.$$

(3)

When the parameter settings are taken as a = 1, b = 2.4, and c = 0.6, the system in Equation (1) has only one equilibrium point. In order to obtain the eigenvalues of the equilibrium point and determine its stability, the Jacobian matrix at the equilibrium point S₀ (0, 0, 0) can be obtained, giving the following:

$$J=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& a\\ -1& -c+b{\left|y\right|}^{b-1}\mathrm{sgn}(y)& -1\end{array}\right]$$

(4)

Thereafter, the characteristic equation can be written as

$${\lambda }^3+{\lambda }^2+c\lambda +1=0$$

(5)

from which the eigenvalues are found to be ${\lambda }_1=-1.1968, {\lambda }_2=0.0984+0.9088i, \mathrm{and} {\lambda }_3=0.0984-0.9088i$. Therefore, S₀ is a saddle-focus point of index 2.

3.2. Bifurcation Analysis

To analyze the sensitivity to parameter changes, let a = 1 and c = 0.6, with an initial condition of (0, 0.1, 0). When the parameter b varies within the interval [1.5, 3], its Lyapunov exponents and bifurcation diagram evolve as depicted in Figure 2. It can be observed that when the parameter b increases, the system undergoes multiple transitions between periodic and chaotic states. In particular, periodic orbits are found within the intervals [1.5, 1.809], [1.58, 2.162], and [2.162, 2.248] and are followed by transitions to chaotic states through period-doubling bifurcations. Its typical phase trajectory is shown in Figure 3, and the LEs and D_KY values are presented in

Table 1. Summary of the dynamic parameters obtained with a = 1, c = 0.6, and the initial condition (0, 0.1, 0).

b	Attractor	Lyapunov Exponents	DKY
1.6	period 1	(0, −0.0744, −0.9256)	1
2.0	period 2	(0, −0.1663, −0.8337)	1
2.7	period 3	(0, −0.0029, −0.9961)	1
2.9	chaos	(0.0746, 0, −1.0746)	2.07

. However, as the parameter b approaches 3, the system tends to diverge, leading to the inability to generate chaotic behavior. The sign of the largest Lyapunov exponent shows good agreement with the chaotic regions in the bifurcation diagram.

When the parameter b is fixed at 2.4 and the initial values are (0, 0.1, 0), the Lyapunov exponent spectrum and bifurcation diagram for parameter a varying within the interval [0.2, 1.8] appear as depicted in Figure 4. It is observed that as this parameter increases, the system again undergoes multiple transitions between chaotic and periodic states, yielding the attractor exemplified in Figure 5.

4. Complete Control

4.1. Amplitude Control

When a = 1, following the substitutions $x\to n^{{\displaystyle \frac{1}{b-1}}}x, y\to n^{{\displaystyle \frac{1}{b-1}}}y, z\to n^{{\displaystyle \frac{1}{b-1}}}z. \left(n>0\right)$, Equation (1) can be rewritten as

$$\left\{\begin{array}{l}\dot{x}=y, \\ \dot{y}=az, \\ \dot{z}=-z-cy+n|y{|}^b-x.\end{array}\right.$$

(6)

It can be observed that the coefficient in front of $|y{|}^b$ enables global amplitude control over x, y, and z. In other words, changes in the parameter n scale the state variables x, y, and z proportionally. Since $|y{|}^b$ is the only nonlinear term in the system, the value of n cannot be zero. At the same time, parameter b serves as the bifurcation parameter for the entire system. By adjusting its value, one can effectively control the amplitude effect of the non-bifurcation parameter n. As shown in Figure 6a, when the parameter n takes values within the range [0.1, 10] while other parameters are fixed, the average magnitudes of the state variables x, y, and z decrease as parameter n increases. Figure 6b illustrates the Lyapunov exponent spectrum of the system when n varies over the interval [0.1, 10]. It is confirmed that varying this parameter does not influence the Lyapunov exponents, indicating the absence of geometric changes in the chaotic trajectories. Some examples of the attractors obtained for different settings of the parameter n are depicted in Figure 7.

When $x\to -n^{{\displaystyle \frac{1}{b-1}}}x, y\to -n^{{\displaystyle \frac{1}{b-1}}}y, z\to -n^{{\displaystyle \frac{1}{b-1}}}z.\left(n>0\right)$, Equation (6) can be rewritten as

$$\left\{\begin{array}{l}\dot{x}=y, \\ \dot{y}=az, \\ \dot{z}=-z-cy-n|y{|}^b-x.\end{array}\right.$$

(7)

When the feedback of fractional exponentiation becomes negative, its phase trajectories are as shown in Figure 8, which is equivalent to inverting all the variables, thus achieving symmetry of the attractor around the origin.

4.2. Two-Dimensional Offset Boosting

Let us consider two parameters d₁ and d₂, which provide offset boosting over the system variables x and z. The parameter d₁ controls the offset of the variables x and z in opposite directions, while parameter d₂ directly controls the variable x. These parameters are added, respectively, to the second and third dimensions of Equation (1).

$$\left\{\begin{array}{l}\dot{x}=y, \\ \dot{y}=az+d_1, \\ \dot{z}=-z-cy+{\left|y\right|}^b-x+d_2.\end{array}\right.$$

(8)

When replaced with $x\to x+{\displaystyle \frac{d_1}{a}}+d_2, y\to y, z\to z-{\displaystyle \frac{d_1}{a}}$, Equation (8) reverts to Equation (1), indicating that the introduced constants do not alter the system structure but provide opportunities for offset boosting the variables x and z.

First, when $d_1=0, d_2\ne 0$, offset boosting over the variable x is achieved. The result of the replacement is $x\to x-d_2, y\to y, z\to z$, indicating that a single parameter $d_2$ added to the third dimension of Equation (1) offers independent offset boosting of variable x without any change in the dynamics, as illustrated in Figure 9.

Second, when $d_1\ne 0, d_2=0$, offset boosting along the dimensions of variables x and z is achieved. The result of the replacement is $x\to x-{\displaystyle \frac{d_1}{a}}, y\to y, z\to z+{\displaystyle \frac{d_1}{a}}$, indicating that a single parameter d₁ introduced in the second dimension of Equation (1) performs cross-dimensional interlocked offset boosting over variables x and z in opposite directions, while the variable y remains unchanged, as illustrated in Figure 10.

Third, when $d_1=-a{d}_2\ne 0$, offset boosting along the dimension of z is achieved. The result of the replacement is $x\to x, y\to y, z\to z-d_2$, indicating that the addition of constrained variables in the second and third dimensions allows for independent offset boosting over the variable z, as illustrated in Figure 11.

Finally, when $d_1\ne 0, d_2\ne 0$, the offset boosting in Equation (1) is derived from $x\to x-{\displaystyle \frac{d_1}{a}}-d_2, y\to y, z\to z+{\displaystyle \frac{d_1}{a}}$, resulting in differential offset boosting over the variables x and z, as illustrated in Figure 12.

To better illustrate the aforementioned properties, when $d_1=0, d_2\ne 0$, with changes in parameter d₂, the mean values of each system variable are depicted in Figure 13a. It can be observed that the average value of variable x increases proportionally, while the averages of y and z remain constant, indicating that variations in parameter d₁ do not induce offset boosting over dimensions x and z. Similarly, when $d_1\ne 0, d_2=0$, the mean values of the variables x and z exhibit proportional changes in opposite directions, with variations in parameter d₁, while the mean value of y remains stable, as illustrated in Figure 13b. When $d_1\ne -a{d}_2\ne 0$ and a = 1, it is observed that only the average value of variable z changes, confirming the achievement of independent offset boosting over variable z. When $d_1\ne 0, d_2\ne 0$, the scenario of $d_1=d_2$ is selected, resulting in mean values depicted in Figure 13d. The offset boosting over variables x and z can be adjusted arbitrarily. In conclusion, changes in the proposed parameters do not affect the dynamic behavior of the system, and the resulting flexibility appears remarkable.

5. Analog Electronic Circuit Experiments

5.1. Circuit Design and Experimental Setup

To demonstrate the practical feasibility of the system and its control, an analog computer was designed following the integrated design approach described in Ref. [28]. In other words, the corresponding algebraic relations were implemented by means of circuits based on operational amplifiers, namely integrations. The fractional exponentiation operation at the heart of the nonlinear feedback in the proposed system, which is notoriously difficult to realize in the analog domain, was provided by the 444 circuit, a newly introduced topology described in Ref. [29].

Based on Kirchoff’s current and voltage laws, the circuit equations can be straightforwardly written as

$$\left\{\begin{array}{l}\dot{x}={\displaystyle \frac{y}{R_1{C}_1}}, \\ \dot{y}={\displaystyle \frac{z}{R_3{C}_3}}+{\displaystyle \frac{V_1}{R_3{C}_3}}, \\ \dot{z}=-{\displaystyle \frac{z}{R_5{C}_5}}-{\displaystyle \frac{y}{R_8{C}_5}}-{\displaystyle \frac{x}{R_9{C}_5}}+{\displaystyle \frac{|y{|}^b}{R_6{C}_5}}+{\displaystyle \frac{V_2}{R_7{C}_5}}.\end{array}\right.$$

(9)

Let the characteristic time scale be given by τ = R₁C₁ = R₅C₅ = R₉C₅, where one recovers the control parameters a = τ/(R₃C₃) and c = τ/(R₈C₅). The amplitude boosting parameter corresponds to n = τ/(R₆C₅), while the offset boosting parameters map to d₁ = V₁/(1 V)τ/(R₃C₃) and d₂ = V₂/(1 V)τ/(R₇C₅). Nominally, n = 1 and d₁ = d₂ = 0. All parameters can, therefore, be expressed in terms of ratios of RC time constants. The corresponding circuit is shown in Figure 14, wherein the labeled voltages correspond to the state variables x, y, and z in Equation (1). Due to the inherent characteristics of the jerk system, positive integration was employed for integrating the x and y variables, while an inverse integration circuit was instantiated to integrate z. For this circuit, we initially set R₁ = R₂ = R₃ = R₄ = R₅ = R₆ = R₇ = R₉ = 100 kΩ, R₈ = 167 kΩ, and C₁ = C₂ = C₃ = C₄ = C₅ = 1 μF, giving τ = 0.1 s and an oscillation frequency in the order of 1 Hz, a = 1, and c = 0.6 consistent with the previous sections.

With respect to the fractional exponentiation operation, as described in Ref. [25], the stages visible in Figure 15 implement, in order, the calculation of the absolute value, the logarithm, and the exponential functions. As long as R₁₆/R₁₉V_p = R₂₃/R₂₀V_p = 1 V, the exponent is given by b = (R₁₇ + R₁₈)R₂₂/[R₁₇(R₂₂ + R₂₁)]. Here, we set R₁₉ = R₂₀ = 1.5 MΩ, R₁₆ = R₂₃ = 100 kΩ, V_p = 15 V, thus fulfilling the condition. Moreover, R₁₇ = 50 Ω, R₁₈ = 100 Ω, R₂₁ = 20 Ω, R₂₂ = 80 Ω, yielding b = 2.4. The relatively large values of R₁₉, R₁₆, R₂₀, and R₂₃ and small values of R₁₇, R₁₈, R₂₁, and R₂₂ served to minimize the errors in the logarithm and exponential relationships while avoiding excessive currents and minimizing the impact of bias currents and other instabilities. The other resistors were set to R₁₀ = R₁₁ = R₁₂ = R₁₃ = R₁₄ = R₂₄ = R₂₅ = 10 kΩ and R₁₅ = 1 kΩ, alongside V₄ = 0.6 V, to ensure the minimum value remains above zero.

All operational amplifiers were type ADA4000 or equivalent, all bipolar transistors were type 2N2222 or equivalent, and all diodes were type 1N4148 or equivalent. A dual power supply set to ±15 V was provided. Following initial experiments, it was discovered that some stages suffered from instability, leading to self-oscillation; to alleviate this problem, 1 nF capacitors were installed between the output and inverting input of U₄, U₅, U₆, U₇, U₈, and U₉. A two-layer PCB board, having a size of 80 × 80 mm, was designed, as visible in Figure 16. The voltages corresponding to the state variables were sampled at 5 kSa/s, 12-bit for 1 million points, using a digitizing oscilloscope.

5.2. Results

Before investigating the dynamic behavior of the system as a function of the control and boosting parameters, the performance of the analog circuit, obtained by calculating the absolute value and fractional exponent, as shown in Figure 15, was measured. For this purpose, a triangular wave having a frequency of 7 Hz and peak-to-peak amplitude of 4 V was generated externally and supplied to R₁₀, while the output of U₉ was recorded. As visible in Figure 17, a stable and highly symmetrical response was observed. The corresponding power relationship was fitted using the Levenberg–Marquart method, yielding an estimate of 2.409 (2.405–2.414), in striking agreement with the theoretical prediction.

The initial investigation focused on the effect of the amplitude boosting parameter n, with the resistor R₆ being varied while maintaining the absence of any offset boosting. To preserve chaoticity, it was necessary to adjust one feedback resistor, designated as R₈ = 141 kΩ, resulting in c = 0.7. As demonstrated in Figure 18, four levels of amplitude boosting were considered, ranging from R₆ = 45 kΩ (n = 2.2) to R₆ = 142 kΩ (n = 0.7). Overall, the scaling effect was clearly evident. The morphology of the phase portraits exhibited a satisfactory degree of consistency, and there was a comparable level of agreement with the results from the integration of Equation (1), as shown in Figure 1. The time series demonstrated the alternation of large and small amplitude cycles with pronounced residual periodicity, including the occurrence of sporadic bursts. It is noteworthy that at the smallest setting of n (corresponding to the largest size of the attractor), distortion became more evident, manifesting as compression of the large fluctuations in the time series and enlargement of the cavity within the chaotic orbit. For values of n below the considered range, divergence and saturation occurred, whereas, for values above it, large swings caused the onset of slow sustained self-oscillations unrelated to the chaotic dynamics.

Second, the effect of the offset boosting parameter d₂, which directly controls the variable x, was investigated by adjusting the voltage V₂ while retaining V₁ = 0 V. To retain chaoticity, it was necessary to further slightly adjust R₈ = 150 kΩ and to find a compromise between range and attractor size, giving R₆ = 69 kΩ (n = 1.45). As demonstrated in Figure 19, the analysis encompassed four levels of amplitude boosting on the x variable, ranging from V₂ = 0 V (d₂ = 0) to V₂ = −3 V (d₂ = −3). Despite minor discrepancies, the time series and phase portraits exhibited consistency with those observed in Figure 18. As the offset was increased, a marked leftward shift along the x dimension was observed. The dynamics, as evident in the time series and morphology of the phase portraits, remained largely consistent, despite the apparent shrinkage of the cavity seen in large settings. However, a notable non-ideal effect of the boosting was the generation of substantial and isolated fluctuations, classified as (super)extreme events, which resulted in the transient ejection of the trajectory from the vicinity of the attractor. For even stronger boosting, divergence eventually ensued, whereas for boosting in the opposite direction (rightward), a tendency to reduce the level of chaoticity could be noticed.

These findings, when considered as a whole, substantiated the viability of analog electronic fractional exponentiation within the specified framework and the effective integration of amplitude and offset boosting strategies. The deviations from ideal behaviors, which are prevalent in analog chaotic circuits, could be reasonably ascribed to the combined impact of parametric tolerances, limited precision of the amplification stages, and constraints imposed by the finite swing accessible to the system variables due to the limited power supply voltages.

6. Discussion and Conclusions

In this paper, a three-dimensional jerk system with fractional exponentiation was introduced, thus providing a novel candidate to the gallery of chaotic systems featuring the properties of amplitude control and two-dimensional interlocked offset boosting. The design of the corresponding circuit was made possible by a recently published analog computer element known as the 444 circuit, which provides fractional exponentiation by combining exponential and logarithmic operations in a compact design. A comprehensive experimental evaluation was undertaken using a physically realizable analog electronic circuit and confirmed the viability of the approach. This work exemplifies a method by which fractional exponentiation can provide an effective channel for easily re-scalable chaos generation without influencing the geometric properties of the chaotic trajectories. The proposed design may facilitate future applications of chaotic signal generators where signal versatility is important, such as in large-scale model network construction and practical scenarios such as mixing and steering control. In many cases of information processing, chaos is often used instead of true random numbers [30]. This nonlinear feedback of fractional exponentiation can be widely applied for the generation of chaos or hyperchaos in jerk systems and other dynamic systems. Therefore, we can imagine that this specific module of the 444 circuit has significant value in chaos generation and also has great potential for the controllability design of chaos.

7. Patents

A Chinese national patent number “ZL 2024 1 0519263.7” entitled “A product operation circuit with adjustable fractional powers”, covering the 444 circuit, was filed by the Nanjing University of Information Science & Technology.