Three-Saddle-Foci Chaotic Behavior of a Modified Jerk Circuit with Chua’s Diode

This paper investigates the chaotic behavior of a modified jerk circuit with Chua’s diode. The Chua’s diode considered here is a nonlinear resistor having a symmetric piecewise linear voltage-current characteristic. To describe the system, we apply fundamental laws in electrical circuit theory to formulate a mathematical model in terms of a third-order (jerk) nonlinear differential equation, or equivalently, a system of three first-order differential equations. The analysis shows that this system has three collinear equilibrium points. The time waveform and the trajectories about each equilibrium point depend on its associated eigenvalues. We prove that all three equilibrium points are of type saddle focus, meaning that the trajectory of (x(t),y(t)) diverges in a spiral form but z(t) converges to the equilibrium point for any initial point (x(0),y(0),z(0)). Numerical simulation illustrates that the oscillations are dense, have no period, are highly sensitive to initial conditions, and have a chaotic hidden attractor.


Introduction
Nowadays, chaos theory is an important subject dealing with physics, mathematics, and engineering. A chaos system is a nonlinear dynamical system that has a non-periodic oscillation of waveforms. It is sensitive to initial conditions and has the self-similarity property. A significant development of chaos theory is the discovery of the celebrated Chua's system by L.O. Chua in 1983. This system was described by a set of three first-order ordinary differential equations (ODEs). Chua's discovery has encourged others to look for more chaotic systems, for example, systems of the type Rössler, jerk [1,2], circulant [3,4], hyperjerk [5,6], and hyper chaotic [5,7,8]. In addition, several chaotic circuits have been investigated, for example, Lorenz-based chaotic circuits [9,10], Chua' circuits [11][12][13][14], Wien-type chaotic oscillator [15], and chaotic jerk circuits [16][17][18][19]. Chaos theory has increasingly attracted much attention due to its wide applications in physical/natural/health sciences and engineering, for example, communication systems, weather forecasting, image encryption [20], celestial mechanics [21], population models [22], hydrology [23], cardiotocography [24], and dynamical disease [25]. Chaos theory as formulated for physical dynamic systems turns out to be useful in social science. For example, chaos theory can be applied to a simple nonlinear model concerning arms race; see, for example [26,27]. The works [28,29] substantiate the chaotic phenomena in dynamic love affair models.
L.O. Chua [14] investigated the chaotic theory for a simple famous circuit in Figure 1, known nowadays as Chua's circuit. The circuit consists of only resistors, capacitors, and a nonlinear resistor. The nonlinear resistor, also called Chua's diode, consists of many op-amps. Many researchers discussed several ways to modify the classical Chua's circuit to a more complicated circuit having chaotic phenomenon. Morgul [30] used an inductorless realization of a Chua's diode consisting of the Wien-bridge oscillator, coupled in parallel with the same nonlinear resistor used in the classical Chua's diode. Numerical experiments illustrated similar chaotic behavior. Aissi and Kazakos [31] modified the Chua's circuit by replacing the op-amps in Chua's diode with RC op-amps. Stouboulos et al. [32] modified the oscillator so that it consists of a nonlinear resistor and a negative conductance, demonstrating the birth and catastrophe of the double-bell strange attractor for different values of frequency. Kyprianidis [33] investigated the anti-monotonicity of the Chua's circuit, which is the creation of forward period-doubling bifurcation sequences followed by reverse period-doubling sequences. The work [34] of Kyprianidis and Fotiadou shows a possible way to replace the piecewise linear characteristic of the Chua's diode with a smooth cubic polynomial. Recently, the work [35] investigates chaotic behavior of the classical Chua's circuit with two nonlinear resistors. The existence of two nonlinear resistors in that case implies that the system has three equilibrium points. In 2011, Sprott [19] studied a simple chaotic jerk circuit, as shown in Figure 2, consisting of only five electronic components: two capacitors, an inductor, an adaptive resistor and a nonlinear resistor. His work shows a chaotic behavior of the trajectories around the equilibriums of the system, and launches a quest for other circuits that chaotically oscillate. Indeed, this circuit can be formulated into a third-order ODE consisting of a nonlinear term, called a "jerk"or the third-order derivative of a variable. According to much recent interest about chaotic oscillators based on jerk equations, this paper investigates the chaotic behavior of a new chaotic jerk circuit. We modify the chaotic jerk circuit in [19] so that there is a Chua's diode connected parallel to the nonlinear resistor as in Figure 3. The existence of Chua's diode discriminates the proposed system to the system [19]. The voltage-current characteristic of the Chua's diode satisfies a symmetric piecewise linear relation. To describe our system (see Section 2), we apply fundamental laws in electrical circuit theory to formulate a mathematical model in terms of a third-order (jerk) nonlinear ODE, or a system of three first-order ODE. The analysis in Section 3 shows that this system has three collinear symmetric equilibrium points. The time waveform about each equilibrium point depends on its associated eigenvalues. We prove that all three equilibrium points are of type saddle focus node, meaning that the trajectories of (x(t), y(t)) diverge in a spiral form but z(t) converges to the equilibrium point for any initial value (x(0), y(0), z(0)). Numerical simulation in Section 4 illustrates the chaotic phenomenon, including time waveforms, trajectories about each equilibrium point, effects of changing initial points, and existence of a chaotic hidden attractor. Finally, we summarize the paper in Section 5. In particular, we compare our work to [14,19].

Formulation of a Modified Chaotic Jerk Circuit with Chua's Diode to a System of ODEs
In this section, we formulate a mathematical model for a modified chaotic jerk circuit with Chua's diode in terms of a system of ODEs concerning a piecewise linear function and exponential term. We divide the circuit into four parts, as illustrated in Figure 3. Our analysis is based on fundamental theory of electrical circuit analysis such as Ohm's law, Kirchhoff's current law (KCL) and Kirchhoff's voltage law (KVL).
For Part 1, using KCL and the current-voltage equation for the capacitor, we have . Without loss of generality, we may normalize the value of R 1 C 1 to be 1 ms and we thus havė Similarly, for Part 2 we reachv For Part 3, we have by KCL that i R 3b Setting the time constants R 3a C 3 and R 3b C 3 to be 1ms, we geṫ The circuit in Figure 4 is a more complicated one since it consists of two nonlinear resistors. For the nonlinear resistor on the left, we have by Ohm's law that where v e is the voltage of the op-amp on the left hand side. Combining these three equations to Similarly, for the nonlinear resistor on the right, we obtain that v N R = i y R y where However, as pointed out in [19], the behavior of i N R depends on the voltage v C 1 . Indeed, when v e < v f , the graph of i N R with respect to v C 1 is as follows: From Figure 5, we have where v f ,max is the maximum voltage at the node f . The current i D through the diode D depends on the time-derivative of the voltage v C 1 (see, e.g., [18]) as follows: where k is the Boltzmann constant and T is the absolute temperature of the P-N junction. Let us denote α := kT. Of particular interest is that the chaos persists when α tends to zero. Since At Part 4, we use KCL to analyze this part and we get i R 4b = i R 4a . From Parts 2 and 4, we have by Ohm's law that v C 2 /R 4b = v R 4a /R 4a and, thus, the second capacitive voltage is For convenience, denote  Let us rescale the variables v C 1 , v C 2 , v C 3 to new variables x, y, z so that the current i N R is reduced to so that the characteristic in Figure 5 becomes that in Figure 6. Thus, the third-order (jerk) system can be described by the group of Equations (1)-(3), or equivalently, the following system of three first-order ODEs:

Analysis for Chaotic Behavior of the System
In order to analyze the behavior of the dynamical system (6), we need to find all its equilibrium points. Note that Figure 6 illustrates how the three-segment piecewise function g(x) depends on the range of x. The investigation of equilibrium points is thus divided into three cases in Sections 3.1-3.3. We then prove that each equilibrium point is of type saddle focus in Section 3.4. In Section 3.5, we describe how to find an initial point to localize a hidden attractor of the system. Thus, the equilibrium for Case 1 is given by E 1 = (x 1 , y 1 , z 1 ) = ( α 2 1−m 0 , 0, 0). When α tends to 0, the equilibrium point reaches the origin (0, 0, 0).
Thus, the equilibrium for Case 2 is given by When α → 0, we have (x 2 , y 2 , z 2 ) reaches the point ( −m 0 +m 1 1−m 1 , 0, 0). The system (6) can be put in the vector form (7) where We have g(x) = m 1 x + m 0 − m 1 and, thus, the equilibrium point is given by , 0, 0). We also have the vector form (7) where the Jacobian matrix A is the same as that in the previous case, and Thus, the system (6) has three colinear equilibrium points on the X-axis. Note that when α → 0, we have that the points E 2 and E 3 are opposite to each other with respect to the origin E 1 . This observation shows the symmetry of the equilibrium points.

Type of Equilibrium Points
Recall the following theorem: Theorem 1 (see, e.g., [36]). Let A(t) = [a ij (t)] ∈ R n×n be a continuous matrix-valued function on an interval I (i.e., each a ij (t) is a real-valued continuous function on I). Let B(t) ∈ R n be a continuous vector-valued function on I. Then the following initial value problem has a unique solution X(t) ∈ R n on the interval I.
This theorem guarantees the Equation (7) has a unique solution X(t) on any time interval (note that, in this case, A is a constant matrix). Thus in all cases of x, given an initial point (x(0), y(0), z(0)), the trajectory of (x(t), y(t), z(t)) is uniquely determined. The trajectory of (x(t), y(t), z(t)) in a neighborhood of each equilibrium point depends on the signs of the real/imaginary parts of the eigenvalues of the coefficient matrix A.
For Case 1: −1 < x < 1, we have the characteristic equation Since all parameters of the equation are real and the equation degree is odd, we have that a root (says λ 1 ) is real and other roots are a conjugate pair of complex numbers. Note that m 0 < 0 from Figure 6. Now, the product of all roots (eigenvalues) satisfies Since (λ 2 , λ 3 ) is a complex conjugate pair, the real root λ 1 must be negative. Write λ 2 = a + ib and λ 3 = a − ib, where a, b ∈ R. Since the sum of products of two roots of the cubic equation satisfies we get

Solving this quadratic equation to obtain
Since λ 1 < 0 and λ 2 1 − b 2 − 1 < |λ 1 |, we get a > 0. Hence, an eigenvalue is a negative real and two other eigenvalues are a conjugate pair of complex numbers having positive real parts. Therefore, this equilibrium is a saddle focus, and the trajectory of (x(t), y(t)) diverges in a spiral form, but z(t) converges to the equilibrium point for any initial point (x(0), y(0), z(0)).
For Cases 2 and 3, the Jacobian matrices are the same and we have the characteristic equation Since m 1 < 0 (from Figure 6), we obtain the same conclusion as in Case 1, i.e., the equilibrium point is a saddle focus.
We summarize the above discussion in the following theorem: Theorem 2. The system (6) has three equilibrium points, each of which is of type saddle focus. Moreover, the trajectory of (x(t), y(t)) diverges in a spiral form, but z(t) converges to the equilibrium point for any initial point (x(0), y(0), z(0)).
Since the equilibrium points are saddle foci, our system has chaotic behavior.

Localization of a Hidden Attractor of The System
Recall that an oscillation in a dynamical system can be numerically localized if an initial condition from its neighborhood leads to asymptotic behavior. Such an oscillation is known as an attractor, and its attracting set is called the basin of attraction. If the basin of attraction intersects a small neighborhood of an equilibrium point, then such attractor is said to be self-excited; otherwise it is called a hidden attractor. The hidden attractor was discovered in [37] for a generalized Chua's circuit, and then was discovered in the classical Chua's circuit [38].
In order to find a hidden attractor of the system, we will find a suitable initial point (x(0), y(0), z(0)) so that our system will have chaos. First, let us write the system (6) into a first-order vector differential equation where X(t) = [x(t) y(t) z(t)] T ∈ R 3 , A ∈ R 3×3 , r ∈ R 3 , q ∈ R 3 , and q : R → R is a continuous piecewise-differentiable function. Here, (·) T denotes the transposition operation. To find a periodic oscillation, we introduce a coefficient k of harmonic linearization so that the matrix A 0 = A + kqr T of the linear system has a pair of pure-imaginary eigenvalues ±iω 0 for some ω 0 > 0, and the rest of the eigenvalues have negative real parts. Then the system (8) has a periodic solution X(t) such that where the amplitude a is a solution of the integral equation 2π/ω 0 0 (ψ(a cos ω 0 t))a cos ω 0 t − k(a cos ω 0 t) 2 dt = 0.
Denoting φ(σ) = ψ(σ) − kσ, we can write Equation (8) to where is a small positive number, and investigate a periodic solution of the system Let us introduce the describing function We make an invertible linear transformation X(t) = SY(t) where S ∈ R 3×3 is a nonsingular matrix. The following theorem tells us how to choose an initial point in order to get a hidden attractor of the system.

Theorem 3 ([39]).
If there is a positive number a 0 such that Φ(a 0 ) = 0 and b 1 Φ (a 0 ) < 0, then the system (9) has a stable periodic solution with initial point

Remark 1.
In order to obtain the chaotic phenomenon, one can adjust some parameter values of electronics devices in the circuit so that the eigenvalues of the Jacobian matrix satisfy the condition for the type of equilibrium point (see Section 3.4).
In this case, we reach the system The eigenvalues of the system associated with the equilibrium point E 1 are the solutions of the cubic equation We get the following eigenvalues For the case −v f /v e x −1, we can obtain the equilibrium point associated with eigenvalues For the case 1 x v f /v e , the system has the equilibrium point E 3 = (0.452152, 0, 0). Note that E 2 and E 3 have the same eigenvalues since their associated matrices are the same.
From the signs of real/imaginary parts of the associated eigenvalues, we conclude that the three equilibrium points E 1 , E 2 , E 3 are saddle foci. Hence, the proposed circuit has a chaotic behavior.

Time Waveforms and Trajectories of The System
The time waveforms of x(t), y(t) and z(t) are reported in Figures 7-9, where the time interval is in ms. We see that the oscillations in the figures are non-periodic.   The trajectories of x(t), y(t), z(t) in 2D and 3D are numerically simulated in Figures 10-13. We see that the trajectory of (x(t), y(t)) diverges in a spiral form, but z(t) converges to the equilibrium point. The trajectories are dense and seem to have no periodic. Thus, chaotic behavior occurs in the modified jerk circuit with Chua's diode. Moreover, the attractor of the system is shown by the blue lines in Figures 10-13. From the 3D plot in Figure 13, we see that the oscillation does not connect with the equilibrium points E 1 , E 2 , E 3 , thus the system has a hidden attractor.
From Figures 14-16, we see that a small difference in initial points leads to a big difference in oscillations of x(t), y(t), z(t). Thus our dynamical system is highly sensitive to initial conditions, a characteristic of a chaotic system.