**1. 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 [11–14], Wien-type chaotic oscillator [15], and chaotic jerk circuits [16–19]. 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.

**Figure 1.** Chua's circuit [14].

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.

**Figure 2.** A chaotic jerk circuit [19].

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].

**Figure 3.** A modified chaotic jerk circuit with Chua's diode.
