**2. 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

$$\frac{\upsilon\_{R\_1}}{R\_1} = i\_{R\_1} = i\_{\mathbb{C}\_1} = \mathbb{C}\_1\\\frac{d\upsilon\_{\mathbb{C}\_1}}{dt} = \mathbb{C}\_1\\\dot{\upsilon}\_{\mathbb{C}\_1}.$$

Now, since *vR*<sup>1</sup> = *vC*<sup>2</sup> , we obtain *v*˙*C*<sup>1</sup> = *vC*2/(*R*1*C*1). Without loss of generality, we may normalize the value of *R*1*C*<sup>1</sup> to be 1 *ms* and we thus have

$$
\psi\_{\mathbb{C}\_1} = v\_{\mathbb{C}\_2}.\tag{1}
$$

Similarly, for Part 2 we reach *v*˙*C*<sup>2</sup> = *vC*3/(*R*2*C*2). Setting the time constant *R*2*C*<sup>2</sup> := 1 yields

$$
\psi\_{\mathbb{C}\_2} = v\_{\mathbb{C}\_3}.
\tag{2}
$$

For Part 3, we have by KCL that *iR*<sup>3</sup>*<sup>b</sup>* + *iNR* + *iD* = *iR*<sup>3</sup>*<sup>a</sup>* + *iC*<sup>3</sup> . It follows that

$$\dot{v}\_{\text{C}\_{3}} = -\frac{v\_{\text{C}\_{1}}}{R\_{3b}\text{C}\_{3}} - \frac{v\_{\text{C}\_{3}}}{R\_{3a}\text{C}\_{3}} + \frac{i\_{\text{N}\_{R}}}{\text{C}\_{3}} + \frac{i\_{D}}{\text{C}\_{3}}.$$

Setting the time constants *R*3*aC*<sup>3</sup> and *R*3*bC*<sup>3</sup> to be 1*ms*, we get

$$
\psi\_{\mathbb{C}\_3} = \ -\upsilon\_{\mathbb{C}\_1} - \upsilon\_{\mathbb{C}\_3} + R\_{3a}(i\_{N\_R} + i\_D). \tag{3}
$$

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 *vNR* = *iR*3*R*3, *ve* = (*R*2*<sup>c</sup>* + *R*3*c*)*iR*<sup>3</sup> and *vNR* − *ve* = *ixR*1, where *ve* is the voltage of the op-amp on the left hand side. Combining these three equations to get *vNR* = *ixRx* where

$$R\_x = -\frac{R\_{1c}R\_{3c}}{R\_{2c}}.$$

Similarly, for the nonlinear resistor on the right, we obtain that *vNR* = *iyRy* where

$$R\_{\Psi} = -\frac{R\_{4c}R\_{6c}}{R\_{5c}}.$$

Using KCL at node *c*, we have *iNR* − *ix* − *iy* = 0. Then the current *iNR* satisfies the relation

$$v\_{\mathcal{N}\_{\mathbb{R}}} = i\_{\mathcal{N}\_{\mathbb{R}}}(\mathcal{R}\_{\mathcal{X}} + \mathcal{R}\_{\mathcal{Y}}).$$

However, as pointed out in [19], the behavior of *iNR* depends on the voltage *vC*<sup>1</sup> . Indeed, when *ve* < *vf* , the graph of *iNR* with respect to *vC*<sup>1</sup> is as follows:

From Figure 5, we have

$$i\_{\rm N\_R} = \left(\frac{1}{R\_x} + \frac{1}{R\_{4c}}\right) v\_{\rm C\_1} + \frac{1}{2} \left(\frac{1}{R\_y} - \frac{1}{R\_{4c}}\right) \left(\left|v\_{\rm C\_1} + \frac{v\_{f,\rm max}}{v\_f} v\_{\rm C\_1}\right| - \left|v\_{\rm C\_1} - \frac{v\_{f,\rm max}}{v\_f} v\_{\rm C\_1}\right|\right), \tag{4}$$

where *vf* ,*max* is the maximum voltage at the node *f* . The current *iD* through the diode *D* depends on the time-derivative of the voltage *vC*<sup>1</sup> (see, e.g., [18]) as follows:

$$i\_D = k^2 T^2 e^{\phi c\_1/kT} \text{ .}$$

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

$$\lim\_{\alpha \to 0^+} \alpha^2 e^{y/\alpha} = \infty.$$

At Part 4, we use KCL to analyze this part and we get *iR*<sup>4</sup>*<sup>b</sup>* = *iR*<sup>4</sup>*<sup>a</sup>* . From Parts 2 and 4, we have by Ohm's law that *vC*2/*R*4*<sup>b</sup>* = *vR*<sup>4</sup>*a*/*R*4*<sup>a</sup>* and, thus, the second capacitive voltage is

$$v\_{\odot} = \frac{R\_{4b}}{R\_{4a}}v\_{R\_{4a}}.$$

For convenience, denote

$$m\_0 = R\_{34} \left(\frac{1}{R\_x} + \frac{1}{R\_y}\right), \quad m\_1 = R\_{34} \left(\frac{1}{R\_x} + \frac{1}{R\_{4c}}\right).$$

**Figure 4.** Two nonlinear resistors in Chua's circuit.

**Figure 5.** I-V characteristic of nonlinear resistors.

Let us rescale the variables *vC*<sup>1</sup> , *vC*<sup>2</sup> , *vC*<sup>3</sup> to new variables *x*, *y*, *z* so that the current *iNR* is reduced to

$$\mathbf{g(x)} = m\_1 \mathbf{x} + 0.5(m\_0 - m\_1) \left( |\mathbf{x} + \mathbf{1}| - |\mathbf{x} - \mathbf{1}| \right),\tag{5}$$

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:

$$\begin{aligned} \dot{\mathbf{x}} &= y\_\prime \\ \dot{y} &= z\_\prime \\ \dot{z} &= -\mathbf{x} - z + \mathbf{g}(\mathbf{x}) + a^2 e^{\frac{y}{n}} .\end{aligned} \tag{6}$$

**Figure 6.** Changing scales of I-V characteristic of nonlinear resistors.
