*3.5. 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

$$X'(t) = AX(t) + \psi(r^T X(t))q\tag{8}$$

where *<sup>X</sup>*(*t*)=[*x*(*t*) *<sup>y</sup>*(*t*) *<sup>z</sup>*(*t*)]*<sup>T</sup>* <sup>∈</sup> <sup>R</sup>3, *<sup>A</sup>* <sup>∈</sup> <sup>R</sup>3×3, *<sup>r</sup>* <sup>∈</sup> <sup>R</sup>3, *<sup>q</sup>* <sup>∈</sup> <sup>R</sup>3, and *<sup>q</sup>* : <sup>R</sup> <sup>→</sup> <sup>R</sup> is a continuous piecewise-differentiable function. Here, (·)*<sup>T</sup>* denotes the transposition operation. To find a periodic oscillation, we introduce a coefficient *k* of harmonic linearization so that the matrix *A*<sup>0</sup> = *A* + *kqr<sup>T</sup>* of the linear system

$$X'(t) := A\_0 X(t)$$

has a pair of pure-imaginary eigenvalues ±*iω*<sup>0</sup> for some *ω*<sup>0</sup> > 0, and the rest of the eigenvalues have negative real parts. Then the system (8) has a periodic solution *X*(*t*) such that

$$\sigma(t) := r^T X(t) \approx a \cos \omega\_0 t,$$

where the amplitude *a* is a solution of the integral equation

$$\int\_0^{2\pi/\omega\_0} (\psi(a\cos\omega\_0 t))a\cos\omega\_0 t - k(a\cos\omega\_0 t)^2 dt = 0.5$$

Denoting *φ*(*σ*) = *ψ*(*σ*) − *kσ*, we can write Equation (8) to

$$X'(t) = A\_0 X(t) + q\phi(r^T x).$$

Let us change *φ*(*σ*) to  *φ*(*σ*) where  is a small positive number, and investigate a periodic solution of the system

$$X'(t) = A\_0 X(t) + \epsilon \eta \phi(r^T x). \tag{9}$$

Let us introduce the describing function

$$\Phi(a) = \int\_0^{2\pi/\omega\_0} \phi(a \cos\left(\omega\_0 t\right)) \cos\left(\omega\_0 t\right) dt.$$

We make an invertible linear transformation *<sup>X</sup>*(*t*) = *SY*(*t*) where *<sup>S</sup>* <sup>∈</sup> <sup>R</sup>3×<sup>3</sup> 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*<sup>0</sup> *such that* Φ(*a*0) = 0 *and b*1Φ (*a*0) < 0*, then the system* (9) *has a stable periodic solution with initial point*

$$X(0) := \mathcal{S}[y\_1(0) \ y\_2(0) \ y\_3(0)]^T$$

*where y*1(0) = *a*<sup>0</sup> + *O*()*, y*2(0) = <sup>0</sup>*, and y*<sup>3</sup> = *On*−2() *with period O*() + <sup>2</sup>*<sup>π</sup> ω*<sup>0</sup> *.*

#### **4. Numerical Experiment**

In this section, we provide a numerical experiment to illustrate the chaotic behavior of the proposed circuit via MATLAB. Consider the circuit in Figure 3 with the following parameters: *R*<sup>1</sup> = 1 kΩ, *R*<sup>2</sup> = 200 Ω, *R*3*<sup>a</sup>* = 500 Ω, *R*3*<sup>b</sup>* = 500 Ω, *R*4*<sup>a</sup>* = 1 kΩ, *R*4*b* = 1 kΩ, *R*1*<sup>c</sup>* = 250 Ω, *R*2*<sup>c</sup>* = 250 Ω, *R*3*<sup>c</sup>* = 500 Ω, *R*4*<sup>c</sup>* = 750 Ω, *R*5*<sup>c</sup>* = 180 Ω, *R*6*<sup>c</sup>* = 400 Ω, *C*<sup>1</sup> = 1 μF, *C*<sup>2</sup> = 5 μF, *C*<sup>3</sup> = 2 μF, *m*<sup>0</sup> = −0.1768, *m*<sup>1</sup> = −1.1468, and *α* = 0.026077. We set the initial condition to be *X*(0)=(*x*(0), *y*(0), *z*(0)) = (0, −0.7, 0).

**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).*
