**5. Circuit Implementation**

The analog circuit of system (2) is designed as shown in Figure 20 with the circuit equation:

$$\begin{cases} \dot{\mathbf{x}} = -\frac{1}{\mathcal{R}\_1 \mathbf{C}\_1} \mathbf{y} - \frac{1}{\mathcal{R}\_2 \mathbf{C}\_1} \mathbf{x} \mathbf{z} - \frac{1}{\mathcal{R}\_3 \mathbf{C}\_1} \boldsymbol{\mu} \\ \dot{\mathbf{y}} = -\frac{1}{\mathcal{R}\_4 \mathbf{C}\_2} \mathbf{x} + \frac{1}{\mathcal{R}\_5 \mathbf{C}\_2} \mathbf{x} \mathbf{z} \\ \dot{\boldsymbol{z}} = -\frac{1}{\mathcal{R}\_6 \mathbf{C}\_3} + \frac{1}{\mathcal{R}\_7 \mathbf{C}\_3} \mathbf{x} \mathbf{y} \\ \dot{\boldsymbol{u}} = \frac{1}{\mathcal{R}\_8 \mathbf{C}\_4} \mathbf{x} - \frac{1}{\mathcal{R}\_9 \mathbf{C}\_4} \mathbf{y} \end{cases} \tag{8}$$

**Figure 20.** Circuit schematic of system (8).

Totally, the hyperchaotic circuit consists of four channels, which contain the integration, addition, subtraction, and nonlinear operations. The circuit is powered by 18V. The variables *x*, *y*, *z* and *u* in system (2) are the state voltages of the capacitors in four channels. The corresponding circuit element parameters can be selected as *C*<sup>1</sup> = *C*<sup>2</sup> = *C*<sup>3</sup> = *C*<sup>4</sup> = 10*nF*, *R*<sup>2</sup> = *R*<sup>5</sup> = *R*<sup>7</sup> = 4*k*Ω, *R*<sup>3</sup> = *R*<sup>4</sup> = *R*<sup>9</sup> = 40*k*Ω, *R*<sup>1</sup> = 8*k*Ω, *R*<sup>6</sup> = 100*k*Ω,*R*<sup>8</sup> = 80*k*Ω,*R*<sup>10</sup> = *R*<sup>11</sup> = 10*k*Ω. Here, a common time scale of 1000 is applied for better demonstration in the oscilloscope. The phase trajectories in circuit (8) under amplitude control are shown in Figure 21. Circuit experiment proves that the parameter *m* rescales the amplitude of *x*, *y* and *u*. Symmetric attractors are shown in Figure 22.

**Figure 21.** Circuit simulation of system (8) with *a* = 5, *b* = 4, *c* = 1.3, *k* = 0.5, *m* = 1 (green), *m* = 4 (red) under initial condition [1, −1, −1, 1]: (**a**) *x*-*u* plane, (**b**) *y*-*z* plane.

**Figure 22.** Circuit simulation of symmetric attractors in system (8) with *a* = 5, *b* = 4, *c* = 1.3, *k* = 0.5, *m* = 1 under initial conditions IC1= (1, −1, −1, 1)(green), IC2= (−1, 1, −1, −1)(red): (**a**) *x*-*y* plane, (**b**) *x*-*z* plane, (**c**) *y*-*z* plane, (**d**) *x*-*u* plane.

#### **6. Discussion and Conclusions**

A hidden hyperchaotic attractor is found, which has the property of amplitude control and offset boosting. The proposed system shares a symmetric structure, where one can find an independent knob for amplitude control. An extra introduced dimension leaves an opportunity for attractor shifting in phase space by an independent controller. Broken symmetry induced bistability is also well addressed in this work. All the coexisting symmetric attractors governed by the basin of attraction can be rescaled by the non-bifurcation parameter. Numerical and circuit simulation agree with each other proving the properties found in the hyperchaotic system.

**Author Contributions:** Conceptualization, C.L.; Data curation, X.Z.; formal analysis, X.Z.; funding acquisition, C.L., Z.L. and C.T.; investigation, C.L., X.Z. and T.L.; methodology, C.L.; project administration, C.L.; resources, C.L.; software, X.Z.; supervision, C.L.; validation, C.L., X.Z., T.L., Z.L. and C.T.; visualization, X.Z.; writing—original draft, X.Z.; writing—review and editing, C.L., Z.L. and C.T. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported financially by the National Natural Science Foundation of China (Grant No. 61871230, 51974045), the Natural Science Foundation of Jiangsu Province (Grant No.: BK20181410), and a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.

**Conflicts of Interest:** The authors declare no conflicts of interest.
