**1. Introduction**

As we all know, chaos is ubiquitous in nature and human society, and has great potential in engineering applications. However, there exists great challenge in conditioning broadband chaotic signals, and appropriate amplitude and polarity are the key specifications for chaos generation and transmission [1–3], and therefore, recently great efforts have been made on the research of amplitude control and offset boosting. Normally, the amplitude of system variable requires further adjusting a couple of parameters. In many cases, a unipolar signal is more suitable for transmitting in inter-coupled integrated circuits. Such a challenge exists in the conversion from the bipolar signal to unipolar signal. An independent non-bifurcation parameter to rescale the signal without revising the Lyapunov exponents is important for chaos application. Suitable signal control saves the modulator in chaos-based applications [4,5], including amplitude control [6,7] and offset boosting [8,9].

In addition, hidden attractors exist in chaos, but one cannot find them from the neighborhood of any equilibrium point. Thus, it is of great value in theoretical and physical significance and engineering application to study the realization method of hidden attractors. The Chua system, Lorenz-like systems, and the chaotic systems with stable equilibria [10–15], line equilibria [16–18], or no equilibria [19–24] give us impressive points. Hyperchaos with higher complexity is beneficial to secure communication, so some research extends to hyperchaos. A hyperchaotic system with a hidden attractor was proposed by Wang et al. [25]; Chlouverakis and Sprott [26] claimed the simplest hyperchaotic snap system in algebra; and Yuan et al. [27] showed a memristive multi-scroll hyperchaotic system. Other many hyperchaotic systems have come out of the Lorenz-like system [28–31]. Some other hyperchaotic ones have been proposed, including a memristive hyperchaotic system [32,33], fractional order hyperchaotic system [34,35] or hyperchaotic multi-wing system [36,37]. To the best of our knowledge, there is no relevant research on a hyperchaotic hidden attractor with geometric

control. Based on a three-dimensional Lorenz-like system, Wang et al. [38] put forward a hyperchaotic system for producing multi-wing attractors; while in this work, the proposed system has four features as follows:


As shown in Figure 1, the proposed hyperchaotic system has multiple independent geometric controllers including controllers for rescaling amplitude, frequency and offset. Some of the reported 4-D hyperchaotic Lorenz-like systems are listed in Table 1. In the paper, the system controllers are signal controllers and multistability observers as well. In Section 2, the mathematical model of the hyperchaotic system is given. In Section 3, complex dynamic behavior is analyzed. The process of amplitude control and offset boosting is discussed in Section 4. In Section 5, multistability is investigated. In Section 6, the analog circuit is given. Finally, we give the conclusions and discussion.

**Figure 1.** Hyperchaotic attractor with multiple independent controllers.


