## 1 Introduction  
A rolling bearing is a precision mechanical element that is widely used in many mechanical devices, and it is a vulnerable component. In many practical applications, it is necessary to detect the running status of a rolling bearing online to provide a guarantee for the safe and stable operation of mechanical equipment [1, 2]. There are many failure modes of rolling bearings, including pitting, cracking, stripping, scraping and so on. When the bearing runs to these fault parts, it will produce vibration and shock. However, due to the short impact time and scattered energy, in the early stage of the fault, the corresponding fault characteristic signal is very weak and is covered by a large amount of machine running noise, so it is difficult to distinguish the fault signal by the traditional signal processing method. Therefore, it is necessary to preprocess the measured vibration signal to improve its signal-to-noise ratio (SNR) before detecting the fault characteristic signal. The weak signal detection method based on chaotic systems makes use of the characteristics of chaotic systems, such as sensitivity to the initial value, immunity to noise and nonlinear amplification, to achieve signal detection. Many studies have found that signal detection methods based on chaotic systems can detect a signal with SNR lower than \(-30\,\mathrm{dB}\)[3]. This kind of method is completely different from the traditional detection technology. Furthermore, this approach is one of the important research directions of chaos theory that can be applied in practical engineering and thus has broad application prospects. Since the 1990s, many scholars have studied the signal detection method based on chaotic systems and have been devoted to the expansion and supplement of weak signal detection technology [4]. These studies mainly focus on the following three aspects. The first aspect is the research on the signal detection mechanism and the operation state recognition method of the chaotic system. Representative studies are as follows. [PERSON] used chaotic oscillator as a detector for BPSK signals in very low signal-to-noise ratio (SNR) conditions in Ref. [5]. [PERSON] proposed a method based on the chaotic oscillator for weak signal detection using information fusion technology in Ref. [6]. [PERSON] found the analytic response function of a non-isochronous Hopf oscillator to a step stimulus in Ref. [7]. [PERSON] proposed that the transformation of state can be judged by calculating the autocorrelation coefficient of phase space trajectories, and performed a comparison with the maximum Lyapunov exponent method in Ref. [8]. However, these methods are based on the assumption that the frequency of the weak signal is known, which greatly restricts the application of these methods. The second aspect is the research on the measurement method of the weak signal with unknown frequency. To solve this problem, [PERSON] studied the intermittent chaos phenomenon of the Duffing oscillator system and proposed a new method for detecting various parameters of the periodic signal in Ref. [9]. [PERSON] proposed a new method of frequency estimation by searching the maximum value of the phase space trajectory variance based on a genetic algorithm in Ref. [10]. [PERSON] presented a method of detecting the signal frequency by the intermittent chaos in Ref. [11]. However, the structure and parameter settings of chaotic system models used in these studies are very complex. The third aspect is the research on the application and implementation of chaotic detection systems. The representative studies are as follows. [PERSON] tried to use the Duffing oscillator to detect eccentric faults of motors under different load conditions in Ref. [12]. [PERSON] proposed an enhanced differential chaos shift keying (DCSK), based on a first-order hybrid chaotic system, for a high-reliability underwater acoustic communication system in Ref. [13]. Obviously, the above studies are focused on the problems of detecting weak signals using a single chaotic system.