## 1 Introduction  
The main task of weak signal detection is to extract weak signal features from strong background noise. In recent years, various weak signal detection methods have been widely used in radar detection, communication, fault diagnosis and other fields [1; 2; 3]. The commonly used detection methods mainly include wavelet decomposition, neural network detection, stochastic resonance, high-order spectral analysis, etc., which have good effects in signal noise reduction and feature extraction [4; 5; 6; 7; 8; 9; 10]. When the signal-to-noise ratio of the signal is too low, these methods often fail to show good performance. Such a problem also exists in the field of bearing fault diagnosis. Although the frequency of fault signals can be speculated, early weak fault signals are often buried in the complex background noise, which brings great difficulties to its fault diagnosis. As a common important part of mechanical equipment, the failure of bearings easily causes great economic losses and safety risks. Therefore, the feature extraction of weak signals is of great significance and is a major research direction for scholars in the field of signal processing. A chaotic oscillator is widely used in signal processing because of its sensitivity to periodic signals and good anti-noise performance [11]. However, a signal detection method using a Duffing system has room for improvement. In recent years, many scholars have devoted themselves to the study of detecting weak periodic signals by using chaotic vibrators and achieved many achievements. In terms of the model of the system, a double-coupled Duffing system [12], a coupled Duffing and van der Pol-Duffing system [13] and a double-coupled van der Pol-Duffing system [14] are proposed. By establishing a coupled system, there is a certain phase difference between the two coupled systems that can improve the anti-noise ability of the detection system. Scholars coupled the restoring force and damping terms of the system by studying the coupling items of the Duffing system and achieving certain improvement effects [15]. Some scholars modified the resilience items of the Duffing system and improved the detection performance of the Duffing system by using high-order resilience items [16]. When the Duffing system is used for fault detection, there is often a strong interference in the system, resulting in a low recognition rate. When the Duffing system is used for signal detection, its chaos threshold needs to be judged. The commonly used methods to judge the chaos threshold include the Lyapunov index method, the correlation dimension method, the Melnikov index method, etc. [17]. After adding the signal to be tested into the Duffing system as an incentive, direct observation methods and the neural network learning discriminant method can be used to identify the state of the system [18; 19]. These methods have a good effect, can effectively determine the chaos threshold and can judge the state of the system. In the aspect of system state recognition, there are a series of problems; for example, the efficiency of direct observation method may be not high, the neural network learning discriminant may have misjudgment, etc. These problems still need to be solved. By preprocessing signals, the detection performance of chaotic systems can be effectively improved. Feature extraction is a commonly used preprocessing method, mainly including EMD, EWT, VMD, etc. [20; 21; 22]. Using these methods, the tested signal can be effectively decomposed into modal components, signal features can be extracted, and the impact of noise on signal detection can be reduced. In order to improve the detection performance of existing methods, this paper proposes a method for the weak fault identification of rolling bearings based on VMD and a double-coupled Duffing system.