Dynamic analysis of the fractional-order Duffing-Van der Pol oscillator and its application extension

This paper proposes a ship-radiated noise (S-RN) detection method based on fractional-order Duffing coupled Van der Pol oscillator (FODVP) for detecting unknown signal frequency. To begin with, aiming at the low detection accuracy of integer-order Duffing coupled Van der Pol oscillator (IODVP), FODVP is proposed and the selection of oscillator parameters is further analyzed. According to the system performance, FODVP is immune to white noise greater than about − 13 dB. This system’s minimum signal-to-noise ratio threshold detectable is − 69.03 dB, which is 33.98 dB lower than that of IODVP. Then, accurately determining the critical threshold of the system is the key to chaos detection. The entropy value measurement curve is used to quantitatively ascertain the critical threshold and study the positive phase transition detection of the oscillator. Furthermore, the conditions under which fractional-order Duffing coupled Van der Pol oscillator with oscillator array (FODVP-OA) produces intermittent chaos effects are studied. The geometric sequence structure that satisfies the driving force frequency in the oscillator array is determined. Finally, the effectiveness of the proposed detection method is verified by detecting analog and measured signals. The results demonstrate that FODVP-OA presents significant sensitivity to low-frequency signals. This research is the result of the combination of chaotic system and fractional theory, which further promotes the progress of S-RN detection and provides a new idea for detecting other types of signals.