Sensors based on chaotic oscillators have a simple design, combined with high sensitivity and energy efficiency. Among many developed schemes of such sensors, the promising one is based on the Duffing oscillator, which possesses a remarkable property of demonstrating chaotic oscillations only in the presence of a weak sine wave at the input. The main goal of this research was to evaluate the maximal sensitivity of a practically implemented metal detector based on the Duffing oscillator and compare its sensitivity with conventional sensors. To achieve high efficiency of the Duffing-based design, we proposed an algorithm which performs a bifurcation analysis of any chaotic system, classifies the oscillation modes and determines the system sensitivity to a change in different parameters. We apply the developed algorithm to improve the sensitivity of the electronic circuit implementing the Duffing oscillator, serving as a key part of a three-coil metal detector. We show that the developed design allows detecting the presence of metal objects near the coils more reliably than the conventional signal analysis techniques, and the developed detector is capable of sensing a large metal plate at distances up to 2.8 of the coil diameter, which can be considered a state-of-the-art result.

The synchronization of chaotic systems is a fundamental phenomenon in nonlinear dynamics. Most known synchronization techniques suggest that the trajectories of coupled systems converge at an exponential rate. However, this requires transferring a substantial data array to achieve complete synchronization between the master and slave oscillators. A recently developed approach, called time-reversible synchronization, has been shown to accelerate the convergence of trajectories. This approach is based on the special properties of time-symmetric integration. This technique allows for achieving the complete synchronization of discrete chaotic systems at a superexponential rate. However, the validity of time-reversible synchronization between discrete and continuous systems has remained unproven. In the current study, we expand the applicability of fast time-reversible synchronization to a case of digital and analog chaotic systems. A circuit implementation of the Sprott Case B was taken as an analog chaotic oscillator. Given that real physical systems possess more complicated dynamics than simplified models, analog system reidentification was performed to achieve a reasonable relevance between a discrete model and the circuit. The result of this study provides strong experimental evidence of fast time-reversible synchronization between analog and digital chaotic systems. This finding opens broad possibilities in reconstructing the phase dynamics of partially observed chaotic systems. Utilizing minimal datasets in such possible applications as chaotic communication, sensing, and system identification is a notable development of this research.

Developing new features for time-series characterization is a current challenge in data science and machine learning. In this paper, we propose a new metric based on a simple and efficient algorithm, namely, the return map. The return map analysis is well established in the field of non-linear dynamics, in particular, for fitting the parameters of a chaotic system from a waveform, or to attack a chaotic communication channel. We show that our metrics work well for both non-linear dynamics and time-series feature extraction problems in the field of machine learning. In an experiment aiming to classify vibration signals of normal and damaged bearings, we compare our method with two other methods that reported to have excellent accuracy, based on entropy and statistical feature distribution, respectively. We show that our method achieves higher accuracy with almost the lowest time costs, which was confirmed in experiments with two different datasets containing three main classes of bearings: normal, with inner race faults, and with outer race faults, having different damage origins and recorded in various conditions. In particular, for the dataset supplied by Case Western Reserve University, our method reached an accuracy of 100% at signals of 5000 sample points length, with a total time of 0.4 s required for feature estimation, while the entropy-based method reached an accuracy of 95% with a time of 100 s, and a statistical feature distribution method reached an accuracy of 93% with a total time of 1.9 s. Results show that the developed method is better suited to real-time bearing condition monitoring applications than most of the methods reported to date.