In this paper we give approximations to the Mittag-Leffler functions in terms of elementary functions using different methods. This allowed us to establish a practical method we called integerization principle. This principle states that many fractional nonlinear oscillators may be solved by means of the solution to some integer-order Duffing oscillator equation. The accuracy of the obtained results is illsutrated in concrete examples. Formulas for estimating the errors in the approximations are also provided.

The intensive development of nanodevices acting as two-state systems has motivated the search for nanoscale molecular structures whose long-term conformational dynamics are similar to the dynamics of bistable mechanical systems such as Euler arches and Duffing oscillators. Collective synchrony in bistable dynamics of molecular-sized systems has attracted immense attention as a potential pathway to amplify the output signals of molecular nanodevices. Recently, pyridin-furan oligomers of helical shape that are a few nanometers in size and exhibit bistable dynamics similar to a Duffing oscillator have been identified through molecular dynamics simulations. In this article, we present the case of dynamical synchronization of these bistable systems. We show that two pyridine-furan springs connected by a rigid oligomeric bridge spontaneously synchronize vibrations and stochastic resonance enhances the synchronization effect.

The signature of chaotic systems can be characterized either by the sensitivity of the initial conditions or by the change of its parameters. This feature can be used for manufacturing high sensitivity sensors. Sensors based on chaotic circuits have already been used for measuring water salinity, inductive effects, and both noise and weak signals. This article investigates an impedance sensor based on the Van der Pol and Duffing damped oscillators. The calibration process is a key point and therefore the folding behavior of signal periods was also explored. A sensitivity of 0.15 kΩ/Period was estimated over a range from 89.5 to 91.6 kΩ. This range can be adjusted according to the application by varying the gain of the operational amplifier used in this implementation. The development of this type of sensor might be used in medical and biological engineering for skin impedance measurements, for example. This type of chaotic sensor has the advantage of sensing small disturbances and then detect small impedance changes within biological materials which, in turn, may not be possible with other detectors.

This paper introduces the complex Rayleigh-van der Pol-Duffing oscillators (RVDOs), which are hyperchaotic and can be either autonomous or non-autonomous. The fundamental dynamics of the autonomous and nonautonomous complex RVDOs including dissipation, symmetry, fixed points and their stability are studied. These oscillators are found in various important fields of physics and engineering. The paper proposes a scheme to achieve phase synchronization (PS) and antiphase synchronization (APS) for different dimensional models. These kinds of synchronization are considered as generalization of several other types of synchronization. For this scheme we use the active control method based on Lyapunov stability theory. By analytically determining the control functions, the scheme achieved PS and APS. Our scheme is applied to study PS of hyperchaotic behaviours for two distinct hyperchaotic non-autonomous and autonomous complex RVDOs. Additionally, the scheme is employed to achieve APS of chaotic real non-autonomous RVDO and hyperchaotic complex autonomous RVDO, including those with different dimensions. Our work presents numerical results that plot the amplitudes and phases of these hyperchaotic behaviours, demonstrating the successful achievement of PS and APS.

This study uses machine learning to predict the convergence results of the Duffing equation with and without damping. The Duffing equation represents a nonlinear second-order differential equation with interesting behavior in undamped free vibration and forced vibration with damping. Convergence alternates randomly between 1 and -1 in undamped free vibration, depending on initial conditions. For forced vibration with damping, multiple factors influence vibration patterns. We utilize the fourth-order Runge-Kutta method to collect convergence results for both conditions. Machine learning techniques, specifically the long short term memory (LSTM) and LSTM-Neural Network (LSTM-NN) method, are employed to predict these convergence values. The LSTM-NN model is a hybrid approach that combines the LSTM method with the addition of hidden layers of neurons. Both the LSTM and LSTM-NN models are thoroughly explored and analyzed in this research. The research process involves three stages: data preprocessing, training, and verification. The results show that the LSTM-NN model becomes more adept at predicting binary datasets, boasting an impressive accuracy of up to 98%. However, when it comes to predicting multiple solutions, the traditional LSTM method outperforms the LSTM-NN approach.