Frequency formula for a class of fractal vibration system

Four fractal nonlinear oscillators (The fractal Duffing oscillator, fractal attachment oscillator, fractal Toda oscillator, and a fractal nonlinear oscillator) are successfully established by He’s fractal derivative in a fractal space, and their variational principles are obtained by semi-inverse transform method. The approximate frequency of the four fractal oscillators are found by a simple frequency formula. The results show the frequency formula is a powerful and simple tool to a class of fractal oscillators.

A Practical Underwater Information Sensing System Based on Intermittent Chaos Under the Background of Lévy Noise

The Gaussian noise model has been chosen for underwater information sensing tasks under substantial interference for most of the research at present. However, it often contains a strong impact and does not conform to the Gaussian distribution. In this paper, a practical underwater information sensing system is proposed based on intermittent chaos under the background of Lévy noise. In this system, a novel Lévy noise model is presented to describe the underwater natural environment interference and estimate its parameters, which can better describe the impact characteristics of the underwater environment. Then an underwater environment sensing method of dual-coupled intermittent chaotic Duffing oscillator is improved by using the variable step-size method and scale transformation. The simulation results show that the method can sense weak signals and estimate their frequencies under the background of strong Lévy noise, and the estimation error is as low as 0.03%. Compared with the intermittent chaos of the single Duffing oscillator and the intermittent chaotic Duffing of double coupling, the minimum SNR ratio threshold has been reduced by 11.5dB and 6.9dB, respectively, and the computational cost significantly reduced, and the sensing efficiency is significantly improved.

A heuristic review on the homotopy perturbation method for non-conservative oscillators

The homotopy perturbation method (HPM) was proposed by Ji-Huan. He was a rising star in analytical methods, and all traditional analytical methods had abdicated their crowns. It is straightforward and effective for many nonlinear problems; it deforms a complex problem into a linear system; however, it is still developing quickly. The method is difficult to deal with non-conservative oscillators, though many modifications have appeared. This review article features its last achievement in the nonlinear vibration theory with an emphasis on coupled damping nonlinear oscillators and includes the following categories: (1) Some fallacies in the study of non-conservative issues; (2) non-conservative Duffing oscillator with three expansions; (3)the non-conservative oscillators through the modified homotopy expansion; (4) the HPM for fractional non-conservative oscillators; (5) the homotopy perturbation method for delay non-conservative oscillators; and (6) quasi-exact solution based on He’s frequency formula. Each category is heuristically explained by examples, which can be used as paradigms for other applications. The emphasis of this article is put mainly on Ji-Huan He’s ideas and the present authors’ previous work on the HPM, so the citation might not be exhaustive.

Short Pyridine-Furan Springs Exhibit Bistable Dynamics of Duffing Oscillators

The intensive development of nanodevices acting as two-state systems has motivated the search for nanoscale molecular structures whose dynamics are similar to those of bistable mechanical systems, such as Euler arches and Duffing oscillators. Of particular interest are the molecular structures capable of spontaneous vibrations and stochastic resonance. Recently, oligomeric molecules that were a few nanometers in size and exhibited the bistable dynamics of an Euler arch were identified through molecular dynamics simulations of short fragments of thermo-responsive polymers subject to force loading. In this article, we present molecular dynamics simulations of short pyridine-furan springs a few nanometers in size and demonstrated the bistable dynamics of a Duffing oscillator with thermally-activated spontaneous vibrations and stochastic resonance.