Diverse Dynamic Behaviors and Firing Activities of the Modified Fractional-Order Hindmarsh–Rose Neuronal Model Induced by Fractional-Order

It is important to investigate the firing activities of neurons, and previous experimental works have shown that fractional-order neuronal models depict the firing rate of neurons more verifiably. In this study, a modified fractional-order Hindmarsh–Rose neuronal model is proposed, and the dynamics and firing activities are investigated. Some novel phenomenon can be found. First, by analyzing numerically and theoretically, the Hopf bifurcation is found to occur when the external direct current stimulus is chosen appropriately. The effects of fractional-order on the bifurcation are also studied. Second, when injecting a direct current stimulus, compared with the integer-order model, the system has more varying dynamic behaviors and firing pattern transitions. Under different external current stimulus, periodic firing patterns and chaotic firing patterns occur when fractional-order changes, but the regions of chaotic firing patterns are different. In other words, the transition mode of periodic firing and chaotic firing induced by fractional-order is different under different external current stimulus. The two-dimensional colored diagram of firing patterns is also investigated. Finally, when injecting periodic current stimulus, regular/irregular bursting, multiple spiking, regular\irregular square wave bursting, and mixed firing mods are found by setting the appropriate fractional-order, amplitude, and frequency of the external current stimulus. Some firing patterns cannot be found in integer-order models. When the amplitude is chosen at appropriate values, the region of frequency when the system displays the mixed firing modes decreases with increasing fractional-order.

Missing Shapiro steps in topologically trivial Josephson junction on InAs quantum well

Josephson junctions hosting Majorana fermions have been predicted to exhibit a 4π periodic current phase relation. One experimental consequence of this periodicity is the disappearance of odd steps in Shapiro steps experiments. Experimentally, missing odd Shapiro steps have been observed in a number of materials systems with strong spin-orbit coupling and have been interpreted in the context of topological superconductivity. Here we report on missing odd steps in topologically trivial Josephson junctions fabricated on InAs quantum wells. We ascribe our observations to the high transparency of our junctions allowing Landau-Zener transitions. The probability of these processes is shown to be independent of the drive frequency. We analyze our results using a bi-modal transparency distribution which demonstrates that only few modes carrying 4π periodic current are sufficient to describe the disappearance of odd steps. Our findings highlight the elaborate circumstances that have to be considered in the investigation of the 4π Josephson junctions in relationship to topological superconductivity.

On Optimal Truncated Biharmonic Current Waveforms for Class-F and Inverse Class-F Power Amplifiers

In this paper, two-parameter families of periodic current waveforms for class-F and inverse class-F power amplifiers (PAs) are considered. These waveforms are obtained by truncating cosine waveforms composed of dc component and fundamental and either second(k=2)or third(k=3)harmonic. In each period, waveforms are truncated to become zero outside of a prescribed interval (so-called conduction angle). The considered families of waveforms include both discontinuous and continuous waveforms. Fourier series expansion of truncated waveform contains an infinite number of harmonics, although a number of harmonics may be missing. Taking into account common assumptions that for class-F PA the third(n=3)harmonic is missing in current waveform and for inverse class-F PA the second(n=2)harmonic is missing in current waveform, we consider the following four cases: (i)n=k=3,(ii)n=3,k=2,(iii)n=k=2,and (iv)n=2,k=3.We show that, in each of these cases, current waveform enabling maximal efficiency (optimal waveform) of class-F and inverse class-F PA is continuous for all conduction angles of practical interest. Furthermore, we provide closed-form expressions for parameters of optimal current waveforms and maximal efficiency of class-F (inverse class-F) PA in terms of conduction angle only. Two case studies of practical interest for PA design, involving suboptimal current waveforms, along with the results of nonlinear simulation of inverse class-F PA, are also presented.