Threshold for Chaos of a Duffing Oscillator with Fractional-Order Derivative

In this paper, the necessary condition for the chaotic motion of a Duffing oscillator with the fractional-order derivative under harmonic excitation is investigated. The necessary condition for the chaos in the sense of Smale horseshoes is established based on the Melnikov method, and then the chaotic threshold curve is obtained. The largest Lyapunov exponents are provided, and some other typical numerical simulation results, including the time histories, frequency spectrograms, phase portraits, and Poincare maps, are presented and compared. From the analysis of the numerical simulation results, it could be found that, near the chaotic threshold curve, the system generates chaos via the period-doubling bifurcation, from single periodic motion to period-2 motion and period-4 motion to chaotic motion. The effects of fractional-order parameters, the stiffness coefficient, and the damping coefficient on the threshold value of the chaotic motion are analytically discussed. The results show that the coefficient of the fractional-order derivative has great effect on the threshold value of the chaotic motion, while the order of the fractional-order derivative has less. The analysis results reveal some new phenomena, and it could be useful for designing or controlling dynamic systems with the fractional-order derivative.

The Dynamics of the Light Adaptation of the Human Visual System; Subjective and Electroretinographic Analysis

The excitation of the visual system increases with increasing retinal illumination. At the same time, the sensitivity of the system decreases (light adaptation). Higher excitation automatically results in a lower sensitivity. This study investigates whether this parallelism between the excitation and the sensitivity also applies in the dynamic case, that is, during the transition to a higher excitation level after an increase in the retinal illuminance. For this purpose, the courses of the subjective and the electroretinographic threshold during the transitional phase after a step of the adaptation illumination was determined by means of a special light-stimulation apparatus. As a measure of the course of the excitation during this time, the response ERG on the adaptation step was recorded with a special amplifier. The threshold curve always has an overswing, which shows subjectively very strong differences. It can be concluded that the glare caused by a sudden increase in illuminance is subjectively very different. The comparison between the response ERG on the adaptation step and the course of the electroretinographic increment threshold during this time shows a broad agreement between the two courses. It can thus be assumed that the sensitivity of the visual system follows the course of the excitation also in the dynamic case. In addition, the investigation shows that the glare experienced after a step in the illuminance clearly shows great subjective differences.

The development of a threshold curve for the understanding of concussion in sport

Much of what is known concerning human brain injury thresholds is based upon impacts to cadavers and animal models that were used to generate the Wayne State Concussion Tolerance Curve (WSTC) and similar curves. These curves are the foundation for predictive metrics used in standard development as well as helmet design. These curves were based upon a very narrow range of impacts; impacts whose characteristics differ greatly from how the head is impacted in sport. This research examines the uses of time-based curves like the WSTC in the context of understanding mechanisms of brain injury and head protection. Published linear/rotational acceleration magnitude/duration data from Hybrid III laboratory reconstructions of brain injury events were plotted. This research further develops the understanding of injury thresholds in comparison to threshold curves such as the WSTC and Brain Injury Curve Leuven. The data demonstrate the relationships between magnitude and duration of dynamic response on minor traumatic brain injury (mTBI) in sport.