In this paper, we proposed a new fractional two dimensional trigonometric combined discrete chaotic mapping (2D-TCDCM) and a fractional 2-D Kawakami map within Caputo-Hadamard fractional difference. We observed the dynamic behaviours of the proposed Caputo-Hadamard fractional maps, including fractal graph, maximum lyapunov exponent, phase trajectory and randomness test. We illustrate the advantage of using Caputo-Hadamard fractional difference. As a conclusion, we get the condition of the proposed fractional map to behave chaotically with physics background.
The efficient biohybrid photocatalysts were prepared with different weight ratios of Fe2O3 and treated rape pollen (TRP). The synthesized samples were characterized by different analytical techniques. The results showed that carbonized rape pollen had a three-dimensional skeleton and granular Fe2O3 uniformly covered the surface of TRP. The Fe2O3/TRP samples were used for degradation of Methylene Blue (MB) and Escherichia Coli (E. coli) disinfection in water under visible light. The degradation of MB and inactivation of E. coli was achieved to 93.7% in 300 min and 99.14% in 100 min, respectively. We also explored the mechanism during the reaction process, where reactive oxygen species (ROS) including hydroxyl radicals and superoxide radicals play a major role throughout the reaction process. This work provides new ideas for the preparation of high-performance photocatalysts by combining semiconductors with earth-abundant biomaterials.
We demonstrate that strong stimulated Raman scattering in silicon and germanium microresonators can induce stable and breathing dark pulses generation circumventing traditional complex approaches such as pump modulation and mode coupling. Although multi-photon absorption shows a small influence on the detuning value for stable dark pulse excitation, the concomitant free carrier will assist dark pulse excitation and broaden the excitation area of dark pulse thus making it easier to capture stable pulse. Furthermore, dark breather dynamics in Si and Ge are also observed, which shows distinct properties from the dark soliton breathers dominated solely by Kerr effect. Finally, we show that octave spanning mid-infrared microcomb can be generated combining with high-order dispersion engineering, which in turn affects the breathing dynamics of dark pulses. Our findings provide another way for the initiation of dark pulses in Group IV materials and broadband mid-infrared (MIR) microcomb generation for spectroscopy applications.
The motion of a bead on a rotating hoop is a classical problem in mechanics. The problem describes that a bead is placed on the track of a hoop rotating around a vertical axis. Although this issue is researched by lots of scholars, this system is usually analyzed without considering the friction. However, when considering friction, it becomes difficult to analyze the stability. The purpose of this study is to investigate the stability of the system with friction more easily. We derive the equation of motion of the bead and use the Lyapunov stability theorem to analyze the stability. The stability is determined by the angular velocity of the hoop. If the angular velocity is less than or equal to the critical value the bead is asymptotically stable at origin (the bottom of the hoop) otherwise it is asymptotically stable at a non-zero position. In addition, we study the difference between mass point bead and rigid body bead and find that both have identical bifurcation diagram and stability, whereas the trajectories have a difference on the phase plane. In the end, the numerical simulation method is used to verify the conclusion above.
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