The Hopfield neural network (HNN) has been widely used to simulate brain electrical activity due to its flexible topology and rich dynamical behaviors. Moreover, the spectral characteristics of neural activity serve as fundamental signal, offering potential to identify novel targets for neurological disorders and advancing next-generation neuromodulation therapies. However, the frequency-domain analyses of HNN have rarely been reported. To further explore the complex periodic motions induced by harmonic terms, this study employs discrete mapping method incorporating finite Fourier series to analyze bifurcation evolutions and coexistence of firing behaviors, providing the quantitative analysis of their amplitude-frequency characteristics for the first time. Additionally, by investigating the relationship between harmonic amplitudes and phases in coexisting attractors, new perspectives on the study of coexisting attractors are provided. Finally, the simulation results are verified through the hardware circuit built using printed circuit board and a high-performance pseudorandom number generator is designed by leveraging chaotic sequences derived from unstable periodic orbits. This research contributes a new approach for data encryption and communication security in Internet of Things.

Dandelion seeds possess a complex three-dimensional structure and a self-adapted flying ability. To understand this fascinating flight mechanism, a three-dimensional umbrella-shaped model imitating dandelion seeds is proposed. The effects of folding angle and flow velocity on the dandelion drag force during their descent are studied, and it is found that the larger folding angle results in a smaller drag coefficient. Four different vortex structures are revealed depending on the folding angle. The effect of crosswind disturbances on the attitude stability of dandelions is also investigated by changing folding angles. It is found that dandelions with larger folding angles have better attitude stability. The proposed model suggests that when the folding angle is between 20° and 40°, the dandelion seeds might have a good balance between the drag force and attitude stability, which ensures a broad self-adapted flying ability.

A brush seal has the advantages of adapting to different vibration conditions and increasing the stability of the nonlinear rotor system. In this research, the stability and bifurcations of complex vibrations in a brush-seal rotor system are studied. An analytical seal force model is obtained through the beam theory and mutual coupling dynamics of the bristles and the rotor. The interaction between the bristles and the rotor is clearly depicted by a geometric map. Periodic and chaotic vibrations as well as the corresponding amplitude–frequency characteristics are first predicted by a numerical bifurcation diagram and 3D waterfalls. Discrete dynamic eigenvalue analysis is adopted for a detailed investigation of the stability and bifurcations of nonlinear vibrations. Jumping, quasi-periodic, and half-frequency vibrations are warned during the speeding up and down process. Four separate nonlinear vibration evolving routes are discovered. Two period-doubling bifurcation trees evolving to chaos are illustrated for the observation of global and independent periodic vibrations. Nonlinear vibration illustrations are presented through displacement orbits as well as harmonic amplitudes and phases. Chaotic vibration and unstable semi-analytical vibration solutions are compared. The obtained results and analysis methods provide new perspectives on nonlinear vibrations in the brush-seal rotor system.