Clifford systems, Clifford structures, and their canonical differential forms

A comparison among different constructions in $$\mathbb {H}^2 \cong {\mathbb {R}}^8$$ H 2 ≅ R 8 of the quaternionic 4-form $$\Phi _{\text {Sp}(2)\text {Sp}(1)}$$ Φ Sp ( 2 ) Sp ( 1 ) and of the Cayley calibration $$\Phi _{\text {Spin}(7)}$$ Φ Spin ( 7 ) shows that one can start for them from the same collections of “Kähler 2-forms”, entering both in quaternion Kähler and in $$\text {Spin}(7)$$ Spin ( 7 ) geometry. This comparison relates with the notions of even Clifford structure and of Clifford system. Going to dimension 16, similar constructions allow to write explicit formulas in $$\mathbb {R}^{16}$$ R 16 for the canonical 4-forms $$\Phi _{\text {Spin}(8)}$$ Φ Spin ( 8 ) and $$\Phi _{\text {Spin}(7)\text {U}(1)}$$ Φ Spin ( 7 ) U ( 1 ) , associated with Clifford systems related with the subgroups $$\text {Spin}(8)$$ Spin ( 8 ) and $$\text {Spin}(7)\text {U}(1)$$ Spin ( 7 ) U ( 1 ) of $$\text {SO}(16)$$ SO ( 16 ) . We characterize the calibrated 4-planes of the 4-forms $$\Phi _{\text {Spin}(8)}$$ Φ Spin ( 8 ) and $$\Phi _{\text {Spin}(7)\text {U}(1)}$$ Φ Spin ( 7 ) U ( 1 ) , extending in two different ways the notion of Cayley 4-plane to dimension 16.

Dynamic Modeling and Chaotic Analysis of Gear Transmission System in a Braiding Machine with or without Random Perturbation

This paper is aimed at analyzing the dynamic behavior of the gear transmission system in a braiding machine. In order to observe the nonlinear phenomenon and reveal the time-varying gear meshing mechanism, a mathematical model with five degrees-of-freedom gear system under internal and external random disturbance of gear system is established. With this model, bifurcation diagrams, Poincare maps, phase diagrams, power spectrum, time-process diagrams, and Lyapunov exponents are used to identify the chaotic status. Meanwhile, by these analytical methods, spur gear pair with or without random perturbation are compared. The numerical results suggest that the vibration behavior of the model is consistent with that of Clifford system. The chaotic system associated parameters are picked out, which can be helpful to the design and control of braiding machines.