The Soft Point and Its Applications in Body Falling.

From the mesoscopic point of view, a definition of soft point is introduced by considering the attributes of geometric profile and mass distribution. After that, this concept is used to develop the soft matching technique to simulate the chaotic behaviors of the equations. Especially, a tennis model with deformation factor a(t) is proposed to derive a generalized Newton-Stokes equation v′(t) = λ(vT − a(t)v(t)). Furthermore, a concept of duality of deformation factor a(t) and velocity v(t) with respect to the generalized Newton-Stokes equation is established. To solve this equation, two data-driven models of a(t) are provided, one is based on the concept of soft matching, while the other is by using the amplitude modulation. Finally, the related iterative algorithm is developed to simulate the motion of the falling body via the duality of a(t) and v(t). Numerical examples successfully demonstrate the phenomenon of chaos, which consists of the continual random oscillations and sudden accelerations. Moreover, the algorithm is tested by using larger coefficients corresponding to the terminal velocity and shows more satisfactory results. It may enable us to characterize the total energy of the dynamical system more accurately. [ABSTRACT FROM AUTHOR]