Nonsmooth Temporal Decompositions for Smooth and Nonsmooth Processes

Physical and mathematical principles of non-smooth decompositions of time, and their applications are discussed. These decompositions represent the temporal argument through the standard non-smooth motions of mechanical systems with impact interactions. As a result, singularities of behavior of physical systems are eliminated from their dynamics and described by the discontinuities of slope in the time metrics. From the mathematical point of view, the non-smooth decomposition of the temporal argument generates special algebraic forms of the spatial coordinates. In some cases, these forms resemble regular complex numbers, but possess essentially distinct properties. The developed tools can be applied to both smooth and non-smooth dynamical processes in order to ease the corresponding mathematical formulations. At the next stage, different ‘smooth’ iterative and/or numerical methods can be effectively used in both cases. For detailed illustrations, the time argument is decomposed according to impact interactions in a chain of absolutely rigid and perfectly elastic particles. As a result, explicit equations of the stroboscopic mapping for a general class of dynamical systems are obtained. Between the snapshot times, the system motion is approximated analytically by the Lie series. In particular, the Duffing's oscillator with no linear stiffness under the sine-modulated sequence of Dirac's pulses, that is a modified Ueda's model, is considered. In some cases, a slight randomization of the pulse times could significantly suppress the mapping chaos caused by the system nonlinearity. Monte Carlo simulation show also that such a small random irregularity of the input brings out the system orbits more clearly in stroboscopic phase plots of the dynamics. An asymmetric Van der Pol equation under the regular and random pulses is considered as an example of the nerve pulse propagation modeling adopted in Biology. In some cases, irregularity of the pulse times results in more organized structures of the stroboscopic diagrams. Shooting diagrams are invoked also for visualization of the manifolds of periodic motions and bifurcations of nonlinear oscillators under smooth, nonsmooth, and impulsive excitations. The corresponding boundary value problems with no singularities are obtained by introducing the sawtooth time decomposition. It is shown that the temporal mode shape of the loading can be responsible for major qualitative features of the dynamics, such as transitions between the regular and random motions. The important role of unstable periodic orbits and their links with strange attractors are discussed.