Non-Smooth Spatio-Temporal Coordinates in Nonlinear Dynamics

This paper presents an overview of physical ideas and mathematical methods for implementing non-smooth and discontinuous substitutions in dynamical systems. General purpose of such substitutions is to bring the differential equations of motion to the form, which is convenient for further use of analytical and numerical methods of analyses. Three different types of nonsmooth transformations are discussed as follows: positional coordinate transformation, state variables transformation, and temporal transformations. Illustrating examples are provided.Comment: 15 figure

Personalized scaffolding technologies for alveolar bone regenerative medicine

Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/149271/1/ocr12275.pdfhttps://deepblue.lib.umich.edu/bitstream/2027.42/149271/2/ocr12275_am.pd

Non-conventional synchronization of weakly coupled active oscillators

We present a new type of self-sustained vibrations in the fundamental physical model covering a broad area of applications from wave generation in radiophysics and nonlinear optics to the heart muscle contraction and eyesight disorder in biophysics. Such a diversity of applications is due to the universal physical phenomenon of synchronization. Previous studies of this phenomenon, originating from Huygens famous observation, are based mainly on the model of two weakly coupled Van der Pol oscillators and usually deal with their synchronization in the regimes close to nonlinear normal modes (NNMs). In this work, we show for the first time that, in the important case of threshold excitation, an alternative synchronization mechanism can develop when the conventional synchronization becomes impossible. We identify this mechanism as an appearance of dynamic attractor with the complete periodic energy exchange between the oscillators, which is the dissipative analogue of highly intensive beats in a conservative system. This type of motion is therefore opposite to the NNM-type synchronization with no energy exchange by definition. The analytical description of these vibrations employs the concept of Limiting Phase Trajectories (LPTs) introduced by one of the authors earlier for conservative systems. Finally, within the LPT approach, we describe the transition from the complete energy exchange between the oscillators to the energy localization mostly on one of the two oscillators. The localized mode is an attractor in the range of model parameters wherein the LPT as well as the in-phase and out-of-phase NNMs become unstable