Levinson Theorem for Differential Equations with Piecewise Constant Argument Generalized

In this work, it is presented an adaptation of an asymptotic theorem of N. Levinson of 1948, to differential equation with piecewise constant argument generalized, which were introduced by M. Akhmet in 2007. By simplicity and without loss of generality, the case where the argument is delayed is considered. The N. Levinson's theorem which is adapted is that dealt by M. S. P. Eastham in his work which is present in this bibliography. The more relevant hypotheses of this theorem are highlighted an it is established a version of this theorem with these hypotheses for ordinary differential equations. Such a version is that which is adapted to differential equation with piecewise constant argument generalized. The adaptation is proved by mean the Banach fixed point where contractive operator is built form a suitable version of the constant variation formula

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