Nonlinear-damped Duffing oscillators having finite time dynamics

A class of modified Duffing oscillator differential equations, having nonlinear damping forces, are shown to have finite time dynamics, i.e., the solutions oscillate with only a finite number of cycles, and, thereafter, the motion is zero. The relevance of this feature is briefly discussed in relationship to the mathematical modeling, analysis, and estimation of parameters for the vibrations of carbon nano-tubes and graphene sheets, and macroscopic beams and plates.Comment: 15 page

A MATHEMATICAL MODEL OF THE WAY MICROORGANISMS REPRODUCE AT THE EXPENSE OF NUTRIENT CONSUMPTION IN THE CHEMOSTAT

One of the simplest experiments in microbiology is the growing unicellular microorganisms such as bacteria and following changes in their population over a period of time. We discuss the mathematical representation of such an experiment as modeled by [1] (for equations see \u27Additional Files\u27 below) where N(t) represents the density of the microorganism, C(t) is the concentration of the stock nutrient, C0 is the initial concentration, Kmax represents an upper bound for K(C), and for C = Kn, K(C) = (1/2)Kmax , µ is the mortality rate of the microorganism, while D and α are rate parameters. We use dimensional analysis to reduce the number of parameters and also calculate the steady states and investigate their linear stability properties. [1] G.F Gause, The Struggle for Existence. (Hafner Publishing, New York, 1969)