Second-order Nonlinear Dynamic Equations Research Articles

Second-order periodic boundary value problems on time scales

In this paper, we study the existence of solutions of periodic boundary value problem for second-order nonlinear dynamic equations on time scales. For this purpose, we use two methods here. First, sign properties of the Green's function are investigated and existence results of bounded solutions of nonlinear boundary value problems are established as a result of Schauder fixed-point theorem. Second, the monotone method is discussed to assure the existence and uniqueness of solutions of such periodic boundary value problems on time scales.

Deterministic Model of Homophase and Heterophase Fluctuations in a Liquid–Vapor System

Within the framework of the assembly-type catastrophe model, a nonlinear dynamic equation (DE) homogeneous in the parameter ηt with an aftereffect is constructed, in which ηt characterizes the deviation of the reduced density of a thin surface layer on the liquid-vapor interface from the mean density of the vapor-liquid system. This equation is used to treat a second-order nonlinear DE with a variable damping coefficient for a vapor-liquid system excited by periodic “impacts” (acts of evaporation and condensation of molecules). This DE is integrated over a finite time interval to find a two-dimensional mapping whose numerical solution describes the chaotic dynamics of the density in time, including “homophase” and “heterophase” fluctuations. For this system, the bifurcation diagrams are constructed and the Lyapunov exponents are found.