Highly nonlinear differential equations or exponentially singular perturbations (Q1327087)

Ordinary differential equations of the type \(\ddot x = f(t,x, \dot x)^{[1/ \varepsilon]}\), \(\varepsilon\) being a real positive infinitesimal number, are studied. An approximation of the solution \(x_ \alpha(t)\) of the problem \({dx \over dt} = \varepsilon f(x)^{1/ \varepsilon}\), \(x(0) = x_ 0 - \alpha\), which is a generalization of the example \(d\varphi/dt = \varepsilon \exp (-(\varphi-1)/ \varepsilon)\); \(\varphi (0) = 1 - \alpha\) \((\alpha>0)\) by \textit{W. Eckhaus} [Matched Asymptotic Expansions and Singular Perturbations, North-Holland, Amsterdam (1973; Zbl 0255.34002)], is developed. Regions of infinitesimal derivatives and of jumps are discussed. Slow motions of the solutions were defined. Two macroscopes adopted for highly nonlinear equations enable to observe rapid solution motions when \(f\) does not depend on \(\dot x\). A detailed study of the equation family \(\ddot x = f(x)^{[1/ \varepsilon]}\) is presented. Existence and behaviour of the solutions and examples for the solutions having three types of nonuniformity (limited, angular and free layers) are studied. The thickness of the layer and the solutions with a free layer are analyzed. A partial answer is given for the localization of the free layers.