Differential Equations with a Small Parameter Attached to the Highest Derivatives and Some Problems in the Theory of Oscillations

Abstract

This paper presents a brief review, for the most part, of the authors' results concerning systems of differential equations of the form <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\epsilon \dot{x}^1 = f^i(x^1,\cdots,x^k,y^1,\cdots,y^l) i =1, 2,\cdots,k</tex> <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\dot{y}^i = g^i(x^1,\cdots,x^k,y^1,\cdots,y^l) j =1,2,\cdots,</tex> where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\epsilon</tex> is a small positive parameter. The emphasis is on periodic solutions of such systems which are close to discontinuous solutions. Such periodic solutions are mathematical representations of relaxation oscillations which are encountered in various mechanical, electrical and radio systems.