Homogeneous Dynamics in the Plane

Abstract

The fundamental problem in the theory of ordinary differential equations is that of deducing the local or global properties of the solutions from the properties of the governing functions (given vector fields). The subject matter of ordinary differential equations has played a prominent role in both pure and applied mathematics. Evidently, some of the most important ideas in differential and integral calculus, analytical geometry, curve theory, group theory, and topology were developed in attempts to resolve particular dynamic or geometric problems involving differential equations. Like any branch of mathematics with a long history behind it, the study of ordinary differential equations has undergone profound changes. As is well known, the main endeavour was in the beginning and for a long time devoted to quantitative methods and appropriate techniques for actually solving (integrating) many special differential equations. The solutions were expressed either by finite formulas or by expansions in terms of infinite series. By the late nineteenth century (Lipschitz, Peano, Picard), the necessary and sufficient conditions for both the existence and the uniqueness of the solutions, from any initial values, were finally obtained. Moreover, with the existence conditions taken for granted, the main efforts had shifted to the study of the qualitative behaviour of solutions, Poincaré [80], Liapunov [60]. This new approach (“qualitative integration”) utilized the topological properties of the phase space and the analytical properties of the governing functions in order to describe the local and global behaviour of the solutions. From the number of fixed points (singularities) and the behaviour of trajectories in their neighborhoods, crucial information about the geometric (topological) configuration of the phase portrait was obtained. Within recent years, interest in nonlinear differential equations has grown enormously; although many of these are not new, the systematic study of them is somehow a rather new phenomenon.