Systems of Differential Equations

Abstract

Systems of differential equations, often called differentiable dynamical systems, play a vital role in modelling time-dependent processes in mechanics, meteorology, biology, medicine, economics and other sciences. We limit ourselves to two-dimensional systems, whose solutions (trajectories) can be graphically represented as curves in the plane. The first section introduces linear systems, which can be solved analytically as will be shown. In many applications, however, nonlinear systems are required. In general, their solution cannot be given explicitly. Here it is of primary interest to understand the qualitative behaviour of solutions. In the second section of this chapter, we touch upon the rich qualitative theory of dynamical systems. The third section is devoted to analysing the mathematical pendulum in various ways. Numerical methods will be discussed in Chap. 21.