What you will study
Nonlinear ordinary differential equations arise in a wide variety of circumstances: a simple pendulum, oscillations in electrical circuits, oscillations of mechanical structures, molecular vibrations, the motion of particles in accelerators, planetary motion, the effects of strong electromagnetic fields of atoms and molecules. In biology, they occur as models of evolving populations and the spreading of infectious diseases and also in the modeling of neural systems.
The module is based on the set book Nonlinear Ordinary Differential Equations by D. W. Jordan and P. Smith. It is an introduction to some of the basic theory and to the simpler approximation schemes. It deals mainly with systems that have two degrees of freedom, and it can be divided into three parts.
First, the geometric aspects of the two-dimensional phase space are discussed; we show why the fixed points are important and how they can be classified, and the notion of a limit cycle is introduced.
Then we develop schemes by which the solutions of autonomous and non-autonomous equations can be approximated, and so begin to understand how the solutions behave. In this section there is some emphasis on periodically forced nonlinear oscillators and on nonlinear oscillators with periodically time-varying parameters, leading to parametric resonances.
Finally, the stability of these solutions is discussed and various tests for stability are obtained, together with methods to establish the existence of periodic solutions.
You will learn
Successful study of this module should enhance your skills in understanding complex mathematical texts, constructing solutions to problems logically and communicating mathematical ideas clearly.