chapter  4
49 Pages

Nonlinear Dynamics

WithRussell L. Herman

This chapter explores a few different nonlinear systems and introduces some of the tools needed to investigate them. Nonlinear differential equations are either integrable but difficult to solve, or they are not integrable and can only be solved numerically. The chapter explores a simple nonlinear population model. It describes the stability of nonlinear first-order autonomous equations. The chapter aims to the nonlinear pendulum as an example of periodic motion in a nonlinear system. It investigates the nonlinear pendulum equation and determines its period of oscillation. The chapter discusses nonautonomous systems. One of the most important models in the historical study of dynamical systems is that of planetary motion and investigating the stability of planetary orbits. Applications involving differential equations can be found in many physical systems such as planetary systems, weather prediction, electrical circuits, and kinetics. When a small change in the parameter leads to changes in the behavior of the solution, the system is said to undergo a bifurcation.