This course is concerned with the application of methods from dynamical systems and bifurcation theories to the study of nonlinear oscillations. The mathematical models we consider are (fairly small) sets of ordinary differential equations and mappings. As examples we look at nonlinear oscillations: the famous oscillators of van der Pol and Duffing, the Lorenz equations, and a bouncing ball problem. We show that the solutions of these problems can be markedly chaotic and that they seem to possess strange attractors: attracting motions which are neither periodic nor quasiperiodic. In order to develop the idea of chaos, we discuss the Smale horseshoe map and describe the method of symbolic dynamics. Moreover, we analyse global homoclinic and heteroclinic bifurcations and illustrate the results with the examples of the nonlinear oscillators.
John Guckenheimer, Philip Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Applied Mathematical Sciences, Vol. 42, Springer-Verlag, New York, 1983, ISBN 0-3879-0819-6